
(lass T~A/S~/ 

Book /< ? o r_ 



ENGINEERS' AND MECHANICS' POCKET-BOOK. 

CONTAINING ^ 

WEIGHTS AND MEASURES ; RULES OF ARITHMETIC ; 

WEIGHTS OF MATERIALS; LATITUDE AND LONGITUDE; 

CABLES AND ANCHORS I SPECIFIC GRAVITIES; 

SQUARES, CUBES, AND ROOTS, ETC. J 

MENSURATION OF SURFACES AND SOLIDS; 

TRIGONOMETRY; 

MECHANICS; FRICTION; AEROSTATICS; 

HYDRAULICS AND HYDRODYNAMICS; 

DYNAMICS; GRAVITATION; ANIMAL STRENGTH; WIND-MILLS J 

; STRENGTH OF MATERIALS; 

LIMES, MORTARS, CEMENTS, ETC. ; 
WHEELS ; HEAT ; WATER ; GUNNERY ; SEWERS ; COMBUSTION ; 

STEAM AND THE STEAM-ENGINE; 

CONSTRUCTION OF VESSELS ; MISCELLANEOUS ILLUSTRATIONS J 

DIMENSIONS OF STEAMERS, MILLS, ETC.; 

ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS, 

ETC., ETC., ETC. 



Thirty-ninth. Edition, Revised, and Enlarged. 



BY CHA.S. H. HASWELL, 
\\ 

CIVIL, MARINE, AND MECHANICAL ENGINEER, MEMBER OF THE AM. SOC. OF CIVIL ENGINEERS, OF THE 

N. Y. ACADEMY OF SCIENCES, OF THE INSTITUTION OF NAVAL ARCHITECTS, ENGLAND, 

CORRESPONDING MEMBER OF THE AMERICAN INSTITUTE OF ARCHITECTS, 

AND OF THE BOSTON SOCIETY OF CIVIL ENGINEERS, ETC. 



Ad examination of facts is the foundation of science. 



NEW YORK: 
HARPER & BROTHERS, PUBLISHERS, 

FliANKI.lN BQUABB. 

1881. 



By CHAS. H. HASWELL, 

Civil, Marine, and Mechanical Engineer and Surveyor. 



MENSURATION. 






For Tuition and Reference, containing Tables of Weights and Meas- 
ures ; Mensuration of Surfaces, Lines, and Solids, and Conic Sec- 
tions, Centres of Gravity, &c. To which is added, Tables of the 
Areas of Circular Segments, Sines of a Circle, Circular and Semi- 
elliptical Arcs, &c, &c, &c. , &c. By Chas. H. Haswell, Civil and 
Marine Engineer, &c. Fifth Edition. 12mo, Sheep, $1 25 



MECHANICS' TABLES. 






Containing Areas and Circumferences of Circles, Sides of Equal 
Squares; Circumferences of Angled Hoops, angled Outside and In- 
side ; Cutting of Boiler-Plates, Covering of Solids, &c, and Weights 
of various Metals, &c, &c, &c., &c. Miscellaneous Notes, compris- 
ing Dimensions of Materials, Alloys, Paints, Lackers, &c. ; U. S 
Tonnage Act, with Diagrams, &c. By Chas. H. Haswell, Civil 
and Marine Engineer, &c. Second Edition. 12mo, Cloth, $1. 

Published by HARPER & BROTHERS, New York. 

B3T Either of the above Works, or the Pocket-Book, will be sent by MuiL 
postage prepaid, on receipt of the price. 



Entered according to Act of Congress, in the year one thousand eight hundred 

and seventy -six, by 

HARPER & BROTHERS, 

In the Office of the Librarian of Congress, at Washington. 



INSCRIBED 

TO 

CAPTAIN JOHN ERICSSON, LL.D., 

AS A SLIGHT TRIBUTE TO HIS GENIUS, 

AND IN TESTIMONY OF THE GREAT REGARD 

OF HIS FRIEND, 

THE AUTHOR. 



^^C-4 *-'- 



W 



7 



PREFACE 

To the Twenty-first Edition. 



The First Edition of this work, consisting of 284 
pages, was submitted to the Engineers and Mechanics of 
the United States by one of their number in 1843, who 
designed it for a convenient reference to Rules, Results, 
and Tables connected with the discharge of their various 
duties. 

At the period of its first publication, the want of a 
work of this description had long been felt, and this 
was undertaken to meet the requirement, and, from the 
adaptation of its rules to the Metals, Woods, and Manu- 
factures of the United States, it has supplied that want 
to an extent beyond what .was anticipated. 

This Edition was commenced in 1856, is the result of 
much labor, and, in addition to the design of the original 
work, it has been essayed to embrace very general in- 
formation upon Arithmetical, Physical, and Mechanical 
subjects. 

The Tables comprising the Weights of Metals, of Balls, 
Pipes, etc., were computed expressly for this work, from 
specific gravities of the different materials taken for the 
purpose. 

The proportions of the parts of Steam-engines and of 
Boilers will be found to differ in some instances from 
English authorities ; but as they are based upon the ac- 
tual results of successful experience, the author is of the 
conviction that an adherence to them will insure both 
success and satisfaction. 

To the Young Engineer and Mechanic it is recom- 
mended to cultivate a knowledge of Physical Laws and 
to note results, without which, eminence in his profes- 
sion can never be securely attained ; and if this work 
shall assist him in the attainment of these objects, one 
great purpose of the author will be fully accomplished. 



INDEX. 



A. p ag e 

Accelerated Motion 395 

Acids ...100 

Aerodynamics 423 

u Resistance to Plane Surf ace 424 

Aerostatics 390 

" Distance of Sound's 392 

" Foretelling Weather 393 

" Diameter of Balloon 392 

" Rarefaction of Air 392 

" Velocity of Sound 391 

Air and Steam 579 

Alcohol, Proportion of, in Liquors . . 100 

" Dilution of 100 

Algebraic Symbols and Formulae .... IS 

Alimentary Principles 104 

Alligation 57 

Alloys and Compositions 627 

" Melting Points 530 

Almanac 69 

American Gauge 113, 117 

Analysis of Substances and water 77, 540 

" "Food 78,102 

Anchors 133, 135, 138, 139 

" and K edges 133 

" Weights of, etc 133, 138 

Angle Iron 117 

Angles of Equilibrium and Repose 319, 320 

Animal Power 403 

" " Men, Horses, etc.. 403, 405 

Animal Strength 401 

" " Horses 402 

" " Men 401 

Animals, ages of. 129 

Annuities 59 

Arches and Abutments, etc 555 

Arcs, To describe 169 

Arcs, Circular 199 

" Semi-Ellipse 201 

u of a Circle, etc 253 

Area of the Earth 103 

Areas, Links, and Surfaces 244 

Areas of Circles, Eighths 172 

" " To Compute 176 

" and Circumfs of Circles, Tenths 182 
u " " Feet and ins. 191 

44 of Segments of a Circle 205 

" of Zones of a Circle 207 

Arithmetical Progression 52 

Atmospheric Air, Consumption of. . . . 104 

B. 

Babbitt' s Metal 628 

Balances, Fraudulent 145 

Balloon, capacity, etc 162, 392 



Balls, Weight and Dimensions 545 

Barker's Mill 441 

Barometer, Measurement of Height by 391 
Barrel, Dimensions of 102 

A* 



Page 

Beams 461, 464, 466, 469, 479 

Stiffness of 560 

Bells, Weight of 131 

Belts 610 

Beton 503 

Birmingham Gauge 115, 117 

Blacking 625 

Blasting 557 

Blowers, Fan 608 

Blowing Engines 607, 653 

u " Density of a Blast 609 

" " Horses' 1 power 609 

Blowing off 601 

Board and Timber Measure 146 

" Spars and Poles 147 

Boats 651 

Boilers 591 

" Consumption of Fuel 595 

" Heating Surfaces of 592 

" Heating and Orate Surfaces 593 
" Plates, Bolts, and Joints . . . 595 

" Pressure of 595 

" Shellof 493 

" Stay Bolts 596 

" Surfaces and Joints . . . 596, 597 
" Thickness and Diameter of. 595 

" Weights of 601 

Boiler Tubes, Diam. and Weight of. . 119 

Bolts and Nuts, Dimensions, etc. 124, 125 

" Dimensions and Weights of 123 

" Comparison of Proportions 125 

Crass, Weight of 112, llo 

" Plates, weight of 113, 115 

" Thickness of 116 

" Tubes and Pipes, weight of. . 120 

" Wire, weight of 114 

Brass and Copper Tubes, Weights of. 121 

Brazier's Copper 116 

Breast Wheels 435 

Bricks, Volume and Number 132 

Brick Work 456, 657* 

Bridges, Length of, etc 12S, 474, 555 

Browning 622 

Buildings, Capacity of 103 

C. 

Carles and Hawsers 135 

" " Weight of 143 

" " Diameter, etc. 134, 13S 

11 and Anchors 139 

11 Breaking strain 140 

" Circumference of 137, 142 

" " and Length.. 13S 

" Proof and Breaking Strain. . 142 

u To Compute Strain ir>6 

' ' Units of strain 136 

Cables and Wire Rope compared 141 

Calendar, Roman 104 



VI 



INDEX. 



Page I 

Caloric ^ 3 | 

Canals, Flow of Water, etc 375; 

" Resistance of Boats 553 j 

Canal Locks 377 

11 Dimensions 152 

Candles 534 

Capillary Tube 26S 

Cascades and Waterfalls 131 

Cast and Wrought Iron, weight of a 

square foot 118 

Castings, Shrinkage of 560 

Cast Iron Balls, Weight of 109 

" Pipes " 110 

Cast Ikon 495 

" Bars 480 

" Columns 473 

Weight of Ill 

Cattle and Horses, space required . . . 127 

Cattle, Weight of 143 

Cements, Concretes, etc. . 499, 591, 504 

Cements 621, 626 

Central Forces 410 

Centre op Gyration 411 

Centres of Oscillatlon and Per- 
cussion 413 

Centres op Grayity 33S 

Centrifugal Force 410 

Chains, Cables, and Anchors 13S 

Chain Cables, Proof of 142 

" Diameter of 134 

•' Diam. and Length of 

21, 134, 13S, 33S 

Chains and Ropes 338 

Chain Rigging, Links, etc 139 

Chances and Odds 65 

Channels and Bars, Depth* of 9C 

Characters, Explanation of 16 

Charcoal 565 

Chemical Formulae 

Chinese Windlass 32S 

Chronological Eras and Cycles 1' 

Chronology 19, 65 

" A hnanac 69 

1 ' Cycles of Sun and Moon . . 66 

" Dominical Letter 66, 69 

11 Easter and Dates of Days 6S 

Eimcts 67,69 

" Golden Number ^ 

" Moon's Age 68, 70 

" Number of Direction 67 

11 Numbers of the Month. . . 69 

41 Roman hidiction 67 

Churches, capacity of 103 

Circle 250 

Circles 167 

Circular Arcs 199 

Ring 262 

" Spindles 265,281 

Zone* 25S 

Circumference of Circles, Feet <fe ins. 191 
u " Eighths... 177 

" " Tenths IS? 

" "To compute 1S1 



Page 

Cisterns, capacity of. 122 

Clipper Ships 652 

Cloth Measure 21 

Clouds, Classification, etc 393 

Coal 564 

Coal Fields, Area, etc 569 

Coke 565 

Collision or Impact 417 

Columns, Height of 136 

Combination 61 

Combustion 569 

" Volume of Air 571 

44 Gas 571 

*' " products of Coal 572 

* ' Consumption of Fuel . . . 572 

Compass, Mariner's, Degrees of 37 

Compositions and Alloys 626 

Compound Axle 328 

Concrete or Beton 503 

Cones 263, 274 

Congelation, Line of perpetual . . 128, 532 

Conic Sections 2S9 

Conoids 2S5 

Construction of Vessels 611 

11 Area of Sails 616 

" Cost per Ton 617 

" Displacement 613 

" Elements of Vessels and 

Boats 651 

44 Resistance of Hull 617 

" Immersed Surface 612 

" Sails 618 

44 Ship building , . 618 

" Side Wheel and Propeller 614 

Copper, Weight of 112, 118 

" Pipes, weight of 118 

' ' Plates, weight of 113, 1 1 5 

14 Rods and Bolts, iceight of . . 117 
" Tubes and Pipes, ** . . 120 

Wire 114 

Copper and Brass Tubes, weight of. . 121 

Co-secants and Secants 312 

Cosines and Sines 301 

Co-tangents and Tangents 315 

Cotton Factories 657 

Cotton Press 656 

Courses, Race and Trotting 151 

Cranes 335 

Crops, Vegetable 101 

Crushing Strength 471 

Cube 270 

Cubes, Squares, and Rootg 210 

Cube Root, To Extract 50 

Cutter, Steam 649. 

Cut Nails, number in a pound 127 

Cycloid 262 

( 'ycloidal Spindle 266, 2S2 

Cylinder 259, 274 

Cylindrical Ring 262, 2S0 

' » Tube? 472 

' Ungulas 275 



INDEX. 



Yll 



D. Page] 

Dams and Tunnels 55'J 

Days of the Week 68 

Decimal Fractions 43-46 

Deflection of Bars. Beams, etc. . . 476, 418. 
Degrees, Minutes, and Seconds of 

Compass 37 

Departures, Table of 36 

Depth of Harbors, U. S 96 

Detruslve Strength 484, 560 

Diamond Measure 25 

Dilution of Alcohol 100 

Disc, revolving 560 

Discount 59 J 

Dimensions and Details of Steam- 
ers 63S, 

Displacement of Vessels 613 

Distances, of Visible Objects .... 37, 3S 

" Geographical 36 

" Railroad, U. S 95 

" Travelling 91, 92 

" Sailing and Railroad 93, 94, 95 

Distillation 52S 

Distillation of Coals 568 

Domes, Diameter of 131 

" Height of 130 

Dominical Letter, To Ascertain ... 66, 6S 

14 Letters 69 

Drainage of Lands .;. .' 553 

Drawing Paper 2 r 

Dredging Machine 51S, 659 

Dubbing 624 

Ductility of Metals 453 

Duodecimals 46 

Dynamics 394 



Earth, Digging 126 

" Area and Population 103 

u Pressure of 320 

Earthwork 559 

" Proportion of Laborers, etc. 559 

Easter Day, To Ascertain 68 

Elasticity and Strength 447,479 

Elastic Fluids, Laws of 563 

Elementary Bodies 77 

Elevations, Heights of 130 

Ellipse 290 

Ellipses 167 

Ellipsoids 256, 266, 27S, 285 

Elliptic Spindle 283 

Embankments and Walls 322 

" Labor upon 404 

Engines. Elements of G53 

Ensigns, U. S. Pennants and Flags . . 149 

Epacts 67 

" To A ^certain, etc 6S 

Equation of Payments 59 

Equilibrium, Angles of. 319, 320 

" of Forces 319 

Establishment of the Port, U. S. . . . 74, 75 
" " " Europe... 75 



Page 
Evolution 49 

Excavation and Embankments 404 

Explanation of Characters 16 

F. 

Factors, useful 251 

Fan Blower 608 

' ' Horses' power 609 

Fellowship 5$ 

Figure of Revolution, any 267, 287 

Fire-engine, Steam 654, 657* 

Flags, U. S 149 

Flour Mills 556, 655 

Flues and Tubes 488 

Fluids, Motion of Bodies in 444 

" Impulse and Resistance of . . 441 

" Percussion of 443 

Flying 409 

Fly Wheels 416 

Food, 78 

Foods, Value of 102 

Floors, Fire-proof 563 

Forces, Composition and Resolution . 318 

" Equilibrium of 318 

Foreign Measures and Weights 29 

" " " Value of 34 

Formulae and Algebraic Symbols 18 

Fourth and Fifth Powers 242 

Fraudulent Balances 145 

Friction 342 

" of Axles 350,351 

" of Bearings 346 

" and Fluidity of Oils. . . 345 

" Influence of Velocity 34S 

" of Launching 354 

" of Machinery 346 

[ ' of Machinery and Trains 353, .-'54 

" of Mar i7ie Railway 346 

" of Motion 352 

" of Pivots 351, 852 

" Rolling 353 

" and Resistance of Steamers. 346 
" and Rigidity of Cordage.... 349 

" of Steam-engines 345, 347 

a Traction of Carriages 347 

' ' of Unguents 344 

" of Vehicles 347, 348 

" of Water in Pij>es and Sew- 
ers 883,386 

" of Water 3S9 

" Coefficients of 320, 344, 345, 350, 852 

Fuel 564 

" Ash, proportion of H]Q 

" Comjjosition of 566 

" Evaporation of Combustible . . 569 

" Evaporative Powers of 567, 569 

M Relative Value of 568 

M To Compute Consumptio?i of.. 572 
" Weights and Comp'live Value of 565 

G. 

Galvanized Sheet Iron, Thickness, etc. 120 

Gas 586 

" Pipes and Engines 588 

" Illumination 534 



viii 



INDEX. 



Page I Page 

Gas, Composition of London 538* Hawsers and Cables 135 

" Volume of, to produce Light of " " Circumference 

aCandle 536 ! of 137 

Gas Pipes 53S Hawsers, Circumference of 137, 142 

" Thickness of 125 " Hopes and Wire Ropes. . . . 140 

" Volume of Discharge of . . 537 " To Compute Strain 136 

Gauge, American 113 " Units of Strain 136 

M Birmingham 115 " Weight of 143 

Geometrical Progression 53 Heat 51 



Geometry 163 

" To ascertain Distances be- 
tween Inaccessible Objects 171 
Geographical Measures and Dis- 
tances 21, 36 

Gestation, Periods of 104 

Giffard's iDjector 5S9 

Girders. . 461, 464, 467, 465, 469, 470, 475 

" Combined with Chains 474 

" Hollow 4GS 

Girders, Beams, etc., Comparative Re- 
sistance for equal A reas 407 
" " Transverse Strength of 

various Figures 467 

Glass, Window 102 

Globes 494 

Glue 622 

Gnomon 244 

Gold, Thickness of 116 

Governors 423, 5~S 

Grain, Standard Weights of 102 

Gravities of Bodies 154 

Grease 625 1 



Agitation 532 

Boiling Points 529, 530 

Capacity for 526 

Conduction or Convection 527 

Congelation and Liquefaction. 528 

Distillation 528 

Effect upon Bodies 529 

Evaporation 528 

Expansion and Dilatation 530 

" of Air and Water.. SSI 

" of Fluids 531 

Frigorific Mixtures 530 

Mean Temperatures 532 

Melting Points 530 

Radiation of 526 

Reflection 527 

Sensible and Latent 524 

Specific 525 

Thermometers 532 

Ventilation 533 

Volume of a Gas 531 

Volume of Substances 531 

Warming Buildings and Apart- 
ments 533 

Helix 264 



Gravitation 396'Heights by a Barometer 391 

" Velocity of Stream of Water 401 » of Columns, Tower, Domes, 
Gravity, Resistance of, at Inclinations 552 



etc 130 

" of Elevations 130 

11 of Mountains 130 

High Water, Times of, and To Com- 

2mte, etc 72, 76 

11 Bench Marks 75 

11 Times of 72 

( Mills in an Acre of Ground 127 

Shot and Shells 544' Hollow Shafts 4S3, 4S7 

Percussion Caps 546Lr i- e*a 

Percussion Primers 548 *?««*»« **«*■ C ? 6 



Gudgeons 485, 4S8 

Gun Barrels, Length of. 104 

Gunnery 542 

Flight, Time of 544 

Initial Velocity and Ranges 543 
Penetration of Balls. . 544, 545 



" Proportion of Powder to 

Shot 546 

11 Range of Shot, etc 542 

" Ranges of Small Arms . . . 54S 

11 U. S. Small Arms 548 

" Wadding 546, 54S 

" Weight and Dimensions of 

Halls 545 

" Windage 545 

Gunpowder 546 

" Barrels 548 

" Proof of 54S 

Gunter's Chain 21 

H. 

Harbors, U. S., Depth of pg 

Hay 12(5 

Hauling Stones, etc 405 



Horse Power 6C3 

Horse-shoe Nails, Length of 127 

Horse Shoes, Number of, etc 127 

Horses and Cattle, Space required for 127 
" " 'Transportation of 111 

Horses, Weight of 143 

Horses 402, 406* 

" Miscellaneous Performances 407 
" Effect of, upon lioads and 

Canals 553 

Hydraulic Ram 558 

Hydraulics 359 

11 Canal Locks 377 

" Coefficient of Discharge of 

Water 362 

" Coefficient of Discharge of 

various Opening*,ctc. 364, 36S 
" Coefficient of Friction in 

Pipes 3S0 



INDEX. 



IX 



Page; 

Hydraulics, Difference between Lev- Hydr. 
els 360 

" Discharge of Water through 

different Apertures 3G3 

" Discharge of Fluids from 

Reservoirs 359 

* * Discharge of Fluids under 

like Heads 361 

" Discharge of Water by 

Conduits 361 

" Discharge from Com- 

2?ound Reservoirs 362 

" Discharge from Rectangu- 
lar Notches, or Vertical 
Apertures, or Slits. 364, 36S 

" Discharge from a Trian- 
gular Notch 365 

* ' Discharge from Rectangu - 

lar Weirs 364, 368 

" Discharge from Horizon- 
tal Slits 365,368 

" Discharge from Rectangu- 
lar Openings or Sluices 365, 368 

" Discharge from Circular 

Openings or Sluices 365 

" Discharge from short 

Tubes or Mouth-pieces 366, 36T 

' * Discharge from Cylindric- 
al Prolongations or Ajut- 
ages 366 

" Discharge fromConvergent 

and Divergent Tubes. . . 366 

" Discharge from Compound 
Mouth-pieces and Ajut- 
ages 363 

" Discharge from Cylindric- 
al Tubes or Pipes. 367 

" Discharge from Short Cy- 
lindrical Tubes 368 

" Discharge from Circular 

Sluices, etc 369 

" Discharge from Irregular- 
shaped Vessels,as a Pond, 
Lake, etc 3T1 

" Discharge from Semicir- 
cular Sluices 369 

" Discharge from Semicir- 
cular Weirs or Notches. . 369 

" Discharge from Trape- 
zoidal Weirs or Notches. 369 

" Discharge from Trape- 
zoidal Sluices or Slits . . 369 

?• Discharge from Triangu- 
lar Weirs and Notches. . 3T0 

" Discharge from Wedge and 

Pyramidal Vessels 3T0 

" Discharge from Spherical 

and Prismoidal Vessels . 371 

" Discharge through Pipes 

or Canals 371, 376 

" Discharge under Variable 

Pressures 373 

" Discharge from Vessels in 

Motion 375 

c< Discharge from Canal 

Locks 377 



Page 
ulics, Discharge from Over- 

fall Weirs 379 

Discharge by Sluice Weirs 
or Sluices 380 

Discharge from Pipes or 
Sewers 384 

Flow of Water in Beds. . 380 

Forms of Transverse Sec- 
tion of ditto £81 

Flow of Water in Rivers 
and Canals 375 

Friction of a Fluid 361 

Friction in Pipes and 
Sewers 383, 3S6 

Friction upon a Plane 
Surface 389 

Illustrations of Various 
Rules 377 

Jets d'Eau 388 

Miscellaneous Illustra- 
tions 390 

Profiles and Angles of 
Slope 382 

Rise of Water in Rivers 
by Piers, etc 389 

To Compute the Fall of 
Water in a Pond, etc. . . 373 

To Compute the Time and 
Fall of Water under va- 
riable Pressures 374 

To Compute the Elements 
of Flowing Water 375 

To Compute the Mean Ve- 
locity of Flowing Water 
in a River 376 

To Compute the Height 
or Head of Flowing 
Water 377, 3S6 

To Compute the Horses' 
Power to elevate Water. 389 

To Compute the Projec- 
tion of a Jet 387 

To Compute the Projec- 
tion of a Stream from 
a Pipe 3S8 

To Compute the Velocity, 
etc., of Water in Ca- 
nals, Streams, Pipes, 
etc., etc 376 

To Compute the Trans- 
verse Section of Water 
Flowing in Beds 383 

Variable Motion of Wa- 
ter 883 

Velocity of Fluids under 
like Heads 361 

Velocity of Water or of 
Fluids 362 

Velocity of Watt r through 
different Apertures . . . 363 

Velocity of Water from 
Pipes and Sewers 3S7 

Velocity of Flow to clear 
Setcers, etc 3S0 

Volume of Water ]>rr 
Acre of Area 389 



INDEX. 



Page] Page 

Hydrodynamics 426 Iron Plates, Weight of 113, 115, 118 

R/>***r>* If AJ.1 I ron Tubes 120 



Barker's Mill 441 

Breast Wheels 485 

Centre of Pressure 426 

Hydraulic Ram . . 429, 558 
" Operation of 430 
Hydrostatic Press. 429, 656 
Impact and Reaction 

Wheels 438 

Impulse and Resistance 

of Fluids 441 

Memoranda 441 

Overshot Wheels 432 

Percussion of Fluids. . 443 

Pipes 429 

PonceleVs Wheel 437 

Pressure 42T ! 

Sluice and Flood Gates 42S 

Sluices 431 

To Compute the Power 

of a Fall of Water . . 430 
To Compute the Dimen- 
sions of Water-wheels 

and Turbines 440 

Turbines 439 

Undershot Wheels .... 406 1 



Dimensions and Weight of 119 

Iron Wire 114 

Iron Works 553 

J. 

Jet Pump 553 

Jets d' Eau 388 

Jewish Long Measures . . . . 35 

Jumping 405* 

K. 

Kedges and Anchoes 133 

Weight of 133 

Keystones 555 

Knot, Admiralty 29 



Laborers in Bank, Proportion of 559 

Lackers g22 

Lakes, Northern, of the U. S.' .... '. '. '. 131 
Lamps 534-538 



Lath and Plaster, Estimate of Mate- 



rials 



503 



WaterPower 430 | Laths and Lime 132 

Water Wheels 431 Latitudes 421 



" Wheels and Turbines . 440 

Hydrometers 148 

Hydrostatic Press 656 



> Latitudes and Longitudes of Places. 84 

! " " 0/ Observatories 84 

Launch, Steam 649 

Hyperbolic Logarithms 579 1 " Pipes, weight of 122 

Hyperbolic Spindle 285 " Sheet \\ 131 

Hyperboloid. . . ._. 266, 285 Lead^Tubes, Resistance of '. 4" 4 



Hyperboloid of Revolution 287 



Ice, Strength of 559 

Impact or Collision 417 

Impact and Reaction Wheels 43S 

Imperial Gallons 55S 

Impulse and Resistance of Fluids . . . 441 

Inclined Plane 329 

Incubation, Periods of 104 

Independent Steam Fire and Bilge 

Pumps 590 

Ingredients, To Compute the Propor- 
tions of, in a Compound 155 

Inks 622 

Interest, Simple and Compound ... 57, 53 

" and Discount 59 

Involution 49 

Irregular Bodies 2S8 

" Figures 250 

11 Polygons 250 

Iron 495 

" Case Hardening 499 

" Comparative Value of Cast-iron 

and Wood Columns 474 

Iron Bars, Extension of, when sus- 

pended 453 

Iron Pipes, Weight of 118 



Leaping*. 405*, 409 

Levee 325 

Life Boat , C51 

Lifting .■; . 406 

Light 534 

" Combustion, Temperature, and 

Power of Gases 539 

" Comparative and Relative In- 
tensity of. 534 

" Illumination, Gas,' etc 534 

" Intensity of, with different 

Burners 535 

" Relative Consumption and Cost 536 
" Regulation of Tubing and 

Number of Burners 539 

" Volume of Gas discharged 537 

Lightning, Classification of. 293 

Limestones 499 

Lime and Laths 132 

Limes, Cements, 1'omars, etc 499 

" Materials and Labor for 

Laths and Platter 503 

11 Experiments upon Ce- 
ment*, Mortars, etc. 505, 506 

Limits of Vegetation 128 

Line of Perpetual Congelation . . 12S, 532 

Lines, Areas, and Surfaces 244 

Links 262. 280 

Lintels 401 



INDEX. 



XI 



Pag* 
Locomotive Engine. 598 

4 ' Speed of Trains 598 

Log-lines 21 

Longitude, Length of Degrees 39 

11 To Reduce into Time 36 

Lunation of the Moon 71 

Lune 261 

M. 

Machines and Engines, Elements of . 653 

Malleable or Aluminum Bronze 6:9 

Malleability of Metals 453 

Manures 102 

Marine Engines, Side Wheels 639 

" Screw Propeller.. 643 

Mariner's Compass, Degrees, etc 31 

Masonry 132, 505, 503, 559 

Mason's and Dixon's Line 104 

Materials, Strength of 4 IT 

" Girders, Beams, and Tubes 4T0 
" Bars, Hollow Girders or 

Tubes 4T0 

" Beams, Bars, etc 465 

" BrickWork 456 

" Compression of Bars 473 

" Crushing Strength 4T1 

" " " Columns 474 

" Cylindrical Tubes 472 

" Deflection of Bars, Beams, 

etc 476 

" Detrusive Strength 4S4 

" External and Internal Press- 
ure 4SS,491, 493 

" Flanged Beams 47" 

" Floor Beams, Girders, etc.. 464 

" General Deductions. 480 

u Girders, Beams, Lintels. 

etc 461,466,563 

" Girders, Beams, etc. 461,466, 476 

11 Glass Globes 494 

" Gudgeons 485 

({ Hollow Girders and Tubes. 468 
•• Lead Pipe, Thickness of . . 494 

" Memoranda 494 

" Modulus of Elasticity 448 

" Rails, Deflection of, etc 478 

" Resistance of Flues to 

Pressure and Collapse . . 49.3 
" Resistance to End Pressure 462 

" Shafts 482, 485, 4S6 

" " Deflection of 487 

" " and Gudgeons. . 485, 4SS 

" Springs 483 

" Tensile Strength 449 

u To Compute Bursting Press- 
ure 493 

" To Compute Collapsing Re- 
sistance 491 

" To Compute Relative Value 

of Materials 460 

" Torsional Strength 481 

" Transverse Strength 453 

11 Trussed Beams or Girders. 475 

" Tubes and Flues 488 

" Wire Ropes 452 



Page 

Measures and Weights, U. S. 20, 150,630 

" Foreign. 29-36 

of Boards and Timber. . . . 146 

Geographical and Distances 36 

Imperial Gallon 553 

Land 22 

Length 21 

Miscellaneous. 22, 23, 24, 25, 36 

Paper 22,23 

Scripture and Ancient ... 35 

Surface 22 

Time 22 

Value 25 

Volume 23 

Weight 24 

Mechanical Power 325, 334 

Mechanics 318, 324 

Men 401,4'i5 

Men and Women, Weight of Ii3 

Mensuration, Areas, Lines, and 

Surfaces 244 

" Solids 270 

Metals, Ductility and Malleability. . . 453 

" Various Qualities 484 

Mineral Constituents from Soil 101 

" Water..... 540 

Mining and Blasting 557 

Miscellaneous 104 

" Illustrations 630 

" Notes 82,660 

Models 35S 

Modulus of Elasticity 448 

Moon passing Meridian 73 

u Times of New 70 

Moon's Age 6S, 70, 71 

" Period of Lunation 71 

Mortars, Limes, Cements, etc 499 

Motion 394 

Motion of Bodies in Fluids 444 

Mountain, Heights of 130 

Mowing Machine 657* 

N. 

Nails, Length and Number 127 

Natural History, To preserve Objects 625 

Nautical Measures 2.1 

New Moon, Times of 70 

New York City Avenues 629 

Notation 17 

Number of Direction 67 

" of the Month 69 

Nutritious Properties of Vegetables . . 100 

O. 

Observatories, Latitude, etc. , of 8S 

Oceans, Areas of 131 

Octagon, To Describe 167 

Odds and Chances 65 

Oil, Yield of 101 

Opera Houses, Capacity of 103 

Organic Substances 77 

Orthography of Technical Words 

and Terms 666 

Oscillation and Percussion 413 

Overshot Wheels 432 



Xll 



INDEX. 



P. Page | E. Peg* 

Pacing 408*! Race Boat 651 

Paper Measures 22 ' r ace Courses, Weights, Distances, etc. 151 

Paper for Draughtsmen 625| m English, Lengtlis of. . 152 



Parabola 292! 



Race and Trotting Courses 151 



Pa"^^ 

Paraboloid of Revolution 2S6 Railkoads ■ ^* 

Parallelograms 244 1 Railroad Distances 91, 95 

Parallelopipedon 270 " Spikes 125 

Passages of Steamers and Vessels 613 'Rails, If eight of 549 

Pastiles 625 " Deflection and Strength of 4TS 

P&^nts,Dime\^^'ofbm^nm.y. 629 Railway Bridges, Comparison of 474 

Peat " 530 Railway Carriages, Friction of 551 

Railways and Dry Docks 150 

Railways axd Roads 549 

44 Adhesion of Locomotive. 549 



Pendulums 421 

Centre of Gravity of. ... 422 

" Length of 21,421 

Pennants, IT. S., and Flags 149 

Percussion and Oscillation, Cen- 
tres of 413 

Percussion Caps 546 

Percussion of k luids 443 

Performances of Men, Horses, etc. . . . 405 

Permutation 55 

Perpetuities 61 

Pile Driving 419, 629, 656 

Physical Endurance 657 

Piling of Shot and Shells 145 

Pilot Boat 05. 

Pipes, Wrought Iron, Copper, or 

Brass, Weight of 120 

" Tin, Weight of 122 

" Gas 125 

Pipes, Iron and Copper, Weight of... 118 
•' Lead and Tin, " ... 122 

Plane" Trigonometry 296 

Pneumatics 423 

44 Windmills 424 

Poles and Spars 147 

Polygons 165, 169, 247, 250, 26S 

Position 55, 56 

Polyhedrons 271 

Poncelet's Wheel 437 

Population of thej Earth 103 

Power, Concentrated in Moving Bod- 
ies 560 

Powers of Numbers. . 242 

Tableof 51 

Prismoid3 260, 270 

Prisms 259, 270 

Probability 62 

Projection of Water 55S 

Proof of Spirituous Liquors 156 

Properties of Numbers 51 

Proportion— Compound 4S 



Mean 47 R°»"ian Calendar. . 

Pulley 3P,3 

Tumping Engine 65' 

Pumps, Independent Steam, etc 590 

Pyramids 263, 274 

Pyramids of Egypt 102 



Capacities 552 

Fric'don 551 

11 Gravity, Resistance to .. 552 

44 Load, Traction 550 

" Miximum Load 551 

" Points and Crossings . . . 549 

11 Regulations 551 

" Resistance of Train 549 

" Roads 552 

" Sidings, etc 549 

* ' To Compute Power, Speed, 
or lime of Running a 

Lccomotivr, etc 353 

" Tractive Power of Horse 

Teams 553 

" Weight of Rails 549 

Rain, Annual Fall of and Volume. . . 129 

Rarefaction of Air 392 

Reaction Wheels 43S 

Rebate 5S 

Recipes, Miscellaneous 624 

Reciprocals 243 

Rectangle, Rhombus, and Rhomboid . 244 

Regular Bodies 249, 271 

11 " Elements of 272 

11 Polygons 247, 26S 

Revetment Walls 320 

Revolving Disc 560 

Rifle Shooting 620** 

Rigging, Wi ought-iron Chain 139 

Rings 262, 2S0 

Rise and Fall of Tides 75 

River Engines, Side Wheels 646 

Stern Wheels 650 

Rivers, Descent of Western 104 

11 Fall of 629 

" Lengths of 128 

Roads and Railways 549 

Roads, Friction in Proportion to Loads 552 

11 Turnpike 553 

44 and Canals, Effect of Horses . 553 
Rolling Mills 556 



Rat-killing C20" 



104 

Indiction 67 

Ropes, Circumference of 137, 142* 

44 Hawsers and Wire Ropes 140 

44 Unit of Strain 136 

44 Weight of 143 

and Cables, Lengths of 21 



INDEX. 



X1U 



Page 

Ropes, Hawsers, and Cables 135 

" " Breaking Strain . . 140 

n "To Compute Strain 136 

"Ropes, Endless, Friction of 629 

Root, To Extract any 51 

Root, To Ascertain 4th Root 50 

Roots, Squares, and Cubes 210 

Rowing 658 

Running, Man, Horse 405*, 406' 

S. 

Sailing Distances, Ports of IT. S 93 

44 " Ports of the World 94 

Sailing Vessels 652 

" " Passages of 619 

Sails 618 

" Area 616 

Saturation in Marine Boilers 601 

Saw Mills 556, 659** 

Scales 1G5 

Screw 264, 331 

" inferential 333 

Screw Propeller 585, 614 

Screw Threads, Number of, etc 124 

Scripture and Ancient Measures 35 

Sea, Depths of 129 

Secants and Cosecants 312 

Sector of a Circle 254 

Seeds, Number in a bushel, etc 128 

Segment of a Circle, Areas of. . . 205, 254 

Segment of a Sphere 256, 27T 

Semi-elliptic Arcs, Length of s 201 

Sewers, 553 

" Area of Surface of Discharge 554 

Shafts and Gudgeons 4S5 

Shafts, Mining and Tunnel 556 

Shafts, Strength of 4S2, ' " 

Sheathing Copper, Weight of 116 

Sheathing Nails 119 

Sheet Iron, Thickness and Weight of. 116 

Shoemakers' Measure 22 

Shot and Shells, Weight of etc. . 144, 145 

Shrinkage of Castings 560 

Shrouds, Circumference and Num- 
ber of 142 

Sidereal and Solar Time 83 

Sides of Squares 197 

Silver, Thickness of 116 

Sines and Cosines 301 

Siphon, Steam 590 

Skating 406 

Slating and Dimensions of Slates ... 126 

Sluices 365, 380 

Snow Line 12S, 532 

44 Melted 629 

Solar and Sidereal Time 82 

Solders 

Sound 

Sounding 39 

Sounds, Distances at which audible. . 392 

Spars and Poles 147 

Specific Gravities 154, 156 

" Application of the Tables 162 
Spermaceti Candle 629 



B 



Page 

Sphere 255, 277 

Spherical Pyramid 275 

Spherical Sector . . 281 

Spherical Zone 256, 278 

Spheroids 256, 278, 286 

Spikes and Horseshoes, Number, etc. . 127 

Spikes, Ship and Railroad 125 

Spindles 265,281 

Spiral Line 168 

Spirals 264 

Spires, Heights of 120 

Spirituous Liquors, Proof of 156 

Springs 488 

Square and Squares 166, 244 

Square Root, To Extract 50 

Squares, Cubes, and Roots 210 

Squares, Inscnbed and Circumscribed 254 

Stage Coaching and Sleighing 409 

Staining 624 

Stains 625 

Stability 354 

" and Speed of Models 35S 

Statics 318 

Steam 573 

" Acting Expansively 579 

11 Air and Steam 579 

" Effect of Expansion 580 

" Elastic Force, Temperature, 

Volume \ and Density 574 

" Gain in Fuel 580,581 

" Loss of Pressure by Imperfect 

Condensation 578 

" Mechanical Equivalent 578 

11 Sensible and Latent Heat of. . 524 

" Specific Gravity 576 

" Temperature of Water, etc... 577 

" " of Vapors 578 

44 To Compute Pressure and 

Temperature 575 

" To Compute Volume of Water 

in Steam 576 

4 4 To Compute Volume of, to raise 
a Volume of Water to any 

given Temperature 577 

4 4 Velocity of Flow 577, 57S 

Steam-boats, Side Wheels 646 

44 Stern Wheels 650 

44 Passages of 620 

Steam-engine 582 

44 Area of Injection Pipe 606 

44 Blowing Engines 607 

44 Blowing off 601 

44 Boiler Plates, Bolts, etc. 5'.: 5 

44 Boilers 591 

44 Bursting and Collax>s- 
ing Pressure of Boil- 
ers 595 

44 Consumption of Fuel 

per Sq. Foot of Grate 

and Cost 601 

44 Duty 599 

44 Feed Pump Area 607 

44 General Rules 583 

44 GiffarcVs Injector 639 

4 Heating Surfaces £92 



XIV 



IXDEX. 



Page I Page 

Steam - engine, Heating and Grate | Tide Table of U. S 74 

Surfaces 594 Timber, Wood, etc 519 

Injection Pipe Dis- Timher and Board Mea8Ure U6 

t^HzSmZ S' Tim ber, Seasoning d: Prescrvia i 520, 522 

Locomotive 598 „ Comparative Weights of. . . / 162 

pf "flu ,£^'1 : " * U Time, Difference of, at various Places 89 
Portoftte £nfln»ie« and _ „ ' La^Uude ^Observatories in.. 88 

Measure of. 22 

To ascertain Difference of at 

various places 90 

To Reduce into Longitude .... 36 

To Reduce Longitude into 36 

• Tin Pipe, Weight of 122 

Tin Plates, Marks and Weights of... 121 

Tonnage 561 

" Carpenters' and British . . . 562 
" Comparison of Old and Sew 105 

Torsional Strength 4S1 

" " Relative Value of 

different Figures 4S4 

j Towers, Height of 130 

{ Tracing Paper 23 

' Traction 402, 404 

. Transportation of Horses and Cattle. 127 

Transverse Strength 453 

> Trapezium 246 

Trapezoid 247 

; Travelling Distances, U. S. 91 

" Europe 92 

j Triangles 244, 2£ 6 

Water-wheels. . . . ". 5S5, 5So Trigonometry 296 

Weight of Boilers '601 Trotting 407* 

Weights. 599, 600 Trotting and Race Courses 151 



Boilers 591 

Proportionate Weights. 600 
Relative Cost for Equal 

Effects 599 

Safety Valves 5S6 

Screw Propeller 5S5 

Slide Valves, To Com- 
pute Lap, Stroke, Part 
of Stroke any given 
Lap will cut off, 
Breadth of Port, Lead, 
and Distance of Pis- 
ton 5S7, 5S9 

Space occupied 600 

Steam Pump 590 

Steam Siphon 590 

ThicJmess, Pressure, <£ 

Diameter of Boilers. 595 
To Compute the Horses' 

Power to raise Water 605 
Velocity of Discliarge 

of Water 606 

Volume of Water evap- 
orated 606 



Steamers, Dimensions a?ul Details of 63S 
44 Vessels and Boats 651 



Weights, and Distances of 151 

Trussed Beams or Girders 475 

?55tt? I: £g Tcbbd.es 439,440 

g54 Tubes, Wrought-iron, welded, Dimen- 
sions of 119 

Wrought-iron, Diameter and 

Weights of 119 

Wrought-iron, Seamless, Di- 

meiisions, etc 121 

Tubes 463, 470, 472, 4SS, 494 

and Pipes, Weight of 120 

Wrouqht-iron, Copper, or Brass, 

Weight of 120 

Tunnels, Cost of 556 

Length of 131 

Railway and Shafts 559 

and Dams 556 

Turnpike Roads 553 



Steam Fire Engine 

44 HorrER Scow 659 

Steam Vessels, Proportion of Power, 

etc 616 

Steam Wagon 657* 

Steel 453, 497 

Steel Plates, Weiqht of 113 

Steel Wire, Weight of 114 

Stones, etc., Crushing Strength 505 

Strength of Materials 447 

" " Memoranda 494 

Sugar Mills 654 

Surf Boat 651 

Swimming 406 



T. U. 

Tandem 409 Undershot Wheels 436 

Tangents and Cotangents 315 Unguents, Experiments with 344 

Tannin, A verage Quantity of 101 Ungulas 260, 275 

Tensile Strength 449 I 

Tensile Strength of Copper at differ- 
ent Temperatures 452 

Thermometers 532 Value and Weights of Foreicm Coins 20, 23 

72, 629 Varnishes 

Tides, Bench Marks T5 Vegetable Crops, Substances of 101 

Rise and Fall 75, 76 Vegetables, Properties of 100 

44 Time of High Water 72 Vegetation, Limits of 12S 

Stone-gathering 657**1 Velocipedes 657** 



INDEX. 



XV 



Page | 

Ventilation 533 

Vernier Scale 560 

Vessels and Boats, Elements of 651 

Vessels, Construction of 611 

Veterinary 153 

Visible Distance of Objects 37 

Volume and Weights of various Sub- 
stances. 161 

Vulgar Fractions 40 

W. 

Walking 405 

Walls and Embankments 322 

Walls, Thickness of 321 

Warming Buildings and Apartments . 533 

Water 539 

" Jet Pump 558 

44 Lifting Pump 55S 

" Projection of 558 

Waterfalls and Cascades 13T 

Watee Powee 430, 55S 

Watee- wheels 431, 440 

Waves of the Sea 541, 558 

Weather, Foretelling 393 

Wedge 260, 2T1, 331 

Weight and British Value of Foreign 

Coins 28 

Weight and Mint Value of Foreign 

Coins 26 

Weights and Measuees, IT. S. 20 

44 " Foreign ... 29 

Weights, Ancient 

Weights of Anchors and Kedges ... 133 

" of Angle Iron 117 

" of Bells 131 

" of Bolts and Nuts 123,125 

" of Brass 112 

44 of Braziers and Sheathing 

Copper 116 

44 of Cast Iron Ill 

44 of Cast-iron Pipes 110 

" of Cast or Wrought Iron . . Ill 
" of Cast Metals by Patterns. 162 

11 of Cattle and Horses 143 

14 of Qhains 138 

•' of Copper ." 112 

44 of Copper Rods or Bolts. . . 117 
44 of Galvanized Sheet Iron . 120 

11 of Grain 102 

* * of Green and Seasoned Tim- 
ber 162 

" of Ingredients 156 

" of Iron and Copper Pipes. 118 
" of Iron, Copper, Lead, and 

Zinc 118 

•! of Iron, Steel, Copper, and 

Brass Plates 113, 115 

44 of Iron, Steel, Copper, and 

Brass Wire 114 

" of Lead 112 

14 of Lead and Tin Pipes, .. . 122 



Weights of Men and Women 143 

of Boiled Iron 105, 106 

of Seamless Brass and Cop- 
per Tubes 121 

of Sheathing Nails 119 

of Tin Plates 121 

and Volume of Cast-iron 

and Lead Balls 109 

and Volume of Various 

Substances 161 

of Wrought-iron, Copper, 

or Brass Tubes and Pipes 120 
To Ascertain the Weight of 

a Bar, Beam, etc 146 

Weirs, Overfall 379 

Rectangular 36S 

Sluice 3S0 

Wells, Power to raise Water 55S 

Wheel and Axle 327 

Wheel Gearing 509 

General Illustrations 512 

Wheel Work, Complex 32S 

Wheels, Diameter and Pitch of 510, 513 

• Change Wheels 513 

44 Proportions of, etc. . . . 515, 517 

44 To Construct a Tooth 514 

Wind, Velocity and Force of 392 

Winding Engines 517 

Windlass, Chinese 328 

Windmills 424, 559 

Window Glass 102 

Wire Gauges 113, 114, 115, 117 

Wire and Hemp Rope 629 

Wire Rope 137, 140, 452, 494 

44 and Cables, compared . . . 141 

Women, Weight of.. :. 143 

Wood, Impregnation of 521 

Wood, Timber, etc 519, 565 

44 Seasoning and Preserving 520 

Wood Bearing for Shafts 346 

Wood-working Machinery 556 

Wood Work, To Preserve 626 

Woods, Relative Value of .' 474 

44 Weight and Strength of 522 

Work, To Compute done by any Ma- 
chine 563 

Works, American, of Magnitude .... 103 

Writing upon Zinc Labels 625 

Weought Iron 496 

44 Weight of Ill, US 

44 Rigging 139 

44 Tubes and Pipes, Weight of 120 

Y. 
Yacht 645, 652 

Z. 

Zinc, Weight of... 14S 

Zinc Sheets, Thickness and Weight of 102 

Zone, Circular 25S 

Zone of a Circle, Areas of 207 

Zone of a Circular Spindle 2S1 



Base Ball and Cricket, 657** Fly Rod Casting, 406. Ice Boats, 620*. 

Pigeon and Bird Shooting, 620**. Pigeon Flying, 409. Snow Shoes, 406. 



EXPLANATIONS OF CHARACTERS 

Used in Calculations, etc., etc. 

= Equal to, as 12 inches = 1 foot, or 8x8 = 16x4. 

-f- Plus, or More, signifies addition; as 4 + 6 + 5 = 15. 

— Minus, or Less, signifies subtraction ; as 15 — 5 = 10. 

x Multiplied by, or Into, signifies multiplication; as 6x9=72. 
vxd, a.d, or ad, also signify that a is to be multiplied by d. 

-^- Divided by, signifies division ; as 72+-9=S. 

: Is to, : : So is, : To, signifies Proportion, as 2 : 4 : : 8 : 16 ; that is, 
as 2 is to 4, so is S to 16. 

The Vinculum, or Bar, signifies that the numbers, etc., over 
which it is placed, are to be taken together; as 8—2 + 6 = 12, or 
3x5+3=24. 

. Decimal point, signifies, when prefixed to a number, that that num- 
ber has some power of 10 for its denominator; as .1 is ^, .15 is ^%, 
etc. 

<\) Difference, signifies, when placed between two quantities, that 
their difference is to be taken, it being unknown which is the greater. 

' " '" signify Degrees, Minutes, Seconds, and Thirds. 

/_ signifies Angle. 

-t- signifies Perpendicular. 

A signifies Triangle. 

□ signifies Square, as □ inches; and E cube, as cubic inches. 

> 1 ? < L signify Inequality, or greater or less than, and are put be- 
tween two quantities ; as al 6 reads a greater than b, and alb reads 
a less than b. 

,\ signifies Therefore or Hence. 

V signifies Because. 

( ) [ ] Parentheses and Brackets signify that all the figures, etc., 
within them are to be operated upon as if thev were only one; thus, 
(3 + 2)x5 = 25; [8-2]x5 = 30. 

p or 7T is used to express the ratio of the circumference of a circle to 
its diametcr = 3.1416. 

a' a" a"' signify a prime, a second^ a third, etc. 

i qp signify that the formula is to be adapted to two distinct cases. 

V Radical sign, which, prefixed to any number or symbol, signifies 
that the square root of that number, etc., is required ; as V9, or V a + b. 
The degree of the root is indicated by the number placed over the sign, 
which is termed the index of the root or radical; as ty , V ', etc. 

Notes. — The degrees of temperature used are those of Fahrenheit. 

g id the common expression for gravity^ 32. 1G6, 2g— G4.33, y/2g=zS.Q2feet. 



NOTATION. H 

2 , *, etc., set superior to a number, signify the square or cube root, 
etc., of the number ; as 2* signifies the square- root of 2. 

4 4 6 

a , 3 , 3 , etc., set superior to a number, signify the square or cube root 
of the 4th power, etc. 

l \ 3 - 6 , etc., set superior to a number, signify the tenth root of the 
17th power, etc. 

1? 2 , added to or set inferior to a symbol, reads sub 1 or sub 2, and 
is used to designate corresponding values of the same element, as h, 
h L , 7* 2 , etc. 

2 , 3 , 4 , added or set superior to a symbol, signifies that that number, 
etc., is to be squared, cubed, etc. ; thus, 4 2 means that 4 is to be mul- 
tiplied by 4; 4 3 , that it is to be cubed, as 4 3 is=4x 4x4=64. The 
power, or number of times a number is to be multiplied by itself, is 
shown by the number added, as 2 , 3 , % 5 , etc. 

& signifies Dead flat, or the location of the frame of a vessel at its 
greatest transverse section. 

' " set superior to a figure or figures, signify Feet and Inches. 



NOTATION. 




1=1. 7-- 


=VIL 


40=XL. 


2=11. 8 = 


=VIII 


50=L. 


3 = 111. 9 = 


=IX. 


60=LX. 


4=IV. 10 = 


=x. 


70=LXX. 


5=V. 20 = 


=xx. 


80=LXXX. 


6= VI. 3Q= 


=xxx. 


90=XC. 


100=C. 


10,000 = 


=X, or CCIOO. 


500 =D, or 10. 


50,000 = 


=L, or IOOO. 


1000 =M, or CIO. 


60,000 = 


=LX. 


2000 =MM. 


100,000 = 


=C, or CCCIOOO. 


5000 =V, or IOO. 


1,000,000 = 


=M, or CCCCIOOOO, 


6000= VI. 


2,000,000 = 


=MM. 



As often as a character is repeated, so many times is its value re- 
peated. 

A less character before a greater diminishes its value, as IV=I— V, 
or 1 subtracted from 5=4. 

A less character after a greater increases its value, as XI=X+I ? 
or 1 added to 10 = 11. 

For every O annexed, the sum is increased 10 times. 

For every C and O, placed one at each end, the sum becomes 10 
times as many. 

A bar, thus , over any number, increases it 1000 times. 
Illustrations.— 1MQ, MDCCCXL. 18560, XVlIiDLX. 

B* 



18 ALGEBRAIC SYMBOLS AND FORMULAE. 



ALGEBRAIC SYMBOLS AND FORMULAE. 

Where / represents the length, c represents the chord, 

b " breadth, a " area, 

d " depth, r " radius, 

k " height, v l * versed sine, 

h' " h prime, h / " h sub. 

-=sum of the length and the breadth divided by the depth. 



d 
lb 



— product of the length and the breadth divided by the depth. 



— —= difference of the length and the breadth divided by the depth. 
p b 3 — product of the square of the length and the cube of the breadth. 
-^tj— square root of the length divided by the cube root of the breadth. 

— - — = square root of the sum of the length and the breadth di- 
vided by the depth. 

'/■ — r=^=cube root of the difference of h prime and h sub, divided 
V2g 
by the square root of 2g. 

cipv=c greater or less than v. Here there are expressed two values: 
first, the difference between c and v ; second, the sum of c and v. 
In this and like expressions, the upper symbol takes preference of the lower. 

-\/a + (c— ?>y=x. Add the square of the difference between the 
chord and radius to the area, and extract the square root ; the result 
will be equal to x. 

It is frequently advantageous to begin the interpretation of a formula at the right 
hand, as in the above case. 



<J- 



/(x + yY 
lyj- f£ — 1=2. Divide the square of the sum of x andy by the 

square of y ; subtract unity from the quotient ; extract the square root 
of the result ; multiply it by the length, and the product will be equal 

to z. 

2fsin. 75 ) 2 

— : ' £—. Divide twice the square of the sine of the angle of 

, l+(sin.75°) 2 ' 

75° by the square of the sine of the angle of 75° added to unity. 

— ^ \sVTg(Vh-VV) +2.303 c. log.: SV ^r b Lf. 

(SV>) ) JK SV2 f/ h'-b\ 

Multiply S by the V of 2g, and this product by the difference between 
the square roots of A and h prime ; add this to 2.303 times the com- 
mon logarithm of the quotient arising from dividing the product of S 



CHRONOLOGICAL EKAS AND CYCLES. 



19 



into V2ffh diminished by b, by the product of S into V2yh prime di- 
minished by b, and multiply this sum by the quotient of 2a divided by 
the square of the product of S into V2g, which will be equal to t. 

2a +3 cos. 98°= 2a— 3 cos. 82°= twice a diminished by three times 
cosine of 82°. 

Cosine of any angle greater than 90° and less than 270° is always — or negative, 
but is numerically=cosine of its supplement, i. e. the remainder after subtracting 
angle from 180°. 

39.127-. 09982 cos. 2L=7. 

Assume L = 46°. Here, cos. 2 X 46°, being = 92° and less than 270° , becomes — j 
therefore —.09982 and —cos. 90° become + 39.127 + cos. 2L X. 09982 == I. 



CHRONOLOGICAL ERAS AND CYCLES FOR 1880. 

The year 1880, or the 105th year of the Independence of the United States of America, 

corresponds to 
The year 7388-89 of the Byzantine Era; 
" 6593 of the Julian Period ; 
" 5640-41 of the Jewish Era ; 

" 2656 of the Olympiads, or the fourth year of the 664th Olympiad, commenc- 
ing in July, 1S79, the era of the Olympiads being placed at 775.5 
years before Christ, or near the beginning of July of the 393S th 
year of the Julian Period ; 
" 2633 since the foundation of Rome, according to Varro; 
" 2192 of the Grecian Era, or the Era of the Seleucidse ; 
" 1599 of the Era of Diocletian. 
The year 1297 of the Mohammedan Era, or the Era of the Hegira, begins on the 
7th of Februaiy, 1880. 

The first day of January of the year 1880 is the 2,407,716 th day since the com- 
mencement of the Julian Period. 

Dominical Letter . . . ; D C I Lunar Cycle or Golden Number 19 

Epact 18 I Solar Cycle 13 

B mQm Chronology. 

4004. Creation of the World (according to Julius Africanus, Sept. 1st, 5508 ; Samar- 
itan Pentateuch, 4700; Septuagint, 5S72; Josephus, 465S; Hales, 5411). 



2203. 
2090. 
1180. 
1111. 

753. 

576. 

A.D. 

214. 

667. 

991. 
1066. 
1180. 

1383. 
1492. 
1627. 
1752. 

1769. 
ent 



1789. 
1772. 



Deluge (according to Hales, 3154). 

Chinese Monarchy. 

First Egyptian Pyramid. 

Troy destroyed. 

Mariner's Compass discovered. 

Foundation of Rome. 

Money coined at Rome. 

Grist Mills introduced. 

Glass discovered. 

Arabic numerals introduced. 

Battle of Hastings. 

Mariner's Compass introduced in 

Europe. 
Cannon introduced. 
America discovered. 
Barometer and Thermometer invtd. 
New Style, introduced into Britain; 

Sept. 3 reckoned Sept. 14. 
James Watt — First design and pat- 
of a steam-engine having a separate 

" of condensation. 
French Revolution. 
Oliver Evans — Designed the non- 



605. Geometry, Maps, etc., first intro- 
duced. 
289. First Sun-dial. 
219. Hannibal crossed the Alps. 
219. Land Surveying first introduced. 
155. Time first measured by water. 
51. Caesar invaded Britain. 

A.D. 

condensing engine. 1792. Applied 
for a patent for it. 1801. Con- 
structed and operated it. 

1790. Water lines first introduced in the 
models of vessels in the U. S., by 
Orlando Merrill, of Newburyport, 
Mass. 

1797. John Fitch — Propelled a yawl boat 
by the application of steam to 
side wheels, and also to a screw 
propeller, upon the Collect pond, 
New York. 

1807. Robert Fulton — First passenger 
Steam-boat. 

1827. First Rail Road in U. S., from 
Quincy to Neponset, Mass. 



UNITED STATES MEASURES AND WEIGHTS. 

According to J±ct of 1866. 

For Equivalents of Old Measures and Weights to New, see page 630. 

3VIeas-u.res of* Length.. 

Denominations and Values. Equivalents in use. 



Myriameter.. . 

Kilometer 

Hectometer.. . 
Dekameter . . . 
Meter 


10 000 meters. 

1 000 meters. 

100 meters. 

10 meters. 

1 meter. 


6.2137 miles. 

.62137 mile, or 3280 feet and 10 ins, 
328 feet and 1 inch. 
393.7 inches. 
39.37 inches. 


Decimeter 


Xoth of a meter. 


3.937 inches. 


Centimeter . . . 


Xo o tn °f a meter. 


.3937 inch. 


Millimeter 


3^0 oth of a meter. 


.0394 inch. 



Measures of* Surface. 

Denominations and Values. Equivalents in i 



10 000 square meters. 

100 square meters. 

1 square meter. 



Hectare . 
Are 

Centare . 



2.471 acres. 

119.6 square yards. 

1550 square inches. 



IVIeasxires of* "Volume. 

Denominations and Values. Equivalents in use. 



No. of 
Liters. 



Cubic Measure. 



Dry Measure. 



Liquid or Wine 
Measure. 



Kiloliter \ 

or Sterej 

Hectoliter. 

Dekaliter . 

Liter 

Deciliter . . 
Centilliter. 
Milliliter.. 



1000 

100 
10 

1 

1_ 

10 



1 cubic meter 

Yo cubic meter 
10 cubic decimeters 
1 cubic decimeter 
xo cubic decimeter 
10 cub. centimeters 
1 cubic centimeter 



1.308 cubic yards. 

2 bush, and 3.35 pechs, 
9.08 quails. 

908 quart. 
6.1022 cubic inches. 

.6102 cubic inch. 

.061 cubic inch. 



264.17 gallons, 

26 .417 gallons. 
2.6417 gallons, 
1.0567 quarts, 
.845 gill. 
.338 fluid oz. 
21 fluid drm. 



Denominations 


"Weiglats. 

and Values. 


Equivalents in use. 


Names. 


Number of 
Grams. 


Weight of Volume of Water at its 
Maximum Density. 


Avoirdupois 
Weight. 


Millier or Tonneau . 
Quintal 


1 000 000 

100 000 

10 000 

1000 

100 

10 

1 

JL 
10 

100 

1 

1000 


1 cubic meter. 

1 hectoliter. 

10 liters. 

1 liter. 

1 deciliter. 

10 cubic centimeters. 

1 cubic centimeter. 

Xoth of a cubic centimeter. 

10 cubic millimeters. 

1 cubic millimeter. 


2204.6 pounds. 
220 .46 pounds, 
22.046 pounds. 
2.2046 pounds. 
3.5274 ounces. 

.3527 ounce. 
15.432 grains. 
1.5432 grains, 

.1543 grain. 

.0154 grain. 


Myriagram 

Kilogram or Kilo. . 

Hectogram 

Dekagram 

Gram 


Decigram 


Centigram 

Milligram 



For Measuring Surfaces the square Dekametre is used under the term of Are ; 
the Hectare, or 100 ares, is equal to about 2 acres. 

The Unit of Capacity is the cubic Decimetre or Litre, and the series of measures 
is formed in the same way as in the case of the table of lengths. 

The cubic Metre is the unit of measure for solid bodies, and is termed Stere. 

The Unit of Weight is the Gramme, which is the weight of one cubic centimetre 



MEASURES AND WEIGHTS. 21 

of pure water weighed in a vacuum at the temperature of 4° Centigrade, or 39°. 2 
Fahrenheit, which is about its temperature of maximum density. 

In practice, the term cubic Centimetre, abbreviated C. (J., is used instead of Mil- 
lilitre, and cubic Metre instead of Kilolitre. 

According to Previous and Existing Laws. 
MEASURES OF LENGTH. 

The Standard of measure is a brass rod, which, at the. temperature 
of 32°, is the standard yard. 

Lineal. 

12 inches =1 foot. Inches Feet. Yards. Rods. Furl. 

3 feet =1 yard. 36= 3. 

5.5 yards =1 rod. 198= 16.5= 5.5. 

40 rods =1 furlong. 7920= 660 = 220 = 40. 

8 furlongs = 1 mile. 63360=5280 =1760 =320=8. 

The inch is sometimes divided into 3 barley corns, or 12 lines. 
A hair's breadth is the .02083 (48th part) of an inch. 

1 yard is 000568 of a mile. 

1 inch is 0000158 of a mile. 

Ounter's Chain. 

7.92 inches=l link. 

100 links =1 chain, 4 rods, or 22 yards. 
80 chains =1 mile. 

Ropes and. Cables. 
6 feet=l fathom. | 120 fathoms=l cable's length. 

Greogx*aphical and Nautical. 

1 degree of a great circle of the earth= 69.77 Statute miles. 
1 mile =2046.58 yards. 

Log Lines. 

Estimating a mile at 6J39.75 feet, and using a 30" glass, 
1 knot =51. 1629 feet, or 51 feet 1.95 inches. 
1 fathom = 5.11629 feet, or 5 feet 1.395 inches. 

If a 28" glass is used, and 8 divisions, then 
1 knot=47 feet 9.024 inches. | 1 fathom=5 feet 11.627 inches. 

The line should be about 150 fathoms long, having 10 fathoms between the chip 
and first knot for stray line. 

Note. — Bowditch gives 6120 feet in a sea mile, which, if taken as the length, with 
a 28" glass, will make the divisions 47.6 feet and 5:95 feet. 

Cloth. 
1 nail =2.25 inches =.0625 of a yard. 
1 quarter =4 nails. 
5 quarters =1 ell. 

Pendulums. 
6 points = 1 line. | 12 lines = 1 inch. 



22 MEASURES AND WEIGHTS. 



Shoemakers'. 

No. 1 is 4.125 inches in length, and every succeeding number is .333 
of an inch. 

There are 28 numbers or divisions, in two series of numbers, viz., 
from 1 to 13, and 1 to 15. 

Miscellaneous. 

1 palm =3 inches. 1 span =9 inches. 

1 hand =4 inches. I 1 metre =3. 2809 feet. 



MEASURE OF TIME. 



60 seconds =1 minute. 
60 minutes =1 degree. 
360 degrees =1 circle. 



3600=60. 
1296000=21600 = 360. 



Sidereal day=23 h., 56 m., 4.092 sec, in solar or mean time. 
Solar day, mean=24 h., 3 m., 56.555 sec, in sidereal time. 
Sidereal year, or revolution of the earth, 365.25635 solar days. 
Solar, Equinoctial, or Calendar year, 365.24224 solar days. 

1 day = .002739 of a year. | 1 minute = .000694 of a day. 

30°=1 sign. 



MEASURES OF SURFACE. 

144 square inches = 1 square foot. 
9 square feet =1 square yard. 
100 " " =1 square (ArcJiitect's Measure). 



Land.. 
30.25 square yards =1 square rod. 
40 square rods =1 square rood. 
4 square roods \ =± 
10 square chains) 
640 acres =1 square mile. 

203.710326 feet, 69.570109 yards, or 220 by 19S feet square=l acre. 

Paper. 

24 sheets =1 quire. | 20 quires =1 ream. 



Yards Rods. Food*. 

# 1210- 

4840 = 160. 
3097600 = 102400=2560. 



Cap 13 Xl6 inches. 

Demy 15.5 X 18.5 " 

Medium 18 X22 

Koval 19 X24 

Super-royal.... 19 X27 " 

Imperial". 21.25X29 " 

Elephant 22.25x27.75 



Drawing Paper. 

Columbia 23 x 33.75 inches. 

Atlas 26x 33 

Theorem 28 X 34 

Doub.Elephant 26 X 40 

Antiquarian ... 31 x 52 

Emperor 40 X 60 

Uncle Sam 48x120 



Peerless 18 x 52 inches. 



MEASURES AND WEIGHTS. 



23 



Tracing Paper. 



Double Crown 20x30 inches. 

Double D. Crown... 3Qx40 " 
Double D. D. Crown 40 X 60 " 



Grand Royal 18x24 inches. 

Grand Aigle 27x40 u 

Vellum Writing, 18 to 28 in wide. 



1 sheet 
1 quarto 
1 octavo 



Miscellaneous. 

- 4 pages. 1 duodecimo —24 pages. 

= 8 " 1 eighteenmo=36 * 4 

= 16 " 1 bundle =2 reams. 



Roll of Parchments 60 sheets. 



MEASURES OF VOLUME. 

The Standard gallon measures 231 cubic inches, and contains 
8.3388822 avoirdupois pounds, or 58373. troy grains of distilled water, 
at the temperature of its maximum density 39°. 83, the barometer at 
30 inches. 

The Standard bushel is the Winchester, which contains 2150.42 
cubic inches, or 77.627413 lbs. avoirdupois cf distilled water at its 
maximum density. 

Its dimensions are 18.5 inches diameter inside, 19.5 inches outside, 
and 8 inches deep ; and when heaped, the cone must not be less than 
6 inches high, equal 2747.715 cubic inches for a true cone. 



Liq.u.icL. 



4 gills = 1 pint. 
2 pints =1 quart. 
4 quarts =1 gallon. 



Dry. 



2 pints =1 quart. 
4 quarts =1 gallon. 
2 gallons = 1 peck. 
4 pecks =1 bushel. 



32=8. 



Pints. Quarts Gallons. 



16 = 8. 

64 = 32 = J 



Cu."bic. 

1728 cubic inches = 1 foot. 
27 cubic feet = 1 yard. 



Inches. 

4:6656. 



Note. — A cubic foot contains 2200 cylindrical inches, 3300 spherical inches, or 
G600 conical inches. 

Fluid. 



60 minims =1 drachm. 


Minims. Drachms Ounces 


8 drachms = 1 ounce. 


480. 


16 ounces —1 pint. 


7680 = 128. 


8 pints =1 gallon. 


61240 = 1024=128. 



2.4 



MEASURES AND WEIGHTS. 



Miscellaneous. 

1 cubic foot 7.4805 gallons, 

1 bushel. 9.30918 gallons. 

1 chaldron— 36 bushels, or 57.244 cubic feet. 

1 cord of wood 128 cubic feet. 

1 perch of stone 24.75 cubic feet. 



1 load bay or stra\v=36 trusses. 
1 M quills =1200 quills. 

Galls 

Puncheon of Brandy 1 10 to 120 

Puncheon of Rum 100 to 110 

Hogshead ol Brandy 55 to CO 

Pipe of Madeira 92 

Hogshead of Claret 46 

A Hogshead is one half, a Quarter cask is one fourth, and an Octave is one eighth 
of a Pipe, Butt, or Puncheon. 



1 quarter =8 bushels. 
1 sack flours 5 •* 

Galls. 

Butt of Sherry 108 

Pipe of Port 115 

Pipe of Teneriffe 100 

Butt of Malaga 105 

Puncheon of Scotch Whisky. .110 to 130 



MEASURES OF WEIGHT. 

The Standard avoirdupois pound is the weight of 27 7015 cubic 
inches of distilled water weighed in air, at 39°. 83, the barometer at 30 
inches. 

A cubic inch of such water weighs 252.6937 grains. 



Avoirdupois. 

16 drachms — 1 Ounce. Drachms. 

16 ounces =1 pound. 256, 

112 pounds = lcwt. 28672 = 1792. 

20 cwt. =1 ton. 573440=35840 



Ounces. Pounds. 



-2240. 



1 pound = 14 oz. 11 dwts. 16 grs. troy, or 7000 grains. 
1 ounce =18 dwts. 5.5 grains troy, or 437.5 grains. 



Troy. 

24 grains =1 dwt. 
20 dwt. =1 ounce. 
12 ounces =1 pound. 

7000 troy grains 

437.5 troy grains 

1 75 troy pounds 

175 troy ounces 
1 troy pound 
1 avoirdupois pound 



Dwt 



Grains. 

480. 
5760 = 240. 

= 1 lb. avoirdupois. 
= 1 oz. " 
= 144 lbs. 
= 192 oz. " 
= .822857 lb. 

1.215278 lbs. troy. 



20 grains 

3 scruples 

8 drachms 
12 ounces 
45 drops 

2 table spoonsful 



Apothecaries. 



= 1 scruple, 
= 1 drachm. 
= 1 ounce. 
= 1 pound. 

= 1 tea spoonful or a fluid drachm. 
1 ounce. 



Grains. Scruples. Drachms 

60. 

480 = 24. 
5760 = 288 = 96. 



The pound, ounce, and grain are the same as in Troy Weight. 



MEASURES AND WEIGHTS. 



25 



Diamond. 
1 carat= 4 grains. , 16 parts = .8 grains. 

1 grain=16 parts. | 4 grains=3.2 ** 

Miscellaneous. 

Coal and wood per cubic foot. 



lbs. 



Charcoal, hard-wood . . 18.5 

do. pine " 18 

Pine, Virginia . * 21 

do. Southern 25.5 



1 1 ton Scotch coal . . . , = 
1 1 ton R. N. allowance = 



:43 cub. feet. 
:45 " " 



Anthracite, ordinary 50 to 55 

Bituminous " .45 to 55 

Cumberland 53 

Cannel 50.3 

1 ton Welsh coal =43 cub. feet. 

1 ton Newcastle coal. =45 M " 

Lead. 
1 fodder=8 pigs. Roll sheet lead=6.5 to 7.5 feet in width and from 30 
to 35 feet in length. 

MEASURES OF VALUE. 

10 mills =1 cent. 10 dimes =1 dollar. 

10 cents=l dime. 10 dollars=l eagle. 

Standard of gold and silver is 900 parts of pure metal and 100 of alloy- 
in 1000 parts of coin. 

Fineness expresses quantity of pure metal in 1000 parts. 

Remedy of the Mint is allowance for deviation from exact standard fine- 
ness and weight of coins. 

Nickel cent (old) contained 88 parts of copper and 12 of nickel. 

The bronze cent contains 95 parts of copper and 5 of tin and zinc. 

Pure gold, 23.22 grains ■= $1.00. Hence the value of an ounce is 
|20 67.183+. 

Standard gold, $18 60.465+ per ounce. 



WEIGHT AND FINENESS OF U. S. COINS. 
Q-old. 



Coin. 


Weight. 


Weight 
of pure 
metal. 


Coin. 


Weight. 


Weight 
of pure 
metal. 


Dollar 


OZ; 

.05375 

.134375 

.16125 


Gr. 
25.8 
64.5 

77.4 


Gr. 
23.22 
58.05 
69.66 


Half Eagle. . . . 
Eagle 


Oz. 

.26875 
.5375 
1.075 


Gr. 
129 
258 
516 


Gr. 
116.1 


Quarter Eagle . 
Three Dollars . 


232.2 


Double Eagle.. 


464.4 



Silver. 



Dime 

20 Cents.. 

Quarter Dollar. 



,.080375 

.16075 

.2009375 



88.58 


34.722 


77.16 


69.444 


96.45 


86.805 



Half Dollar . . 
Trade Dollar. 
Silver Dollar. 



.401875 


192.9 


.875 


420 


.859375 


412.5 



173.61 

378 

371.25 



Copper. 



Coin. 


Weight. 


Copper. 


Tin and 
zinc. 


Coin. 


Weight. 


Copper. 


Tin and 
zinc. 


One Cent 

Two Cents 


Gr. 
48 
96 


Per cent. 
95 
95 


Per cent. 
5 
5 


Three Cents. . . 
Five Cents. . . . 


Gr. 
30 

7T.16 


Per cent. 

T5 
75 


Per cent. 
25 
25 



Tolerance. Gold Dollar to Half Eagle, .25 grains ; Eagles, .5 grains. 
Silver, 1.5 grains for all denominations. Copper, 1 to 3 cents, 2 grains ; 
5 cents, 3 grains. 

Legal Tender. Gold, unlimited. Silver. For dollars of 412.5 grains, 
unlimited ; for subdivisions of dollar, $10. (Trade dollars [420 grains] 
are not legal tender.) Copper or cents, 25 cents. 

Note.— Weight of dollar up to 1837 was 416 grains; thence to 1S73, 412.5 grains. 
Weight of $1000 @ 412.5 gr. = $859,375 oz 

c 



26 



MEASURES AND WEIGHTS. 



Beitish standards are : Gold, § j of a pound,* equal to 11 parts pure gold and 1 
of alloy ; Silver, |f| of a pound, or 37 parts pure silver and 3 of alloy, =925 fine. 

A Troy ounce of standard gold is coined into £3 17s. lOd. 2/., and an ounce of 
standard silver into 5s. Qd. 1 pound silver is coined into 66 shillings. Copper is 
coined in the proportion of 2 shillings to the pound avoirdupois. 

A pound sterling (1SS0), $4S6.65; hence -^Jq of this = value of one penny = 
2.02770S33 cents. 

"Weiglit, Fineness, and. Mint "Values of* 3Toreign 
Copper, Silver, and. Grold. Coins. 

By Laws of Congress, Regulations of the Mint, and Reports of its Directors. 
The Current Value of Silver Coins is necessarily omitted, as the value of silver is 
a variable element. Hence, in order to compute the current value of a silver coin, 
the price of fine, or a given standard of silver being known, 
Proceed as per following rule to compute value of coins. 

Price of silver should be taken at that of London market for British standard 

(925 fine), it being recognized as standard value, and governing rates in all countries. 

Illustration. — If it is required to determine value of a Mexican dollar in cents. 

Weight S67.5 oz. 903 fine. Value of silver in London 52.75 pence per ounce, = 

106.9616-f- cents. 

867 •SyOO^ 
Then * = .846867— and 106.9616 X .846867 = 90.5832 cents. 



Countries given in 

Country and Denomination. 


Italics I 

Weight. 


mve no 

Fine- 
ness. 


ta Na 

Pure 
silver. 


tional Coi 

Current 

or 
nominal. 


nage. 

J AL U 
G 

U.S. 


E. 
old. 

British. 


A rabia— Piastre or Mocha Dollar 

Argentine Republic — Dollar 

(South American and Foreign Coins.) 
Australasian — Same as British. 

Australia— Sovereign, 1855 

Pound, 1852 


Oz. 

.256.5 
.281 

.397 
.596 

.112 

.803.75 

.S67 

.028.8 
.575 

.187.5 

.027 

.866 
.869 

.801 
.867 

.087 
.866 


Thous 8 . 
916~ 

916.5* 

900 

900 
986 

900 

S70 

916766 
917.5 

926" 

875 
850 
833 

9005 

870 

901 
901 


Grains. 

17L47 
257.47 

347.22 
362.06 

1167 

sT25 

1L34 
353.33 

346.22 

3"f08 
374.63 


$ c. 
83.14 
50.69 

.41 


S c. 

4 85.7 

5 32.37 

2 28.3 


£ s. d. 

19 11.5 
1 1 10.5 


Austria — Kreutzer (copper) 

Florin, new 


.2 


Dollar 4t 




Ducat 


9 4.6 


Belgium — Same as France. 
Bermudas — Same as British. 




Bolivia — Centena 


.75 

.547 

1.01 
1.52 

.9 
.14 


15.59.3 

54.59 
10 90.6 

3 99.97 

14 96.39 

15 59.3 


.37 


Boliviano 




Doubloon, 1827-36 .... 


3 4 1 

.27 


Milreis 


26.92 


20 Milreis, 1S54-56 

Canada — Cent, sterling 


2 4 9.84 
.5 


25 Cents « 

Penny *' 

Shilling " 

Pound, currency 

Cape of Good Hope — Same as 

British 

Central America — 4 Reals 

Dollar 

Doubloon .... 
Chili — Centavo 


.75 

16 5.25 

3 1 5.97 
.45 


Dollar, new 




Doubloon 


3 4 1 


China — Cash Le 


.07 


10 Cents Leang 




Dollar 


— 



* A pound is assumed to be divided into 24 equal parts or carats, hence the propor- 
tion is equal to 22 carats. 



MEASURES AND WEIGHTS. 



27 



Table of "Weight and Mint "Valnes— Continued. 



Country and Denomination. 



Weight. 



Pure 

silver. 



'VALUE. 

Gold. 

U. S. British. 



Cochin China — Mas, 60 s-apeks . . 
10 Mas, 1 quan. . 

Columbia — Centavo 

Peso, new 

4 Escudos 

Doubloon, old 

Costa Rica — Same as Mexico. 
Denmark — Mark, 16 skillings. . . 

Crown 

10 Thalers 

East Indies — See Hindostan. 

Ecuador — Centavo 

Peso 

Egypt — Piastre, 40 paras 

Guinea, bedidlik 

Pound.. 

England — Penny 

Shilling, new 

Half-crown 

Florin 

Sovereign or Pound . . 

France — Centime 

Sou, 5 centimes 

Franc, 100 centimes . . . 

20 u Napoleon, new. 

(25 francs and 20 cent. =.£1 stg.) 

Germany — Groschen, 10 pfennige 

Florin, before 1872 . . 

Thaler 

Ducat 

Greece and Ionian Islands — Same 

as France 

Drachma, 100 lepta . 

Pound 

Guatemala — Same as Mexico. 

Guiana — British, French, and 

Dutch 

Hanse Towns — Mark 

Hindostan — Rupee , 

Holland — Cent , 

Florin or Guilder, 100 
cents 

10 Guilders 

Honduras — Same as Mexico. 
Indian Empire— Rupee, 1 16 annas 

Italy — Same as France 

Lira, 100 centimes 

Japan — Sen 

Itzebu, new 

Cobang, new 

Java — Same as Holland. 

Malta— Scudo, 12 tari 

Mexico — Dollar, new 

Doubloon, new 

20 Pesos, Republic 

Morocco— Ounce, 4 blankeels 

Naples — Scudo 

6 Ducati 

Netherlands— Same as Holland. 



Oz. 



.801 
.433 

.861 



.015 

.427 



.801 

.04 

.275 

.275 

.304 

.182.5 

.454.5 

.363.6 

.256.7 

.032 

.161 

.161 

.207.5 



.34 

.595 

.112 



.010,4 



.012.8 
.374 



.021.6 
.215 



.375 
.16 



.279 
.362 



.867.5 
.867.5 
.081 

.844 
.245 



900 

844 

S70 



900 
895 



900 
755 

875 
875 

924.5 
925 
925 
916.5 



900 

SC9 



900 
900 
9S6 



900 



346.03 



346.03 
14.5 



80.99 
201.8 
161.44 



).55 



146.83 

257.04 



900 
916.5 



900 
899 



91;:. 5 

835 



890 
508 



903 

8T0.5 
873 

830 
996 



164.53 



1G5 
65.12 
119.19 



377.17 



336.25 



. c. $ c. 

6.75 

67.52 

1.01 — 



8.94 



2.02 J 



.2 
1.01 



2.38 



7 55.5 

15 59.3 



7 90 



5 0.52 
4 97.4 



4 86.65 



3 85.5 



2 28.38 



19.3 
5 6.11 



£ s. d. 
3.33 
2 9.33 
.5 

111 0.58 
3 4 1 

4.39 
13.22 

112 5.6 

~".5 



1 6.84 
1 5.3 
1 



10 
.1 
.5 

15 10.26 

1.175 

9 4.63 



9.5 
1 9.6 



23. 1 



40.49 
3 99.7 



4 44 



11.74 

.2 

1 8 
16 5.11 



.5 
18 2.96 



15 6.1 
19 51.5 



3 4 1.S8 

4 2.4 



5 4.4 1 S.75 



* 2.02771 cents. 



t Nominal value=2 shillings sterling. 



28 



MEASURES AND WEIGHTS. 



Table of Weight and. Mint Values- Concluded, 



Country and Denomination. 




Weight. 


Fine- 
ness. 


Pure 

silver. 


Current 

or 
nominal. 


V A L U 
G 

U.S. 


E. 

old. 

British. 


New Brunswick— Same as Canada 
Newfoundland — do. 
New Granada — Dollar, 1857 .... 
Doubloon 


Oz. 

.803 
.867 

.766 

.802 
.867 
.322 

.667 

.16* 
.16 

.8 

.268 

.270.8 

.273 
1.092 
.104 

.112 


Thous 8 . 
896 

858 

900 
900 

868 
835 

875 

835 
835 
900 

896 
896 

750 
750 
900 

999 


Grains. 

341.01 
346.46 

129.06 

277.73 

64.13 
64.13 
345.6 

9S.28 
393.12 


S c. 

22.81 

"777 

6.33 

^83 
4.39 

74.8 
1 


$ c. 
15 37.8 

15 55.7 

4 96.4 

5 10.5 

193.5 
2 31.3 


JE s. d. 
3 3 3 39 


Norway — Same as Denmark. 
Nova Scotia — Same as Canada. 
Paraguay — Foreign coins. 
Persia — Keran, 20 shahis 


11 25 


Peru— Dollar, 1858 






Sol 






Doubloon, old 




3 3 11 22 


Roumania — 2 lei 






Russia — Copeck 


33 


100 copeck, 1 rouble 

Sandwich Islands — U. S. currency 
Sardinia — Lira 




Spain— 100 Centimos, 1 peseta . . 
Dollar, 5 pesetas 


- 


100 Reals 


1 4.8 


10 Escudos 


1 7.32 


St. Domingo — Gourdes, 100 cents 

Sweden — Riksdaler, 100 ore 

Rix-dollar 




Carolin, 10 francs 

Switzerland — Same as France. 

Tunis — Piastre, 16 karubs 

Turkey — Piastre, 40 paras 

Tuscany — Zecchino, sequin 

Tripoli — 20 Piastres, 1 mahbub. . 
Uruguay — Dollar, 100 centimes . . 

England. 
Venezuela — Centavo 


711.42 

5.83 
2.16 
9 6.1 

3 0.89 

~~.5 



To Compute Value of 1 Coins. 

Rule. — Divide product of weight in grains and fineness bj r 480 (grains 
in one ounce), and multiply result by value of pure metal per ounce. 

Or, Multiply weight in ounces by fineness and by value of pure metal 
per ounce. 

Example 1. — When fine gold is $20 67.183-f- per ounce, what is the value of a 
British sovereign ? 

By preceding table, a sovereign weighs .2567 oz. = .2567x480 = 123.216 grains, 
and has a fineness of 916.5 — . 

Hence, m21 ^ 916 - 5 x20.67.1S3+= $4 86.34 

Ex. 2. — When fine silver is $1 15.5 per ounce, what is the value of a United 
States trade dollar. 
By table, a dollar weighs .875 oz., and has a fineness of 900. 
Hence, .875x900x1.15.5=90.95625 cents. 

To Convert TJ. S. to British Currency and. Con- 
trariwise. 

Rule. — 1. Divide cents by 2.02771— (2.02770833), or multiply by 
.49312— (.49311826), and result is pence. 
2. Multiply pence by 2.02771 — , or divide by .49312—, and result is cents. 
Example.— 1. What are 100 cents in pence ? 

100X. 49312— = 49.312— penc, e= 4s. 1.312<f. 
2. What is a pound sterling in cents? 

20x12 = 240 pence, which X2. 02771— = $4 S6.65. 



FOREIGN MEASURES AND WEIGHTS. 

FOREIGN MEASURES AND WEIGHTS. 



29 



MEASURES OF LENGTH. 

British. The Imperial standard yard is referred to a natural stand* 
ard, which is the length of a pendulum vibrating sec- 
onds in vacuo in London, at the level of the sea; meas- 
ured on a brass rod, at the temperature of 62°. 

Admiralty knot=6080 feet, 
French. Old System. (U. S. inches.) 
1 line =12 points 



0.08881 

1 inch =12 lines = 1.06577 

1 foot =12 inches ... = 12.78916 

New System. (U. S. inches.) 
I millimetre = .0393707 inches. 
1 centimetre = .3937079 " 
1 decimetre =3.9370797 " 
1 metre* =39.370797 H 



1 toise=6feet= 76.735 inches. 
1 league= 2280.33 toises (com'n). 
1 league =2000 toises (post). 



Prior to Law of 1866. 
1 decametre = 32.80899 feet. 
1 hectometre = 328.0899 " 
1 kilometre =1093.633 yards. 

i I myriametre^ 6.213825 miles. 

Note In the new French system, the values of the base of each measure — viz., 

Metre, Litre, Stere, Are, and Gramme— are decreased or increased by the following 
words prefixed to them. Thus, 

Milli expresses the 1000th part. Deca expresses 10 times the value. 



Centi 
Deci 



100th"" H.cto "" 100 

10th »J 1 Kilo " 1000 

Myrio expresses 10000 times the value. 



Ta"ble of* Lengtlis of Foreign Lineal HVEeasiAre. 



Place. 


Measure. 


U.S. Inch. 


Place. 


Measure. 


U.S. Inch. 


Abyssinia 

Aleppo and Asia 

Amsterdam 

Antwerp 


Pic, geometrical 
Pic 


30.37 

20.63 

11.144 

11:275 

25. 

12.445 

11.81 

11.48 

30.371 

18. 

12.357 

18. 

12.445 

18. 

25.98 

11.38 

11.23 

18. 

11.128 

25.089 

18.504 

26.89 


China 

Damascus . . . 

Dantzic 

Denmark .... 

Dresden 

Egypt 

Florence 

Frankfort . . . 

Geneva 

Genoa 

Gibraltar 5 . . . 

Greece 

Guinea 

Hamburg .... 

Hanover 

Ionian Isles 8 . 
Japan 

Java 

Leipsic 


Chik or Covid . . 

" Engineer's 

■ •" ■ commercial 

Pic 

Fuss 


13.125 
12.71 


Foot 

FUS3 


14.1 
22.93 


Guz 


11.3 




Fuss 


Fod 


12.357 




it 


Fuss 


11.15 




t( 


Derah 


25.49 


Belgium 1 

Bengal 


Elle 


Braccio 

Fuss, Surveyor's 
Pied 


22.98 


Cubit or Guz. .. 
Fuss 


14.01 
23. OSS 


Birmah ... 


Cubit 


Piede Manuale. . 


13.4SS 


Bohemia 2 


Fuss 


12. 


Bombay 

Brazil 


Hath 




IS. 




Jacktan 

Fuss 


144. 


Bremen 


Fuss 


11.279 


Brunswick 


Schuhor Fuss.. 
Cubit 




11.49 


Calcutta 


Foot , . . . 


12, 


Canary Isles* . . 
Candia 




Ink or Tattamy. 
Fan 


74.S24 


Pic or Ell 

Covid 


12.4- 
12.357 


Ceylon 

Constantinople. 


Foot 


Pic 


Fuss 


11.14S 



* According to Captain Kater's comparison, and the one adopted by the U. S. Ord- 
nance Corps = 39.3707971 inches, or 3.2S0S99 feet. 

c* 



30 



FOREIGN MEASURES A^D WEIGHTS. 

Ta"ble of* Lengths of Foreign Lineal Measures. 
Continued. 



Place. 


Measure. 


U. S. Inch. | 


Place. 


Measure. 


U. S. Inch. 




Covid 


1S.6 
11.167 
12. 

11.12S 
15.62 
25. 

20.592 
21. 
13.1S 
10.3S1 
12.353 
21.441 
38.27 
14.032 
7.SS2 
13.33 
12.357 
12.357 
10.79 
11.592 
9.8 
1.75 
13.75 
28. 


Sardinia .... 

44 

Saxony 

Siam 

Sicily 

Smyrna 

Spain 

(4 
(4 
(4 

Sweden 

Switzerland . 

Tripoli 

Turin 

Turkey . 

Tuscany 

Utrecht 

Venice 

44 

Vienna 

Warsaw .... 
Zurich 


Oncia 


1.686 


Malta 


Pie 


Liprando 

Fuss 


20.23 


Mauritius 9 . . . 


Foot 


11.14S 


Mexico 10 .... 


Pie 


Ken 


39. 


Milan 1 ! 


Foot 


Palmo 


9.53 




Guz 


Pic 


26.48 


Modena 


Piede 


Foot 


11.128 




Cubit or Canna. 




66.768 


Moscow 


Palmo Mayor . . 
Vara 


8.34 


Naples 

Norway 

Parma 


Palmo 


33.384 


Fod 


Fot 


11.657 


Pie 


Fuss (Berne) . . . 

" (Geneva).. 

Pic or Dreah . . . 

Fuss 


11.81 


Persia 




23.028 


Poland 


Foot 


21.75 


Portugal 


Palmo da junta. 
Foot 


13.4S8 


Pic great 

Foot 


27.9 


Prussia 




11.94 


Rhineland . . . 


Foot 




10.74 


Riga 

Rome 

u * 


44 


Pie 


13.68 


Pie, commercial 
Palmo 


Braccio Grosso . 

Braccio 

Fuss 


26.9 
39.371 


Russia ...... 


Verschok 

Foot 


12.45 


44 


Foot (Cracow). . 

44 


14.03 


n 


Archine 


11.812 



I 



Tal)le of* Lengtlis of* Foreign Ptoad. Measures. 



Place. 



Measure. 



U. S.Yards.l 



Place. 



Measure. 



U. S. Yards. 



Arabia 

Austria 

Baden 

Belgium 1 . . 



Bengal 

Birmah 

Bohemia 2 . . 

Brazil 3 

Bremen . . . 
Brunswick 
Calcutta. . . 
Ceylon 9 . . . 

China 

Denmark . . , 

Dresden 

Egypt 

England.. . 
Flanders. . . 
Florence. . . 
Francet . . . 

Genoa 

Germany . . 

Greece 

Guinea 

Hamburg. . 
Hanover.. . 
Hungary . . 

India 

Italy 

Japan 



Mile 

Meile (post) 

Stunden 

Kilometre 

Meile 

Coss 

Dain 

League (16 to 1°) 

" (IS to 1°) 

Meile 



Coss 

Mile 

Li 

Miil 

Post-meile 

Feddan 

Mile 

Mijle 

Miglio 

Kilometre 

Mile (post) 

Mile (15 to 1°). 

Stadium 

Jacktan 

Meile 



Wussa. 
Mile... 
Ink. . . . 



2146. 
S297. 
4SG0. 
1093.63 
2132. 
2000. 
4277. 
75S7. 
6750. 
6865. 
11S16. 
2160. 
1760. 
60S.5 
8238. 
7432. 

1.47 
1760. 
1093.63 
1809. 
1093.6 
S527. 
8101. 
10S3.33 

4. 
8238. 
8114. 
9139. 

24.89 
2025. 

2.038 



Leghorn .... 

Leipsic. ...... 

Lithuania . . . 

Malta 

Mecklenburg 
Mexico 10 .... 

Milanii 

Mocha 

Naples 

Netherlands . 

Norway 

Persia 

Poland 

Portugal 



Prussia . 

Rome . . . 



Russia . 



Sardinia . 
Saxony . . 
Siam 
Spain 



Sweden 

Switzerland . 

Turkey 

Tuscany 
Venice 



Miglio 

Meile (post) 



Canna 

Meile 

Legua 

Miglio 

Mile 

Miglio 

Mijle 

Mile 

Parasang 

Mile (long) 

Mitha 

Vara 

Mile (post) 

Kilometre 

Mile 

Verst 

Sashine 

Miglio 

Meile (post) 

Roenung 

League, legal . . 
" common 

Milla 

Mile 

Meile 

Berri 

Miglio 

Miglio 



1809. 
7432. 
9781. 

2.29 
8238. 
403S. 
1093.63 
2146. 
2025. 
1093.63 
12182. 
6076. 
8100. 
2250. 

3.609 
8238. 
1093.63 
2025. 
1166.7 

2.33 
2435. 
7432. 
4333. 
4638. 
6026.24 
1522. 
11660. 
8548. 
1S2S. 
ISO!). 
1900. 



Carara, Palmo, 9.6 ins. 



t 1.60931 kilometres=l mile. 



FOREIGN MEASURES AND WEIGHTS. 



31 



MEASURES OF SURFACE. 
French. Old System. 
1 square inch =1.13587 U. S. inches. 

1 toise =6.3946 U. S. feet. 

1 arpent (Paris) =900 square toises=4089 square yards. 
1 arpent (woodland)= 100 square royal perches =6 108.24 square yards. 
■New System. 
1 are — \ square decametre =1076.4309 square feet. 
= 100 square metres= 119.6033 square yards. 
1 decare=10 ares. | 1 hectare=100 ares — 2.4711 acres. 

1 square metre= 1550.0599 square inches, or 10.7643 sq. feet. 
1 centiare = 10 7643 square feet. | 1 deciare = 11.9603 square yards. 
Ta"ble of Ijengtlis of Foreign IVIeasixres of Surface. 



Place. 



Square 
Yards. 



Place. 



Measure. 



Square 
Yards. 



Amsterdam . 

Austria . 

Baden 



Berlin 

Bremen 

Brunswick . . . 
Canary Isles 4 . 

Ceylon 

Denmark 

Egypt 

England 

Geneva 

Hamburg 
Hanover 



Morgen. 
Joch . . . 
Viertel . 
Morgen. 



(small) . 



Fanegada 

Acre 

Skieppe 

Feddan al ris'h, 

Acre 

Arpent 

Scheffel 

Morgen 



9722. 
6884. 
1076.4 
4305.6 
3054. 
3070. 
2990. 
2420. 
4S40. 
329.75 
2674. 
4840. 
6179. 
5026.34 
3131.5 



Ionian Isles. 

Modena 

Naples 

Portugal . . . 

Prussia 

Rome 

Russia 

Spain 



Sweden 

Switzerland . 

Turin 

Tuscany 

Vienna 

Zurich 



Misura 

Biolca 

Moggi 

Geira 

Morgen 

Pezza 

Dessatina 

Fanegada (max) 

El Area . 

Tunnland . . 
Juchart (tillage) 
Giornata . . . 
Quadrato . . . 

Joch 

Juchart .... 



1445. 
3392. 
4165. 
7004. 
3054. 
3160. 
13067. 
7682. 

119.6 
5872. 

425.9 
4546 7 

407.2 
GSS4. 

425.9 



MEASURES OF VOLUME. 

British. The Imperial gallon measures 277.274 cubic inches, con- 
taining 10 lbs. avoirdupois of distilled water, weighed in 
air, at the temperature of 62°, the barometer at 30 inches. 
6.2355 gallons in a cubic foot. 
Imperial bushel =221 8. 192 cubic inches. 
* Heaped bushel— 19.5 inches diameter, cone 6 inches high 

= 2815.4872 cubic inches. 
For Grain — 8 bushels =1 quarter; 1 quarter =10.2694 

cubic feet. 
Coal, or heaped measure — 3 bushels = 1 sack ; 12 sacks = 

1 chaldron. 
1 chaldron =58.656 cubic feet, and weighs 3136 pounds. 
Fkench. Old System.— \ Boisseau = 13.01 litres = 793.963 cubic 
inches, or 3.437 gallons. 
1 pinte =0.931 litres, or 56.816 cubic inches. 
1 cubic inch, 1.06577 3 = 1.20157 U. S. inches. 

1 cubic foot = 2091.8667 U. S. inches. 

13.08516 hectolitres... = 1 chaldron. 

* When heaped in the form of a true cone. 



32 



FOREIGN MEASURES AND WEIGHTS. 



French. New System. — Decilitre — 6.1027 U. S. cubic inches. 

Litre == 1 cubic decimetre, or 61.0271 cubic inches= 
1.05675 U. S. quarts. 

Decalitre =610.271 cubic inches. 

Kilolitre = 35.3166 cubic feet. 

Docistere = 3.53166 cubic feet. 

Stere (a cubic metre) = 35.3166 cubic feet = 6 1027 0963 cubic in. 
Decastere = 353.166 " 

Note.— -For the Square and Cubic Measures of other countries, take the length of 
the measure in table, page 29-30, and square or cube it as required. 



Table of Volume of Foreign Liquid. Measures. 

Place. Measure. Cub. Inch Place. Measure. | Cub. Inch 



Amsterdam . 



Antwerp 

Arabia 

Austria 

Bavaria 

Berlin 

Bombay 

Brazil 

Bremen 

Brunswick 

Canary Isles* . 

Candia 

Ceylon 

China 

Cognac 

Cologne 

Constantinople 
Denmark 



Anker 

Wine Stekan. 

Stoop 

Gudda 

Mass 

Eimer 

Anker 

Parrah 

Medida 

Stiibehen 



Dresden . . 
Egypt.... 
Frankfort 
Florence. . 



Germany(Baden) 

Geneva 

Genoa .• . 



Gresce . . 
Hamburg 



Hanover. 
Havana . 



jArroba 

Mistate 

Parrah 

Tau 

Brandy velte. 

jViertel 

JAlmud 

Anker 

! Pot 

Eimer 

Ardeb 

Viertel 

Wine Barile . 
Oil " . 

Stutse 

Setier 

Wine Barile . 

Pinte 

Kila 

Stiibehen 

Ohm 



Holland* 

Hungary 

Ionian Isles 8 

Java 7 

Leghorn 



Leipsic . . . 
Lisbon. . . . 
Lucerne . . 

Madras . , . 



Stiibehen 

: Arroba 

j Wine Arroba , 
Kan 



tamer 

Dicotoli 

Kanne 

Oil Barile. . . 
Wine " ... 

Eimer 

Aliunde 

Ohm 

Marcal 



2331. 
1183.6 

168. 

554.5 
S6.3 
3914.8 
2285.7 
6721. 1 

165.5 

195.9 

227. 

949. 

631. 
1558.4 

332.7 
4454.6 

363.1 

319.4 
2299. 

5S.9 
4627.6 
1 858. 

437.53 
2781.8 
2225.6 

915.1 

2760. 

452S.6 

90.6 

2030.1 

220.9 
8836. 
94^0.6 

237.2 

947. 
2781. 

61.027 
3454.4 
34.6 

111. 
2225.6 
2T81.8 
4127.6 
1009.5 
3162.8 

749.8 



Madeira . 



Malaga . . . 

Malta 

Marseilles 
Milan ii... 

Mocha 

Modena. . . 
Nantes . . . 
Naples . . . 



Alquiere . 
Almude . . 
Arroba . . . 
Caffiso . . . 
Millerolle. 

Pinte 

Gudda . . . 

Fiasco . . . 

Wine Barique 

u Barile. 

Oil Stajo 

Kanna 



Norway . . . 

Oporto j Almude 

Persia Artaba 

Poland Garnice 

Prussia Anker 

Eimer 

Ohm 

Anker 

Wine Barile 
Oil " 
Boceale 

Rotterdam Ohm 

Russia Vedro 



Riga . 
Rome 



Sardinia Barile 

Saxony Eimer 

Slam Sesti 

Sicily |Oil Caffiso... 

u jSalma (Mes'a) 

Smyrna Almud 

Spain Arroba 

" Oil Arroba... 

u Quartillos 

Sweden Kanna 

Syria j Almud 

Switzerland ! Eimer (Berne) 

Tripoli Barile 

Trieste jEimer 

Turkey | Almud . 



Tunis Oil Barile 

Tuscany < u * 

" Fiasco 

Venice 1 * Pinta 

Vienna iEimer 

" jMass 

Ziiricli I " 



504.69 
1009.38 

965.3 
1270. 
3922.4 
61.03 

5545 

127. 
14638.9 
2544. 

617.6 
1276.5 
1530.71 
4013. 

97.05 
2096. 
4192. 
S3S4. 
2387.33 
C560. 
3506.8 

111.2 
9236.3 

750.1 
4528.6 
4627.6 

739.4 

66?. 
523\4 

S19.4 

980.7 

770.6 
SO. 65 

15S. 

319.4 
2547.4 
3956. 
3452.6 

319.4 
1157. 
2225. G 

139.1 

61.3 

3454. 

86.3 

99.83 



Notb.— In Bengal and Calcutta the measures are by weight. 



FOREIGN MEASURES AND WEIGHTS. 



33 



Plaee. 



Table of Foreign Dry Measures. 

Measure. Cubic Inch. Place. Measure. 



Cubic In. 



Abyssinia.. . 

Africa 

Alexandria. . 

Austria 

Algiers 

Amsterdam . 

Asia 

Azores 



Ardeb . 



Barbary 

Bavaria 

Brazil 3 

Brunswick 

Belgium i 

Berlin 

Bombay 

Bremen 

Cadiz 

Canadaia 

Candia 

China 

Constantinople 

Corsica 

Dresden 

Dantzic 

Denmark 

Egypt 

Florence 

Frankfort 

Geneva 

Genoa 

Greece 

Germany 

Holland^ 

Hanover 

Hamburg 

Ionian Isles 8 . . 

Leghorn 

Lisbon 



Rebele . . . 
Metze . . . 

Zarni 

Mudde. . . 

Sesti 

Alquiere . 

Sack 

Temer . . . 
Scheffel . . 
Alquiere . 

Himt 

Litron . . . 
Scheffel.. 
Parah . . . 
Scheffel.. 
Fanega . . 
Minot . . . 
Carga . . . 

Tau 

Killow... 

Stajo 

Scheffel . . 



277. 

277. 
9592.2 
3T52.7 
1220.5 
6786. 

739.39 

731. 
4947. 
1637.7 
13569. 
2240. 
1897.9 

61.027 
31 SO. 
6721.12 
4520. 
3438.8 
2381.5 



Lisbon 

Leipsic 

Madeira .... 

Malaga 

Modena 

Malta 

Milan" . 

Majorca .... 

Madras 

Norway 

Naples 

Netherlands . 

Oporto 

Persia ...... 

Poland ..... 

Prussia 

Parma 

Rome 



Riga 

Rotterdam . . . 

Russia 

Sardinia 

Spain 



443. 

2023. 

6014. 

6340.3 

3354. 

Tonne 8487.6 Sicily. 

Ardeb 10869.2 

Stajo 14S7. 1 Smyrna . 

Malta 6902.4 Sweden 

Coupe 4739. Siam 

Miria 7366.6 Saxony 

Kila 2030. Scotland 

Malter (Baden) 9154. Switzerland. 

Kop ; 61.027 Tripoli. 

Himt 1897.9 Tuscany 

Scheffel 6429.5 Turkey . 

Chilo 2218.2 Zurich. 

Stajo 1487. 1 Venice^ » 

Alquiere 825.2 Vienna , 

Notes In Arabia the Tomand measures 168 lbs. avoirdupois of rice. 

In Bengal and Calcutta the measures are by weight. 



Fanega 

Scheffel 

Alquiere 

Fanega 

Sacco 

Salma 

Soma 

Quarten 

Marcal 

Spann 

Tomolo 

Mudde 

Alquiere 

Artaba 

Zorrec 

Scheffel 

Stajo 

Rubbio 

Quarta 

Loop 

Saik 

Tschetwerik . . 

Mina 

Cahiz 

Fanega 

Salma, gros . . . 
lc general. 

Killow 

Tunna 

Sesti 

Scheffel 

Firlot 

Maas (Berne) . . 

Temen 

Stajo 

Killoio 

Mutt 

Soma 

Metzen 



3300. 

6340.3 

684. 

343S.8 

8597.7 
17676.8 

6103. 

4296. S 
749 8 

4469.6 

3122. 

6103. 

1041.7 

4013. 

3120. 

3354. 

3134.7 
17968.3 

4492.1 

8978. 

6361. 

1600. 

7366.6 
41266. 

343S.8 
21010. 
16924.8 

2023. 

8940. 
739.4 

6340.3 

2197. 
854.9 

1637.7 

1487.1 

2023. 

4:^8. 

6103. 
3753. 



MEASURES OF WEIGHT. 

British. 1 troy grain = .003961 cubic inches of distilled water. 
1 troy pound =22.815689 cubic inches of water. 
1 avoir, drachm =27.34375 troy grains. 



1 clove = 7 pounds. 
1 sack wool=364 u 



1 truss straw=36 pounds. 
1 sack flour =28.2 " 



1 grain 
1 gross 



1 quarter flour=4 pounds 5 oz. S.25 dr. 
French. Old System. 
— 0.8188 grains troy. I 1 once =1.0780 oz. avoirdupois. 



, =58.9548 



1 livre 
New System. 



-. 1.0780 lbs. 



Milligramme = .01543 troy gr's. 
Centigramme = .15433 " 
Decigramme =1.54331 " 



Gramme = 1 5. 433 1 6 tr. gr, 

Decagramme. = 154.33159 " 
Hectogramme= 1543.3159 " 



34 



FOREIGN MEASURES AND WEIGHTS. 



1 kilogramme = 2.204737 lbs. avoirdupois. 

1 myriagramme =22.04737 " 

1 millier = 1000 kilogrammesr=l ton sea weight. 

453.5688 grammes^. 4535688 kilogramme = 1 pound avoirdupois. 
372.2223 " =.3732223 " =1 pound troy 

Table ofValue of Foreign Weights. 



Place. 


Weight. 


Pounds 
Avoirdu- 
pois. 


Place. 


Weight. 


Pounds 
Avoirdu- 
pois. 




Liter 


.6857 

.6S57 

.0571 
16.974 
2.S29 
1.003 
2.2 

.93 
1.19 
3. 

2 553 
2.S43 
1.2S5 

.6195 
1.09 
1.3333 
1.2343 
2.2047 
1.8667 
2.0533 
74.667 
82.123 

.311 
3.3333 

.7 

.5533 
1.0986 
1.2531 
1.03 
1.00S 
1.8667 
2.0533 
82.123 
74.667 
1.014S 
1.1650 
500. 

.0S33 
1.3333 

.5156 
2.8 

2.2047 
5.2439 

2.82ro 

1.1029 
1.0309 
.0257 
1.008 

.74S6 
1.0314 
1 2143 


Germany 

Greece 

Guinea 

Hamburg 

Hanover 

Hollands 

Japan 7 

Java 


Unze , 


0657 


Africa 


Rottoli 

Wakea 


Pound 

Benda 

Pfund 


.8S11 


ii 


1417 


Aleppo 

it 


Batman 

Oke 


1.06S5 


u 


1 0731 


Amsterdam . . . 

u 


Pound (old) 

" (Flem.) . 
Rottoli 

Maund 


Pouden 

Catty 


2.2057 
1 3 


Alexandria .... 


Catty 


1.3333 


Algiers 

Arabia 


Leghorn : 

Leipsic 

Madeira 

Madras 

Malta 

Milani i 

Mocha 

Modena 

Morea 

Morocco 

u 

Munich 

Naples 

Norway 


Libbra 

Pfund (comm'n) 


.74S6 
1.0309 




.jatty 


j 0119 


" (Ottoman). 


Oke 


Vis 


3 125 


Pfund 


Rottoli 

Libbra 

Maund 

Libbra 

Pound 


1 333 


u 


Mark 


2 2046 


B:\rbary 


Rottoli 

Catty 


3. 

.7046 




Pfund 


1.1014 


Belgium 1 

Bengal 




Pound (conmi'l) 

" (market) 

Pfund 


1 19 


Seer (Factory) . 


1.785 

1 ^-66 


u 

u 
B-rlin. 


Maund (Fact'ry) 

u 

Pfund ...... .' .' . 


Rottoli 

Rotolo (piccolo). 

Skalpund 

Mark 


1.9643 

LOOT 




Vis 


" 

Parma 

Persia 

u 

j Portugal 

P: ussia 

j Rome 

j Rotterdam . . . 

Russia 

Sardinia 

Saxony 

SMraz 

1 Siam 


465 


Bombay 


Seer 


Libbra 

Rattel (shirez) . 

Dirhem 

Found 

Pfund , 


.7197 


Mark 


2.1136 




Pfund 


0214 


Bologna 


Pound 

Pfund 


1.0119 
1.0311 


Cairo 


Rottoli 

Seer (Factory). . 


Libbra 

Funt 


.7477 


Calcutta 


1.0895 
90 9 6 


u 

U 


Maund (Fact'ry) 
Libra 


Rottolo 

Pfund 


1.0483 
1.0309 


Canary Isles 4 


Batman 

Catty 


12 6816 


Rottoli 

Tael or ounce . . 
Catty 


2.5S3 


Ceylon 


Sicily 

! " 

Smyrna 

Spaing 

1 Sumatra 

Sweden 

Switzerland . . 

Tripoli 

Tunis 

Turin 

Turkey 

it 
u 


Libbra 

Rottolo (grosso). 
Oke 


.7 
•1.925 


41 


2.S29 


Cologne 

Constantinople. 


Mark 


Cantaro 

Libra 


127.3 


Oke 


1.0164 


Kilogramme . . . 

Rottoli 

Oke 


Catty 


1.333 




Skalpund 

Pfund 


.9376 


Damascus 


1.1514 






Rottol 


1.097 


Dresden 


Pfund 




1.11 


East Indies 


Sicca or Tola . . 
Rotl 


Libbra 

Rottolo 

Almud (oil) 

Oke 


.813 
1.2729 


Florence 


Libbra 

Pfund 


22.6S5 

2 8286 




Pfund 


Libbra 

Pound 

Pfund 


74S6 


(.. npa 

Germany (Bad.) 
" < " ) 


Rottolo 

Mark 


1.04S3|j Venice i» 

.5155 Vienna 

1.1029 [1 Warsaw 


2.2046 
1.235 


Pfund 


" 


.8909 






FOREIGN MEASURES AND WEIGHTS. 35 

!N"otes to tlie preceding Tables. 

1 . The measures and weights of Belgium are the same as those of France and Holland. 

2. The measures and weights of Bohemia are the same as those of Austria. 

3. The measures and weights of Brazil, with some additions, are the same as those 
of Portugal. 

4 The measures and weights of the Canary Isles are the same as those of Spain, 
with some variations. 

5. The measures and weights of Gibraltar are the same as those of England. 

6. The measures and weights of Holland are the same as those of France and Bel- 
gium. 

7. The weights of Japan are nearly the same as those of China. 

8. Since 1817 the measures and weights of the Ionian Isles are the same as those 
of England, with Italian designations. 

9. The measures and weights of Mauritius are the same as those of England and 
France. 

10. The measures and weights of Mexico are the same as those of Spain, with some 
additions difficult to obtain. 

11. The measures and weights of Milan and Venice are the same, and are those of 
France. 

12. The measures and weights of Canada, and all the British Possessions in North 
America, are the same as those of Great Britain, but the U. S. gallon and bashel are 
most in use. 



SCRIPTURE AND ANCIENT MEASURES. 



Scripture Long Measures. 

Inches. 

Digit = 0.912 

Palm.; = 3.648 

Span =10.944 



Feet. Inches. 

Cubit = 1 9.888 

Fathom = 7 3.552 



Egyptian Long Measures. 

Feet. Inches. I Feet. Inches. 

Nahud cubit 1 5.71 | Royal cubit 1 8.66 

" Grecian Long Measures. 

Feet. Inches. 



Feet. Inches. 

Stadium = 604 4.o 

Mile =4835 



Digit = 0.7554 

Pous (foot) = 1 0.0875 

Cubit = 1 1.5984| 

Attic or Olympic foot = 12.108 inches. 

Pythic or natural foot = 9.768 " 

Ancient Greek (16 Egyptian fingers) 11.81 " 

Keramion or Metretes". .. 8.488 gallons, 

Jewish Long Measures. 

Feet. 



Cubit = 1.824 

Sabbath day's journey =3648. 



Mile (4000 cubits) .... =7296 

Day's journey 33.164 miles. 



Roman Long Measures. 



Inches. 

Digit = .72575 

Uncia(inch).... = .967 
Pes (foot) =11.604 



Feet. Inches. 

Cubit = 1 5.406 

Passus ± 4 10.02 

Mile (millarium) = 4842 



Roman Weight. 

Pounds. 

Ancient libra 7094 



3G 



FOREIGN MEASURES AND WEIGHTS. 



( 8.2* 
t 9. It 



Ancient 

Troy grains. 

Attic obolus 

(51.9* 

" drachma -^54.6f 

(69. X 

Lesser mina 3.892 

Greater mina ^ of drachma. 

Talent =60 minse =56 lbs. avoir- 
dupois. Troy graing . 
Drachm =146.5 



Weigh, ts. 

Troy grains 

8.326* 
8.985* 
9.992* 
( 51.9* 
1 62.5f 
54. t 
(415.1* 

Ounce. -<437.2t 

(431. 2$ 
Pound 12 Roman ounces. 



Egyptian mina 

Ptolemaic " 

Alexandrian " 

Denarius (Roman) .. 

" (Nero) .... 



Arabian foot =1.095 

Babylonian foot =1.140 

Egyptian finger = .0614. 



Miscellaneous. 

Feet. 

Hebrew foot =1.212 

" cubit =1.817 

" sacred cubit =2.00? 



GEOGRAPHICAL MEASURES AND DISTANCES. 
To Reduce Longitude into Time. 

Rule. — Multiply the degrees, minutes, and seconds by 4, and the 

product is the time. 
Example. — Required the time corresponding to 50° 31'. 

50° 31' 
4 

3/i. 22' 4". 

To Fted/uce Time into Longitude. 
Rule. — Reduce the hours to minutes and seconds, divide by 4, and 
the quotient is the longitude. 
Or, multiply them by 15. 

Example. — Required the longitude corresponding to 5h. S' 11.2". 

h. m. i. m. t. 

5 8 11.2. = 308 11.2'", whichH-4rrT7° 2' 45.5". 
Or, multiplying by 15 : 

5h. 8m. 11.2s. Xl5 = 77° 2' 45.5". 

Table of" Departures for a Distance rnn of 1 Mile. 

Course. | Departure. Course. | Departure. | Course. Departure. 



3.5 points. 
4. 



.773 
.707 



4.5 points. 
5. " 



.634 
.556 



5.5 points. 
6. 



.471 
.383 



Thus, if a vessel holds a course of 4 points, that is, without leeway, for the distance 
of 1 mile, she will make .707 of a mile to windward. 

Or, a vessel sailing E.N.E. upon a course of 6 points for 100 miles will make 3S.3 
mile.? (100X.3S3) longitude. 



Christian!. 



t Arbuthnot. 



t Paucton. 



GEOGRAPHICAL MEASURES AND DISTANCES. 



37 



Table showing the Degrees, Minutes, and. Seconds 
of each Point of the Mariner's Compass with the 
Meridian. 



NOKTH. 


South. 


Points. 


O ' // 


Sin. A.* | 


Cos. A.* 


Tan A.* 


N, 


* S 


.25 

.5 

.75 


2 48 45 
5 37 30 
8 26 16 


.0489 

.0S8 

.1467 


.9988 
.9152 
.9891 


.0491 
.09P5 




.1484 


N. by E 

N. by W. ... 


S.E. byE. .. ) 
S. byW j 


1. 

1.25 

1.5 

1.75 


11 15 
14 3 45 
16 52 30 
19 41 15 


.195 
.2429 
.2903 
.3368 


.9808 
.97 
.9569 
.9415 


.1989 
.2504 
.3004 

.3578 


N.N.E 

N.N.W 


S.S.F ( 

S.S.W \ 


2. 

2.25 
2.5 
2.75 


22 30 
25 18 45 
28 7 30 
30 56 15 


.3827 
.4275 
.4714 
.5141 


.9239 
.90 

.8819 

.8577 


.4142 

.4729 
.5345 
.5994 


N.E. by N... 
N.W.byN.. 


S.E. byS. .. ( 
S.W. byS. .A 


3. 

3.25 
3.5 
3.75 


33 45 

36 33 45 
39 22 30 
42 11 15 


.5556 
.5957 
.6344 
.6715 


.8315 

.8032 

.773 

.7409 


.6682 
.7410 
.8207 
.9063 


N.E 

N.W 


S.E \ 

S.W \ 


4. 

4.25 

4.5 

4.75 


45 

47 48 45 
50 37 30 
53 26 15 


.7071 
.7404 
.773 

.8032 


.7071 
.6715 
.6344 

.5957 


1. 

1.103 
1.218 
1.34S 


N.E. by E... 
N.W. by W.. 


S.E. byE. .. ( 
S.W. by W.. 1 


5. 

5.25 
5.5 
5.75 


5G 15 
59 3 45 
61 52 30 
64 4i 15 


.8315 

.S577 
.8819 
.904 


.5556 
.5141 
.4714 
.4275 


1.497 
1.668 
1.871 
2.114 


E.N.E 

W.N.W 


E.S.E ) 

W.S.W \ 


6. 
, 6.25 
6.5 
6.75 


67 30 
70 18 45 

73 7 30 
75 56 16 


.9239 
.9415 
.9569 

.97 


.3827 
.33G8 
.2903 
.2429 


2.414 
2.795 

3.296 
3.941 


RbyN 

W. by N. . . . 


E. by S ) 

W. byS i 


7. 

7.25 

7.5 

7.75 


78 45 
81 33 45 
. 84 22 30 
87 11 15 


.9808 
.9891 
.9952 

.9988 


.lf5 
.1467 

.C98 
.0489 


5.027 

6.741 

10.153 

20,555 


East or West 


East or West 


8. 


90 


1. 


.0000 


CO 



* A, representing course or points from the meridian. 

Table of the Visible Distance of Objects in Statute 

Miles. 



Height in 


Distance in 


Height in 


Distance in 


Height in 


Distance in 


Height in 


Distance in 


Feet. 


Miles. 


Feet. 


Miles. 


Feet. 


Miles. 


Feet. 


Miles. 


*.582 


1. 


11 


4.36 


30 


7.18 


150 


16.05 


1 


1.31 


12 


4.54 


35 


7.76 


200 


18.54 


2 


1.85 


13 


4.71 


40 


8.3 


300 


22.7 


3 


2.27 


14 


4.9 


45 


8.8 


400 


26.2 


4 


2.62 


15 


5.07 


50 


9.37 


500 


29.3 


5 


2.93 


16 


5.24 


55 


9.72 


1000 


41.45 


6 


3.21 


17 


5.4 


60 


10.14 


2000 


58.61 


7 


3.47 


18 


5.56 


70 


10.97 


3000 


71.79 


8 


3.7 


19 


5.72 


80 


11.72 


4000 


82.9 


9 


3.93 


20 


5.86 


90 


12.43 


5000 


92.68 


10 


4.15 


25 


6.55 


100 


13.1 


1 mile. 


95.23 



* For a Statute mile the cunrature = 6.99 inches. 



38 



GEOGRAPHICAL MEASURES AND DISTANCES. 



and if for ICO miles, 



The difference in two levels is as the square of their distance. 

Illustration. — If the height is required for 2 mile3, 
12 : 22 : : 6.99: 2T.96 inches; 

12 : 1002 : : 6.93 : : 1.103+ write*. 

The difference in two distances is as the square root of their heights* 
Illustration. — If the distance is required for 3 feet, 

V.5S2=.T63 : 1 : ; ^3=1-732 : 2.2T miles; 
and if for 8 feet, 

V.582=.T63 : 1 : : VS— 2.82S : 3.70 miles. 

Ta"ble of the Visible Distance of Objects in. Geo- 
graphical or Nautical Miles. 



Height in 


Distance in 


Height in 


Distance in 


Height in 


Distance in 


Heijrht in 


Distance in 


Feet. 


Miles 


Feet. 


Miles. 


Feet. 


Miles 


Feet. 


Miles. 


* .663 


1. 


11 


4.08 


30 


6.74 


150 


15.07 


1. 


1.23 


12 


4.26 


35 


7.28 


200 


17.4 


2. 


1.74 


13 


4.43 


40 


7.78 


300 


21.32 


3. 


2.13 


14 


4.6 


45 


8.25 


400 


24.64 


4. 


2.46 


15 


4.77 


50 


8.7 


500 


27.52 


5. 


2.75 


16 


4.92 


55 


9.13 


1000 


38.92 


6. 


3.01 


17 


5.07 


60 


9.53 


2000 


55.04 


7. 


3.25 


18 


5.22 


70 


10.29 


3000 


67.41 


8. 


3.48 


19 


5.36 


80 


11.01 


4000 


77.84 


9. 


3.69 


20 


5.5 


90 


11.68 


5000 


87.03 


10. 


3.89 


25 


6.15 


100 


12.31 


1 mile. 


89.43 



* For a Geographical or Nautical mile, the curvature = 7.C62 inches. 

Illustration. — If a man at the foretop-gallant mast-head of a ship, 100 feet from 
the water, sees another and a large ship u hull to," how far are the ships apart ? 

A large ship's bulwarks are at least 20 feet from the water. 

Then, by table, 100 feet =12.31 

20 " = 5.50 

Distance 17.81 miles. 

Note The .0763 pait should be added for horizontal refraction. 

When an observation for distance is taken from an elevation, as from 
a light-house or a vessel's mast, of an object that intervenes between 
the observer and the horizon, or contrariwise, the observer being at a 
horizon to the elevated object ; the distance of the observer from the 
intervening object can be determined by ascertaining or estimating its 
distance from the horizon or elevation, as the case may be, and sub- 
tracting it from the whole distance between the observer and the point 
from which the observation is taken, and the remainder will give the 
distance of the object from the observer. 

In this case, however, the distance of the intervening object can not 
be computed unless the height of it is known or may be estimated. 

Illustration. — The top of the smoke-pipe of a steamer, assumed to be 50 feet above 
the surface of the water, is in range with the horizon from an elevation of 100 feet; 
what ia the distance to the steamer ? 

100 feet =12.31 

50 " = S.iTO 

3.01 miles. 



GEOGKAPHIQAL DISTANCES AND SOUNDING. 



39 



Lengths of* a Degree of Longitude on the parallels 
of* Latitnde, for each Degree of JLatitnde from the 
Equator to the Pole. 



Lat. 


Miles. 


| Lat. 


Miles. 


| Lat. 


Miles. 


Lat 


Miles 


Lat. 


Mile's. 


1° 


59.99 


19° 


56.73 


37° 


47.92 


55° 


34.41 


73° 


17.54 


2 


59.96 


20 


56.38 


38 


47.28 


56 


33.45 


74 


1654 


3 


59.92 


21 


56.01 


39 


46.63 


57 


32.68 


75 


15.53 


4 


59. S5 


22 


55.63 


40 


45.96 


58 


31.79 


76 


14.52 


5 


59.TT 


23 


55.23 


41 


45.28 


59 


30.9 


77 


13.5 


6 


59.67 


24 


54.81 


42 


44.59 


60 


30. 


78 


12.4S 


7 


59.55 


25 


54.38 


43 


43.88 


61 


29.09 


79 


11.45 


8 


59.42 


26 


53.93 


44 


43.16 


62 


28.17 


80 


10.42 


9 


59.26 


27 


53.46 


45 


42.43 


63 


27.74 


SI 


9.38 


10 


59.09 


28 


52.97 


46 


41.68 


64 


26 3 


82 


8.35 


11 


5S.89 


29 


52 48 


47 


4\92 


65 


25.36 


83 


7.31 


12 


58.69 


30 


51.S6 


48 


40.15 


QQ 


24.4 


84 


6.27 


13 


58.43 


31 


51.43 


49 


39.36 


67 


23.41 


85 


5.23 


14 


58.22 


32 


50.88 


50 


38.57 


68 


22.48 


86 


4.18 


15 


57.95 


33 


50.32 


51 


37.76 


69 


21.5 


87 


3.14 


16 


57.67 


34 


49.74 


52 


36.94 


70 


20.52 


SS 


2. 


17 


57.38 


85 


49.15 


53 


36.11 


71 


19.53 


89 


1.05 


18 


57.06 


36 


48.54 


54 


35.27 


72 


IS. 54 


90 


.00 



Note. — Degrees of longitude are to each other in length as the Cosines of their lat- 
itudes. 



SOUNDING. 
To Reduce a Sounding to Low Water. 



h/ 180f\ ., 

-^l=FCOS.— J=A; 



h representing vertical rise of tide, and h' sounding or depth at low water, 
in feet ; t time between high and low water, and t' time from time of 
sounding to low water, in hours. 

— cos. when <90°, and -f cos. when >90°. 

Example Low water occurring at 3.45, and high water at 10.15 P.M., a sounding 

taken at 5.30 P.M. was 18.25 feet; what was the depth at low water, the vertical 
rise being 10 feet ? 

h = 1 feet ; t'— 5h. 30m. — 3/>. 45m. == 1 h. 45m. = 1.75 hours, 

t = lOh. 15m. — 3/*. 45m. = 6/*. 30m. ==: 6.5 hours. 

Then ~ (l if cos. ^y^) = 5(1- 4S° 27' 24")= 5x (1 —.633186) = 1.6S407/**, 

Sounding 18.25 feet. 

Reduction 1.68407 u 

IGMSmfeet* 



40 VULGAR FRACTIONS. 



VULGAR FRACTIONS. 

A Fraction, or broken number, is one or more parts of a Unit. 

Illustration. — 12 inches are 1 foot. 

Here, 1 foot is the unit, and 12 inches its parts ; 3 inches, therefore, are the one 
fourth of a foot, for 3 is the quarter or fourth of 12. 

A Vulgar Fraction is a fraction expressed by two numbers placed one above the 
other, with a line between them ; as, 50 cents is the -^- of a dollar. 

The upper number is termed the Sumerator, because it shows the number of parts 
used. 

The lower number i3 termed the Denominator, because it denominates, or gives 
name to the fraction. 

The Terms of a fraction express both numerator and denominator; as, 6 and 9 are 
the terms of J. 

A Proper fraction has the numerator equal to, or less than the denominator; as, J, 
etc. 

An Improper fraction is the reverse of a proper one ; as, ^, etc. 

A Mixed fraction is a compound of a whole number and a fraction ; as, 5^, etc 

A Compound fraction is the fraction of a fraction ; as, J of J, etc. 

AComplex fraction is one that has a fraction for its numerator or denominator, or 

both ; as, ]L-> or JLi °r — , or ?!, etc 

6 4 | 6 

Note. — A Fraction denotes division, and its value is equal to the quotient obtained 
by dividing the numerator by the denominator; thus, ^ is equal to 3, and -^ is 
equal to 4i. 



REDUCTION" OF VULGAR FRACTIONS. 

To Ascertain the greatest Number that ^v^ill clivicie Two 

or more Numbers A^itlioixt a Remainder. 

Rule.— Divide the greater number by the less ; then divide the divisor by the re- 
mainder; and so on, dividing always the last divisor by the last remainder, until 
there is no remainder, and the last divisor is the greatest common measure required. 

ExAiirLE. — What is the greatest common measure of 1903 and 936? 
£30) 190S (2 

1872 

36) 936 (26 
72 

216. Hence 36 = greatest common measure. 

To Ascertain tlie least Common. IMXiltiple of Two or more 

Numbers. 

Rft.e.— Divide the given numbers by any number that will divide the greatest 
number of them without a remainder, and set the quotients with the undivided num- 
bers in a line beneath. 

Divide the second line as before, and so on, until there are no two numbers that 
can be divided; then the continued product of the divisors and last quotients will 
give the multiple required. 

Examine. — What is the least common multiple of 40, 50, and 25 ? 
5) 40 . 50 . 25 



5) 8 . 10 . 5 
2) 8 . 2 . 1 



1. Then 5x5x2x4x1x1=200. 



VULGAR FRACTIONS. 41 

To Reduce Fractions to tlieir lowest Terms. 
Rule— Divide the terms by any number or series of numbers that will divide 
them without a remainder, or by their greatest common measure. 
Example.— Reduce J|g of a foot to its lowest terms. 

72 ^_ 10 — £2_ -j- 8 = J? s -4- 3 = |, or 9 inches. 

960 96 18 4' 

To Redrice a Mixed Fraction to its Equivalent, an Im- 
proper Fraction. 

Note Mixed and Improper fractions are the same : thus, 5|— y>. 

RUT.E Multiply the whole number by the denominator of the fraction and to the 

product add the numerator ; then set that sum above the denominator. 

Example.— Reduce 23§ to a fraction. 

23x6+2 140 
6 ~~ 6 

Ex. 2 Reduce ^^ inches to it3 value in feet. 

123 -=- 6 = 20 J ; that is, 1 foot 8=L inches. 

To Rednce a Whole Number to an Equivalent Fraction 
having a given Denominator, 

Rule. Multiply the whole number by the given denominator, and set the product 

over the said denominator. 
Example. — Reduce S to a fraction the denominator of which shall be 9. 
8 X 9 = 72 ; th.n -^ the result. 

To Reduce a Compound Fraction to an Equivalent Sim- 
ple one. 

Rule. Multiply all the numerators together for a numerator, and all the denom- 
inators together for a denominator. 

Note. When there are terms that are common, they may be omitted. 

Example.— Reduce \ of \ of J to a simple fraction. 

\ x "I x I" — A ~ i' 0r ' i x f x f =f &> hy cancelin 9 * he 2 's and 3's. 

Ex. 2. — Reduce ^ of ^ of a pound to a simple fraction. 



To Reduce Fractions of different Denominations to Equiv- 
alent ones having a Common Denominator. 

Rule. — Multiply each numerator by all the denominators except its own for the 
new numerators : and multiply all the denominators together for a common denom- 
inator. 

Note.— In this, as in all other operations, whole numbers, mixed, or compound 
fractions, must first be reduced to the form of simple fractions. 

2. When many of the denominators are the same, or are multiples of each other, 
ascertain the least common multiple of the denominators, and then multiply the 
terms of each fraction by the quotient of the least common multiple divided by ita 
denominator. 



Example. — Reduce J, g, and J to a common denominator. 
1X3X4-12) 

2X2X4=16 >=Ji=£|=Ji. 
3X2X3 = 18 ) 

2X3X4 = 24 
The operation may be performed mentally thus: 
Reduce J, j|, §, and j| to a common denominator. 

i=V- 4=S- 8=1- 

D* 



42 VULGAR FRACTIONS. 

To Reduce Complex Fractions to Simple ones. 

Rule.— Reducs the two parts both to simple fractions; then multfply the numera. 
tor of each by the denominator of the other. 

Example. — Simplify the complex fraction — — . 

4 

2|= f 8x 5 = 40_5 

4 |_^k 3x24=: 72 — 9* 



ADDITION OF VULGAR FRACTIONS. 

Rule.— If the fractions have a common denominator, add all the numerators to- 
gether, and place the sum over the denominators. 

Note. — If the fractions have not a common denominator, they must be reduced to 
one. Also, compound and complex must be reduced to simple fractions. 

Example. — Add i and J together. 
Ex. 2.— Add J of J of T % to 2 J of |. 

2 * 4 X 10 — 80" 

91 n f3_17 v 3_51 Then 18i 51_1 13i 

-8 01 4 — ~8" X 5 — 32* Xn ' 80 + 32— L 160* 



SUBTRACTION OF VULGAR FRACTIONS. 

Rule. — Prepare the fractions the same as for other operations, when necessary; 
then subtract the one numerator from the other, and set the remainder over the com- 
mon denominator. 

Example. — What is the difference between ^ and ^? 

5_1_4 
q <= 6 G_ 6 ' 

Ex. 2.— Subtract j from J. 

6X9=54) 

3 X S = 24 V _ 54 _ 24 _ 3 

n — ( — 72 T2— 72* 

8X9 = 1±> 



MULTIPLICATION OF VULGAR FRACTIONS. 

Rule. — Prepare the fractions as previously required; multiply all the numerators 
together for a new numerator, and all the denominators together for a new denomin- 
ator. 

Example. — What is the product of J and | ? 

4 x 9 — 3 (J — 4* 

Ex. 2 What is the product of 6 and | of 5? 

^Xf of 5 = « x™ = %°- = 20. 



DIVISION OF VULGAR FRACTIONS. 

Rule. — Prepare the fractions as before; then divide the numerator by the numer- 
ator, and the denominator by the denominator, if they will exactly divide; but if not, 
invert the terms of the divisor, and multiply the dividend by it, as in multiplication. 

Example.— Divide ^ 5 by |. 

^ • 3 — 3 — *3 * 



DECIMAL FRACTIONS. 43 



APPLICATION OF REDUCTION OF VULGAR FRACTIONS. 

To Ascertain the Value of a Fraction in Farts of a whole 

Number. 

Rule.— Multiply the whole number by the numerator, and divide by the denomin- 
ator ; then, if any thing remains, multiply it by the parts in the next inferior denom- 
ination, and divide by the denominator, as before, and so on as far as necessary ; so 
■hall the quotients placed in order be the value of the fraction required. 

Example. — What is the value of J of J of 9 ? 

1 of 2-2 an d 2 v 9 — 18 — 2 

Ex. 2. — Reduce j| of a pound to an avoirdupois ounce. 
3 
1 
4) 3 (0 lbs. 

16 ounces in a lb. 
4) 48 (12 ounces. 
Ex. 2 Reduce ^ of a day to hours. 

_3_ w 24 !LJ? — frJL hni/m 

To Reduce a Fraction from one Denomination to anotlier. 

Rule. — Multiply the number of the required denomination contained in the given 
denomination by the numerator if the reduction is to be to a less name, but by the 
denominator if to a greater. 

Example.— Reduce J of a dollar to the fraction of a cent. 

|xioo= ^!^¥- 

Ex. 2. — Reduce i of an avoirdupois pound to the fraction of an ounce. 

l X lG = l|- = f = 2|. 
Ex. 3.— Reduce ^ of a cwt. to the fraction of a lb. 

fx4x2S=^ 4t = 3_ 2 _. 

Ex. 4. — Reduce g of £ of a mile to the fraction of a foot. 

"I of I = & X5280 = £2-^° = 26_4fi 
Ex. 5. — Reduce 1 of a square inch to the fraction of a square yard. 



l v120fi — 1— — 2592 

For Rule of Three in Vulgar Fractions, see page 46. 



DECIMAL FRACTIONS. 



A Dectmal Fraction is that which has for its denominator a unit (1), with aa 
many ciphers annexed as the numerator has places ; it is usually expressed by set- 
ting down the numerator only, with a point on the left of it. Thus, ^ is .4; ^p^ 
is .85; i o°o^% is .0075 ; and xoVcfo o is - 00125 - When there is a deficiency of figures 
in the numerator, prefix ciphers to make up as many places as there are ciphers in 
the denominator. 

Mixed numbers consist of a whole number and a fraction ; as, 3.25, which is the 
same as 3. T %%, or Jfjg . 

Ciphers on the right hand make no alteration in their value; for .4, .40, .400 are 
decimals of the same value, each being ^j, or ^. 



44 DECIMAL FRACTIONS. 

ADDITION OF DECIMALS. 

Rule. — Set the numbers under each other according to the value of their places, as 
in whole numbers, in which position the decimal points will stand directly under each 
other; then begin at the right hand, add up all the columns of numbers as in inte- 
gers, and place the point directly below all the other points. 
Example.— Add together 25.125 and 293.7325. 
25.125 
293.7325 



31S.8575 sum. 






SUBTRACTION OF DECIMALS. 

Rule Place the numbers under each other as in addition ; then subtract as in 

whole numbers, and point off the decimals as in the last rule. 

Example. Subtract 15.15 from S9.175D. 

S9.1759 

15.15 

74.0259 remainder. 



MULTIPLICATION OF DECIMALS. 

Rule. — Place the factors, and multiply them together the same as if they were 
whole numbers ; then point off in the product just as many places of decimals aa 
there are decimals in both the factors. But if there are not so many figures in the 
product, supply the deficiency by prefixing ciphers. 
Example.— Multiply 1.56 by. 75. 

1.56 
.75 
730 
1092 
1.1700 product. 

BY CONTRACTION. 

To Contract the Operation so as to retain only as many 
Decimal places ill the ^Product as may Toe thought nec- 
essary. 

Rule. — Set the unit's place of the multiplier under the figure of the multiplicand, 
the place of which is the same as is to be retained for the last in the product, and dis- 
pone of the rest of the figures in the contrary order to what they are usually placed 
in ; then, in multiplying, reject all the figures that are more to the right hand than 
each multiplying figure, and set down the products, so that their right-hand figures 
may fall in a column directly below each other, and increase the first figure in every 
line with what Avould have arisen from the figures omitted; thus, add 1 for every 
result from 5 to 14, 2 from 15 to 24, 3 from 25 'to 34, 4 from 35 to 44, etc., etc., and 
the sum of all the lines will be the product as required. 

Example.— Multiply 13.57493 by 46.20517, and retain only four places of decimals 
in the product. 

13.574 03 
71 502.64 



54 2«9 72 




8 144 96 4- 2 foi 


IS 


27150 + 2 " 


18 


6 79-f4 " 


35 


144-1 » 


5 


94-2 " 


01 



627.23 20 

None.— When the exact result is required, increase the last figure with what would 
have arisen from all the figures omitted. 



REDUCTION OF DECIMALS. 45 

DIVISION OF DECIMALS. 

Rule. — Divide as in whole numbers, and point off in the quotient as many places 
for decimals as the decimal places in the dividend exceed those in the divisor; but if 
there are not so many places, supply the deficiency by prefixing ciphers. 

Example. — Divide 53 by 6.75. 

6.75) 53.00000 (=7.S51+. 

Here 5 ciphers were annexed to the dividend to extend the division. 

BY CONTRACTION. 

Rule. — Take only as many figures of the divisor as will be equal to the number of 
figures, both integers and decimals, to be in the quotient, and ascertain how many 
times they may be contained in the first figures of the dividend, as usual. 

Let each remainder be a new dividend ; and for every such dividend leave out one 
figure more on the right-hand side of the divisor, carrying for the figures cut off a3 
in Contraction of Multiplication. 

Note. — When there are not so many figures in the divisor as there are required to 
be in the quotient, continue the first operation until the number of figures in the di- 
visor are equal to those remaining to be found in the quotient, after which begin the 
contraction. 

Example.— Divide 2508.92806 by 92.41035, so as to have only four places of deci- 
mals in the quotient. 

92.410315) 250S.928I06 (27.1498 4 608 

1S4S 207 + 1 3 696 

912 
832+4 

80 
74;+ 2 

6 




REDUCTION OF DECIMALS. 
To Reduce a "Vulgar Fraction to its Equivalent Decimal. 

Rule. —Divide the numerator by the denominator, annexing ciphers to the numer- 
ator as far as may be necessary. 

Example.— Reduce # to a decimal. 

5 5) 4.0 

.S* 

To Ascertain the Value of a Decimal in Terms of an I life* 
rior Denomination. 

Rule Multiply the decimal by the number of parts in the next lower denomina- 
tion, and cut off as many places for a remainder, to the right hand, as there are 
places in the given decimal. 

Multiply that remainder by the parts in the next lower denomination, again cutting 
off for a remainder, and so on through all the parts of the integer. 
Example.— What is the value of .875 dollars ? 
.875 
100 

Cents, 87,500 

10 

Mills, 5. 000 = 87 cents 5 mills, 
Ex. 2.— What is the volume of .140 cubic feet in inches? 
.140 

1728 cubic inches in a cubic foot 
241.920 cubic inches. 
Ex. 3.— What is the value of .00129 of a foot ? .01548 inches. 



40 



DUODECIMALS. 



To Reduce Decimals to Equivalent Decimals ofh.igb.er 
Denomination, s . 

Rule.— Divide by the number of parts in the next higher denomination, contim* 
ing the operation as far as required. 
Example. — Reduce 1 inch to the decimal of a foot. 
121 1,00000 

I .0S333 +foot 
Ex. 2.— Reduce 14" 12'" to the decimal of a minute, 
14" 12"' 
60 

GO 14.2" 

| .23666' 4- minute. 

When there are several numbers, to be reduced all to the decimal of the highest. 

Rule. — Reduce them all to the lowest denomination, and proceed as for one de- 
nomination. 
ExAiiPLE. — Reduce 5 feet 10 inches and 3 barleycorns to the decimal of a yard. 
Feet. In. Be. 
5 10 3 
12 



s 

12 


70 

3 

213. 

71. 


3 


5.1*166 




1.1722 + yards. 





RULE OF THREE IX DECIMALS. 

Rule. — Prepare the terms by reducing the vulgar fractions to decimals, compound 
numbers to decimals of the highest denomination, the first and thild terms to the 
game name ; then proceed as in whole numbers. See Rule, page 4S. 
ExAMrLE. — If J a ton of iron cost ^ of a dollar, what will .625 of a ton cost ? 

.5 : .75 : : .625 
.625 
.5) .4CS75 

.i'375 dollars. 



2=.T5 i 



DUODECIMALS. 

In Duodecimals, or Cross Multiplication, the dimensions are taken in feet, inches, 
and twelfths of an inch. 

Rule. — Set the dimensions to be multiplied together one under the other, feet 
under feet, inches under inches, etc. 

Multiply each term of the multiplicand, beginning at the lowest, by the feet in the 
multiplier, and set the result of each immediately under its corresponding term, car- 
rying 1 for every 12 from one term to the other. In like manner, multiply all the 
multiplicand by the inches of the multiplier, and then by the twelfth parts, setting 
the result of each term one place farther to the right hand for every multiplier. The 
sum of the products is the result. 



MEAN PROPORTION. — RULE OF THREE. 47 



Example. — Multiply 1 foot 3 inches by 1 foot 1 inch. 

Feet. Ins. 
1 3 

V 1 

1 3 

1 3 



1 foot. 4 ins. 3 twelfths of an inch. 

Pboof 1 foot 3 inches is 15 inches, and 1 foot 1 inch is 13 inches; and 15x13 

as 195 square inches. 

Ex. 2. — How many square inches are there in a hoard 35 feet 4 J inches long and 
12 feet 3 J inches wide ? 

Feet. Ins. Twelfths. 
35 4 6 
12 3 4 

6 



424 


C 





8 


10 


1 




11 


9 



434 3 11 
Ex. 3.— Multiply 2") feet 6j inches by 40 feet 6 inches. 

By duodecimals, Am. 831 feet 11 inches 3 twelfths equal 831 square feet 
and 135 square inches. 

By decimals 40 feet 6 ins. td 40.5 

20 " 6J " —20.541666 , etc. 
S3 1.937499 feet. 

144 
134.999850 square ins. 

"Value of Duodecimals in Square Feet and Indies. 

Sq Ft. Sq Ins. 



i rooj, 

1 Inch 


l 


1 Twelfth 


12 

— _1_ 
— 144 



Sq.Ft. Sq Ins 
or 144 
" 12 
" 1 



Jjj of 1 twelfth ±± T ^8 or -033333, etc. 
$ of ^ of - = '£$#! « .006944, etc. 



Illustration What number of square inches are therein a floor 100 feet 6 inches 

broad and 25 feet 6 inches and 6 twelfths long? 

2566 feet 11 ins. 3 twelfths = 2566 feet 135 ins. 



MEAN PROPORTION. 

Mean Proportion is the proportion to two given numbers or terms. 
Rule. — Multiply the two numbers or terms together, and extract the square root 
of their product. 
Example. — What is the mean proportionate velocity to 16 and SI ? 
16XS1 — 1296, and ^/\ 296 — 36 mean velocity. 



RULE OF THREE. 

The Rule of Three teaches how to compute a fourth proportional to three given 
numbers. 

It is either Direct or Inverse. 

It is Direct when more requires more, or less requires less; thus, if 3 barrels of 
flour cost $18, what will 10 barrels cost? 

In this case the Proportion is Direct, and the stating must be, 
As 3 : 10 : : 18 : 60. 

It is Inverse when more requires less, or less requires more; thus, if 6 men build 
a certain quantity of wall in 10 days, in how many days will S men build the lile 
quantity? Or, if 3 men dig 100 feet of trench in 7 days, in how many days will 2 men 
perform the same work ? 



48 COMPOUND PROPORTION. 

Here the Proportion is Inverse, and the stating must be, 
As S : 6 : : 10 : 7.5. 
2:3:: 7 : 10.5. 
The fourth term is always ascertained by multiplying the 2d and 3d terms togeth- 
er, and dividing the product by the 1st term. 

Of the three given numbers necessary for the stating, two of them contain the sup* 
position, and the third a demand. 

Rule.— State the question by setting down in a straight line the three necessary 
numbers in the following manner: 

Let the 3d term be that of supposition, of the same denomination as the answer, 
or 4th term is to be, making the demanding number the 2d term, and the other num- 
ber the 1st term when the question is in Direct Proportion, but contrariwise if in 
Inverse Proportion ; that is, let the demanding number be the 1st term. 

Multiply the 2d and 3d terms together, and divide by the 1st, and the product will 
be the answer, or 4th term sought, of the same denomination as the 2d term. 

Note. — If the first and third terms are of different denominations, reduce them to 
the same. If, after division, there be any remainder, reduce it to the next lower de- 
nomination, divide by the divisor as before, and the quotient will be of this last de- 
nomination. 

Sometimes two or more statings are necessary, ivhich may always be known by the 
nature of the question. 

Example. —If 20 tons of iron cost $225, what will 500 tons cost ? 

Tons Tons. Dolls. 
20 : 500 : : 225 
500 

2|0) 11250|0 

5G25 dollars. 
Ex. 2.— If 15 men raise 100 tons of iron ore in 12 days, how many men will raiso a 
like quantity in 5 days ? 

Davs. Pays. Men. Men. 
As 5 : 12 : : 15 : 36 
Ex. 3.— A wall that is to be built to the height of 3G feet, was raised 9 feet high by 
1C men in 6 days; how many men could finish it in 4 days at the same rate of work- 
ing? 

Days Davs. Men. Men. 
4 : 6 * : : 1G : 24 
Then, if 9 feet requires 24 men, what will 27 feet require ? 
9 : 27 : : 24 : 72 men. 




COMPOUND PROPORTION. 

Compound Proportion is the rule by means of which such questions as would re- 
quire two or more statings in simple proportion (Rule of Three) can be resolved in 
one. 

As the rule, however, is but little used, and not easily acquired, it is deemed pref- 
erable to omit it here, and to show the operation by two or more statings in Simple 
Proportion. 

Example. — How many men can dig a trench 135 feet long in 8 days, when 16 men 
can dig 54 feet in 6 days? 

Feet. Feet. Men. Men. 

First As 54 ; 135 : : 16 : 40 

Days. Davs. Men. Men. 

Second As 8:6":: 40 : 30 

Ex. 2.— If a man travel 130 miles in 3 days of twelve hours each, in how many 
days of 10 hours each would he require to travel 360 miles ? 
Miles. Miles Days. Days. 

First As 130 : 360 : : 3 : S.307 

Hours. Hours. Days. Davs. 

Second As 10 : 12 : : S.307 : 9.96S4 

Ex. 3.— If 12 men in 15 days of 12 hours build a wall 30 feet long, 6 wide, and 3 
deep, in how many days of S hours will 60 men build a wall 3 JO feet long, S wide, and 
6 deep? 120 days. 



INVOLUTION. EVOLUTION. 



49 



Or, "by Cancelation, 

Rule.— On the right of a vertical line put the number of the same denomination 
as that of the required answer. 

Examine each simple proportion separately, and if its terms demand a greater an- 
swer than the 3d term, put the larger number on the right, and the lesser on the left 
of the line; but if its terms demand a less answer than the M term, put the smaller 
number on the right and the larger on the left of the line. 

Then Cancel the numbers divisible by a common divisor, and evolve the 4th term 
or answer required. 

Take the preceding, example first : 3d term, or term of supposition of the same de- 
nomination as the required answer. 16 men. 

135 feet require more men than 5±feet. 
Sdays " less " G days. 

Statement. 

16 
54 135 
8 6 2X5X3=30 men. 

Ex. 3 3d term, 15 days. 

60 men require less days than 12 men. 

8 fiours u more u 12 hours. 
800 feet " \» " BO feet. 

8 u u « u e " 

6 n it u ft 3 ft 



Result by Cancelation. 

m s 

f 3 



Statement. 




15 


60 


12 


8 


12 


30 


300 


6 


8 


3 


6 



Result bv Cancelation. 

n 3 
n * 



* 



3 X4X 10 = 120 days. 



INVOLUTION. 

Involution is the multiplying any number into itself a certain number of times. 
The products obtained are called Powers. The number is called the Boot, or first 
power. 

When a number is multiplied by itself once, the product is the square of that 
number ; twice, the cube ; thres times, the biquadrate, etc Thus, of the number 5. 
5 is the Root, or 1st power. 
& 5x5 = 25 " Square, or 2d power, and is expressed 5 2 . 

5x5x5 =s 125 u Cube, or 3d power, and is expressed 5 3 . 
5x5x5x5 = 625 u Biquadrate, or 4th power, and is expressed 5*. 
The lesser figure set superior to the number denotes the power, and is termed the 
Index or Exponent. 
Example.— What is the cube of 9 ? 729. 



Ex. 2— What is the cube of |? 

Ex. 3 What is the 4th power of 1.5? 



64* 

5.0625. 



EVOLUTION. 

Evolution is ascertaining the Root of any number. 

The sign y/ placed before any number indicates that the square root of that num.* 
b«r is required or shown. 
The same character expresses any other root by placing the index above it. 

Thus, V25 = 5, .and 4+2 = V36. 

And, V27 == 3, and V C4 = 4. 
Roots which only approximate are called Surd Roots. 

E 



50 EVOLUTION. 



TO EXTRACT THE SQUARE ROOT. 

Rule.— Point off the given number from units' place into periods of two figures 
each. 

Ascertain the greatest square in the left-hand period, and place its root in the 
quotient ; subtract the square number from this period, and to the remainder bring 
down the next period for a dividend. 

Double this root for a divisor ; ascertain how maay times it is contained in the 
dividend, exclusive of the right-hand figure, which, when multiplied by the numbei 
to be put to the right hand of this divisor, the product will be equal to, or the nexf> 
less than the dividend ; place the result in the quotient, and also at the right hand 
of the divisor. 

Multiply the divisor by the last quotient figure, and subtract the product from the 
dividend ; bring down the next period, and proceed as before. 

Note. — Mixed decimals must be pointed off both ways from units. 
Example. — What is the square root of 2 ? 
II 2.060606 (1.414,+. 

11 1 
241100 



4| 


)6 


Ex. 2.- 


—What is the square root of 144? 


281 

1 


400 

2S1 


11 144 (12 
l| 1 


2S24 
4 


11900 
11296 

"col 


22 044 
44 


2828 




00 



SQUARE ROOTS OF VULGAR FRACTIONS. 

Rttle. — Reduce the fractions to their lowest terms, and that fraction to a decimal, 
and proceed as in whole numbers and decimals. 

Note.— When the terms of the fractions are squares, take the root of each and set 
one above the other ; as, § is the square root of gg. 

Ex ample.— What is the square root of ^ ? .8660254. 

To Ascertain the 4th Root of a Number. 
Rule.— Extract the square root twice, and for the 8th root thrice, etc., etc. 



TO EXTRACT THE CUBE ROOT. 

Rule. — From the table of roots (page 210) take the nearest cube to the given num- 
ber, and call it the assumed cube. 

Then, as the given number added to twice the assumed cube, is to the assumed 
cube added to twice the given number, so is the root of the assumed cube to the re- 
quired root, nearly; and by using in like manner the root thus found as an as- 
sumed cube, and proceeding as above, another root will be found still nearer ; and in 
the same manner as far as may be deemed necessary. 

Example— What is the cube root of 10517.9 ! 

Nearest cube, page 210 ; 10648, root 22. 

10648. 10517.9 

2_ 2 

21296 21035.8 
10517.9 1064S. 



31S13.9 : 316S3.S : : 22 : 21.9 -f. 

To Ascertain or To Extract the Square or Cube Roots of 
Roots, Whole Numbers, and. of Integers and. Decimals, 
see Table of Squares and Cubes, and Rules, p. 210-S4r3. 



PROPEETIES OF NUMBERS. 



51 



To Extract any Root whatever. 

Let P represent the number. j Let A represent the assumed power, r its root. 

n u the index of the power. | R " the required root of P. 
Then, as the sum of n -f- 1 X A and n — lxP is to the sum of n-f-lXP and n — 1 
X A, so is the assumed root r to the required root R. 

Example What i3 the cube root of 1500? 

The nearest cube, page 210, is 1331, root 1 1. 

P = 1500, ?i = 3, A = 1331, r = ll ; 

then, n + 1 X A = 5324, n + 1 X P = 6001) 

n — 1XP =3000, w — lxA = 2662 

8324 J 8662": : 11 : 11.446+. 

To A.® certain tlie Root of an Even Power greater th.an 
those given in the Talkie of Square and. Cxi/be Roots. 

Rule — Extract the square or cube root of it, which will reduce it to half the given 
power ; then the square or cube root of that power reduces it to half the same power ; 
and so on until the required root is obtained. 

Illustration. — Suppose a 12th power is given ; the square root of that reduces it 
to a 6th power, and the square root of a 6th power to a cube. 

Example What is the biquadrate, or 4th root, of 2560000? 

V2560000=:1600, and ^1600=: 40. 



PROPERTIES OF NUMBERS. 



1 . A Prime Number U that which can only be measured (divided without a re- 
mainder) by 1 or unity. 

2. A Composite Number is that which can be measured by some number greater 
than unity. 

3. A Perfect Number is that which is equal to the sum of all its divisora or ali- 
quot parts ; as 6 == &, jj, §. 

4. If the sum of the digits constituting any number be divisible by 3 or 9, the 
whole is divisible by them. 

5. A square number can not terminate with an odd number of ciphers. 

6. No square number can terminate with two equal digits, except two ciphers or 
two fours. 

7. No number the last digit of which is 2, 3, 7, or 8, is a square number. 



Ta"ble of the first Nine Powers of tlie first Nine 

Numbers. 



1st. 


2d. 
1 


3d. 


4th. 


5th. 


6th. 


7th. 


8th 


9th. 


1 


1 


1 


1 


1 


1 


1 


1 


2 


4 


8 


16 


32 


M 


12S 


256 


512 


3 


9 


27 


81 


243 


729 


21S7 


6561 


19683 


4 


16 


64 


256 


1024 


4096 


16384 


C5536 


262144 


5 


25 


125 


625 


3125 


15625 


78125 


390625 


1953125 


6 


36 


216 


1296 


7776 


4G65G 


279936 


1679616 


10077696 


7 


49 


343 


2401 


16807 


117649 


823543 


5764801 


40353G07 


8 


64 


512 


4096 


3276S 


262144 


2097152 


16777216 


134>17728 


9 


81 


729 


6561 


59049 


531441 


4782969 


4G040T21 


337420489 



ARITHMETICAL PROGRESSION. 



ARITHMETICAL PROGRESSION. 

Arithmetical Progression is a series of numbers increasing or decreasing by a 
constant number or difference; as, 1, 3, 5, 7, 9, 15, 12, 9, 6, S. The numbers which 
form the series are called Terms; the first and last are called the Extremes, and the 
others the Means. 

)Y7ien any three of the folloiting elements are given, the remaining tico can be as- 
certained, viz. : The First term, the Last term^the Sumber of terms, the Common 
Difference, and the Sum of all the terms. 

To Ascertain the Last Term, "When the Thirst Term, the 
Common Difference, and. the Number of Terms are 
given. 

Rule Multiply the number of terms less 1, by the common difference, and to the 

product add the first term. 

Example. — A man traveled for 12 days, going 3 miles the first day, S the second, 
and so on ; how far did he travel the last day ? 

12 — 1 X 5 = 55, and 55 -f- 3 — . 58 miles. 

To Ascertain the Common Difference, When the Number 
of Terms and. the Extremes are given. 

Rule. — Divide the difference of the extremes by 1 less than the number of terms. 
Example. — The extremes are 3 and 15, and the number of terms 7; what is the 
common difference? 

15-3-4-(7-l)=^ = 2. 
o 

To Ascertain the Sum of all tlie Terms, "When the Ex- 
tremes and. Number of* Terms are given. 

Rule Multiply the number of terms by half the sum of the extremes. 

Example. — How many times does the hammer of a clock strike in 12 hours ? 



12X (12 -f 1 -•- 2) =78 times. 

To Ascertain the Number or Terms, When the Common 
Difference and. the Extremes are given. 

Rule. — Divide the difference of the extremes by the common difference, and add 
1 to the quotieut. 

Example. — A man traveled 3 miles the first day, 5 the second, 7 the third, and so 
on, till he went 57 miles in one day ; how many day3 had he traveled at the close 
of the last day ? 

57 — 3^-2 = 27, and 27 -f 1 = 28 days. 

To Compute two Arithmetical IVIeans between two given 
Extremes. 

Rule. — Subtract the less extreme from the greater, and divide the difference by 
3, the quotient will be the common difference, which being added to the less extreme, 
or taken from the greater, will give the means. 
Example.— Ascertain two arithmetical means between 4 and 16. 
16 — 4 -4- 3 = 4 com. dif. 
4-}- 4— 8 one mean. 
16 — 4=12 second mean. 

To Compute any Number ofArithmetical Means "between 
two Extremes. 

Rule. —Subtract the less extreme from the creator, and divide the difference by 
1 more than the number of means required to be ascertained, and then proceed as in 
the foregoing rule. 






GEOMETRICAL PROGRESSION. 53 



GEOMETRICAL PROGRESSION. 

Geometrical Progression is any series of numbers continually increasing by a 
constant multiplier, or decreasing by a constant divisor, as 1, 2, 4, 8, 16, and 15, 7.5, 
3.75. 

The constant multiplier or divisor is the Ratio. 

When any three of the following elements are given, the remaining tico can be as- 
certained, viz. : The First term, the Last term, the dumber of Terms, the Ratio, 
and the Sum of all the Terms. 

To Compxite tlie Last Term, 'When, tne First Term and. 
the Ratio are Equal. 

Rule. — Write a few of the leading terms of the series and place their indices over 
them, beginning with a unit. Add together the most convenient indices to make 
the index to the term required. 

Multiply the terms of the series of these indices together, and the product will be 
the term required. 

Or, multiply the first term by the ratio raised to a power, denoted by the number 
of terms less 1. 

Example. — The first term is 2, the ratio 2, and the number of terms 13 ; what is 
Hie last term ? 

Indices, 12 3 4 5 
Terms, 2, 4, 8, 16, 32. 

Then 5 + 5 + 3 = 13 = sum of indices, and 32x32x8 = 8192 = last term. 

Or, 2x213-1 — 8192. 

Ex. 2. The price of 12 horses being 4 cents for the first, 16 for the second, and 64 
for the third ; what is the price of the last horse? $167,772.16. 

"When the First Term and. the R-atio are Different. 

Rule. — Write a few of the leading terms of the series, and place their indices over 
them, beginning with a cipher. Add together the most convenient indices to make 
an index less by 1 than the term sought. 

Multiply the terms of these series belonging to these indices together, and take the 
product for a dividend. 

Raise the first term to a power, the index of which is 1 less than the number of 
terms multiplied ; take the result for a divisor ; proceed with their division, and the 
quotient will give the term required. 

Example — The first term is 1, the ratio 2, and the number of terms 23; what is 
the last term ? 

Indices, 12 3 4 5 
Terms, 1, 2, 4, 8, 16, 32. 
Then 5 -f- 5 + 5 + 5 -f- 2. = 22 = sum of indices, and 32 X 32 X 32 X 32 X 4 = 4194304, 
and 4194304 -4- the 5th power (6 — 1) of 1 = 1 = 4194304. 
Or, 1X223-1 — 4194304. 

Ex. 2. If one cent had been put out at interest in 1630, what would it have amount- 
ed to in the year 1834, if it had doubled its value every 12 years? 

1834 — 1630 jb 204, which-f- 12 = 17, and 17 + 1 = IS = number of terms. 
Indices, 12 3 4 7 
Terms, 1, 2, 4, S, 16, 128. 
Then 7 + 4 + 3 + 2 + 1 = 17, and 128x16x8x4x2x1 = 131072, and 131072 -M, 
the 4th power (5 — 1) of 1 = $ 1 . 310. 72. 

Ex. 3. If a man were to work 20 days, for 4 cents for the first day, 12 for the sec- 
ond, and 36 for the third, and so on, what would be the amount of his pay upon the 
last day ? 

Indices, 12 3 4 5 6 
Terms, 4, 12, 36, 108, 324, 972, 2916. 
Then 6 + 5 + 4 + 3 + 1 =19 = sum of indices, and 2916x972x324x108x12 = 
1190155742208, and this sum -r- the 4th power (5—1) of the first term = 266, and 
1190155742208 -r- 256 = 4649045868 cents. 

E* 



54 



GEOMETRICAL PROGRESSION. 



To Compnte the Snm of the Series, '^hen tlie Thirst Term, 
the K-atio, and. tlie Number of Terms are given. 

Rule.— Raise the ratio to a power the index of which is equal to the number of 
terms, from which subtract 1 ; then divide the remainder by the ratio less 1, and 
multiply the quotient by the first term. 

Example.— The first term is 2, the ratio 2, and the number of terms 13 ; what is 
the sum of the series ? 

2:3-1 = 8192 — 1 = 8191, and 8191 ^-(2 — 1) = 1 = 8191, and 8191x2 = 10382. 

Ex. 2 — If a man were to buy 12 horses, giving 2 cents for the first horse, G cents 
for the second, and so on, what would they cost him ? $5,314.40. 

When the Last Term is given. 

Rule.— Multiply the last term by the ratio, and from the product subtract the 
first term ; then divide the remainder by the ratio less 1. 

Example.— The first term is 1, the ratio 2, and the last term 131072 ; what is the 
eum of the series ? 

131072 X 2 — 1 = 262143, and 262143 -^-2^1 = 262143. 

To Compute tlie Ratio, When the First Term, tlie Last 
Term, and. the Number of Terms are given. 

Rule.— Divide the last term by the first, and the quotient will be equal to the 
ratio raised to the power denoted by 1 less than the number of terms; then extract 
the root of this quotient. 

Example.— The last term, or greatest extreme of a geometrical progression, i3 .46, 
the first, or least term, .005, and the number of terms 40 : what is the ratio? 
.46 
-^r = 92, or the 39th power (40 — 1) of the ratio; then log. 92«— 1 =log. of 

92 = 1.9637S8, which -H- 30; 
.0503535 = .112293. 






: .0503535, and the number corresponding to the log. of 



To Compute the Number of Terms, When the Ratio, the 
First, and. the Last Terms are given. 

Rule.— Divide the logarithm of the quotient of the product of the ratio and the 
last term, divided by the first term, by the logarithm of the ratio. 

Example.— The ratio is 2, and the first and last terms are 1 and 131072; what is 
the number of the terms ? 



log. 



2X131072 



:log. 262144 — 5. 41S54, and5.41S54- 



rn 5.41S54 „„ 
- 1Og -° f2 = 30l03- = 18 - 



Table of Geometrical Progression, 

Whereby any questions of Geometrical Progression and of Double Ratio may be 
solved by Inspection, the number of terms not exceeding 56. 



1 


so 
1 


Ived b 
15 


2 


2 


16 


3 


4 


17 


4 


8 


18 


5 


16 


19 


C 


32 


20 


7 


64 


21 


8 


T8 


22 


9 


256 


23 


10 


512 


24 


11 


1024 


25 


12 


2048 


26 


13 


4096 


27 


14 


S192 


28 



16384 


29 


32768 


30 


65536 


31 


131072 


32 


262144 


33 


52428S 


34 


1048576 


35 


2017152 


36 


4194304 


37 


83S86C8 


38 


16777216 


39 


33554432 


40 


67108864 


41 


134217728 


42 



26S435456 

536870912 

1073741S24 

21474S3C4S 

4294967290 

8589934502 

17179S69184 

3435:)73S36S 

68719470736 

13743S053472 

274S770O0O44 

540755S13SS8 

1099511627776 

2199023255552 



4398046511104 

879609G022208 

175921S6I '44416 

351843720S8S32 

7030S744177064 

1407374SS355328 

2^1474970710656 

502949953421312 

1125S999r6*42624 

2251799813685248 

4503590627370496 

900719925474; >992 

1S014308509481984 

36028797018963968 



Illustrations The 12th power of 2 = 4096, and the 7th root of 128 = 2. 



PERMUTATION. POSITION. 



55 



PERMUTATION. 

Permutation is a rule for ascertaining how many different ways any given num- 
ber of numbers of things may be varied in their position. 

The permutation of the 3 letters abc, taken all together, are 6 ; taken two and two, 
are 6; and taken singly, are 3. 

Rule Multiply all the terms continually together, and the last product will give 

the result required. 

Examine. — How many variations will the nine digits admit of? 
1X2X3X4X5X6X1X8X9 = 362880. 

Wlierx. only part of the JS"nm"bers or Elements are taken 
at once. 

Rule. — Take a series of numbers, beginning with the number of things given, de- 
creasing by 1 , until the number of terms equals the number of things or quantities to 
be taken at a time, and the product of all the terms will give the sum required. 

Example. — How many changes can be rung with 4 bells (taken 4 and 4 together) 
out of 6? 6X5X4X3 = 360. 



When several of the Elements are alike 

Rule. — Ascertain the permutations of all the numbers or things, and of all that 
can be made of each separate kind or division ; divide the number of the permuta- 
tions of the whole by the product of the several partial permutations, and the quo- 
tient will give the number of permutations. 

Example. — How many permutations can be made out of the letters of the word 
persevere (9 letters, having 4 e's and 2 r's) ? 

1x2x3x4x5x6x7x8x9 = 362880 ; 
1X2X3X4 = 24; 1x2 = 2, and 24x2 = 48 ; and 3628S0 ■+ 48 = 7560. 

Tatole of Permutations, 
Whereby any questions of Permutation from 1 may be solved by Inspection, the 



1 


1 1 


5 


120 


9 


2 


2 


6 


720 


10 


3 


6 


7 


5040 


11 


4 


24 1 


8 


40320 


12 



number of terms not exceeding 20. 

6227020800 I 17 

8717S291200 18 

1307674368000 19 

20922789888000 | 20 



3628S0 


113 


362SS00 


14 


39916800 


15 


479001600 


1 16 



355687428096000 

6402373705728000 

121645100408832000 

2432902008176640000 



POSITION. 

Position is of two kinds, Single and Double, and is determined by the number 
of Suppositions. 

Single Position. 

Rule. — Take any number, and proceed with it as if it were the correct one; 
then, as the result is to the given sum, so is the supposed number to the number re- 
quired. 

Example A commander of a vessel, after sending away in boats J, J, and J 

of his crew, had left 300 ; what number had he in command ? 
Suppose he had 600. 



J of 600 is 100 

\ of 600 is 150 450 

150 : 300 : : 600 : 1200 men. 
Ex. 2. — A person being asked his age, replied, if \ of my age be multiplied by 2, 
and that product added to half the years I have lived, the sum will be 75. How old 
was he ? 37.5 years. 



50 FELLOWSHIP. DOUBLE FELLOWSHIP, 



Doiible ^Position. 

Rule. — Take any two numbers, and proceed with each according to the conditions 
of the question ; multiply the results or errors by the contrary supposition ; that i3, 
the first position by the last error, and the last position by the first error. 
If the errors are too great, mark them -|- ; and if too little, — . 
Then, if the errors are alike, divide the difference of the products by the difference 
of the errors ; but if they are unlike, divide the sum of the products by the sum of 
the errors. 

Example. — F asked G how much his boat cost ; he replied that if it cost him 6 
times as much as it did, and $30 more, it would have cost him $300. What was 
the price of the boat ? 

Suppose it cost 60 30 

6 times. 6 times. 

360 "180 

and 30 more and 30 more. 

390 ~2L0 

300 300 

~90-f- ~90"— 

30 2d position. 60 1st position. 

90 2700 54~i0 

90 5400 

ISO) 8100 (45 dollar?. 
Ex. 2.— What is the length of a fish when the head is 9 inches long, the tail as 
long as its head and half its body, and the body as long as both the head and tail. 

6 feet. 



FELLOWSHIP. 

Fellowship is a method of ascertaining gains or losses of individuals engaged in 
joint operations. 

Single ITellcrvv-sliip. 

Rule — As the whole stock is to the whole gain or loss, so is each share to the gain 
or Loss on that share. 

Example. — Two men drew a* prize in a lottery of $9500. A paid $3, and B $2 
for the ticket ; how much is each share ? 

5 : 9500 : : 3 : 5T00, A's share. 
5 : 95'JO : : 2 : 3S00, B's share. 

Dou."ble iF'ello^wsliip, 

Or Fellowship with Time. 

Rule. — Multiply each share by the time of its interest in the Fellowship; then, aa 
the sum of the pro lucts is to the product of each interest, so is the whole gain or lo&3 
to each share of the gain or loss. 

Example. — A ship's company take a prize of $10,000, which they divide according 
to their rate of pay and time of service on board. The officers have been on board 6 
months, and the crew 3 months ; the pay of the lieutenants is $100, ensigns $50, and 
crew $10 per month; and there are 2 lieutenants, 4 ensigns, and 50 men; what id 
each one' s share ? 

2 lieutenants $100 = 200x6 = 1200 

4 ensigns 50 = i0 >X0 = 1200 

50 men 10 = 500x3 — 1500 

3i)00 

Lieutenants -3900 : 1200 : : 10,000 : 3076 92 -h 2 = 1538.46 dolls. 

Ensians 3900 : 1200 : : 10,000 : 3070.92 -f- 4= 760.23 « 

Men". 3900: 1500 : : 10,000: 3846.16-^50 = 76.92 " 



ALLIGATION. — SIMPLE INTEREST. 57 



ALLIGATION. 

Alligation is a method of finding the mean rate or quality of different materials 
when mixed together. 

To Compute the IVIeaxi JPrice of the ilVTixttire. 

Rule. — Multiply each quantity by its rate, divide the sum of the products by the 
sum of the quantities, and the quotient will be the rate of the composition. 

Example. — If 10 lbs. of copper at 20 cents per lb., 1 lb. of tin at 5 cents, and 1 lb. of 
lead at 4 cents, be mixed together, what is the value of the composition? 
10X20 = 200 
IX 5= 5 
IX 4= 4 
12 ) 20J (17.416 cents. 

To Ascertain -what Qnantity of eacli Article must "be 
taken, "When tlie ^Prices and. Mean. 3?rice are given. 

Rule. — Connect with a line each price that is less than the mean rate with one or 
more that is greater. 

Write the difference between the mixture rate and that of each of the simples op- 
posite the price with which it is connected ; then the sum of the differences against 
any price will express the quantity to be taken of that price. 

Example How much gunpowder, at T2, 54, and 48 cents per pound, will compose 

a mixture worth 60 cents a pound ? 



(48 \ 12, at 48 cento. 

60^54x/ 12, at 54 cents. 

(72; 12 -f 6 = 18, at 72 cents. 



Then 12x48 -f 12x54 + 18x72 = 2520 , and 2520 -f- 12 + 12 + 12 -f 6 = 60 cents. 

Note. — Should it be required to mix a definite quantity of any one article, the 
quantities of each, determined by the above rule, must be increased or decreased in 
the proportion they bear to the defined quantity. 

Thus, had it been required to mix 18 pounds at 48 cents, the result would be 18 at 
48, 18 at 54, and 27 at 72 cents per pound. 

When the -whole Composition is limited. 

Rule As the sum of the relative quantities, as ascertained by the above rule, is 

to the whole quantity required, so is each quantity so ascertained to the required 
quantity of each. 

Example. — Required 100 pounds of the above mixture. 
Then 12 + 12 + 18 = 42. 

Then 42 : 100 : : 12 : 28.571. 

42 : 100 : : 12 : 28.571. 
42 : 100 : : 18 : 42.857. 



SIMPLE INTEREST. 



To Compute the Interest on any Griven Snm. for a ^Period 
of One or more Years. 

Rule — Multiply the given sum or principal by -the rate per cent, and the num- 
ber of years ; point off two figures to the right of the product, and the result will give 
the interest in dollars and cents for 1 year. 

Example — What is the interest upon $1050 for 5 years at 7 per cent. ? 
1050X7X5= 36750, and 367.50 = $367.50. 

When the Time is less than One Year. 
Rule. — Proceed as before, multiplying by the number of months or days, and di* 
viding by the following units ; viz. ; 1 2 for months, and 365 or 366, as the case may 
be, for days. 



58 



COMPOUND INTEREST. REBATE. 



Example.— What is the interest upon $1050 for 5 months and 30 days at 7 pel 
cent. ? 

5 months and 30 days = 1S3 days, 
1050x7x183 iL 36S5) and 36<S5 _ $36 S5> 
3bo 
The interest upon any sum at 6 per cent.=l per cent, for 2 months. 
The interest at 5 per cent, is Jth less than at 6 per cent. 
The interest at 7 per cent, is JVli greater than at 6 per cent. 
The operation of computing interest may be performed thus ; 
Taking the preceding example — 2 months =1 per cent. =10.50 
2 " =1 « 10.50 

1 " =J " 5.25 

30 days = 1 month == 5.25 

31.50 

Add Jth for 7 per cent. — . 5.25 

$36.75 
Xote.— The difference between this amount and the preceding arises from 1S3 
days being taken in the one case, and half a year, or 182. 5 days, in the other. 



COMPOUND INTEREST. 



If any Principal be multiplied by the amount (in the following table) opposite the 
years, and under the rate per cent., the sum will be the amount of that principal at 
compound interest for the time and rate taken. 

Example. — What is the amount of $500 for 10 years at 6 per cent. ? 

Tabular amount 1.790S4, and 1.790S4X500 — 895.42 dollars. 

Ta"ble snowing tlie Value of SI, etc., for any Num- 
ber of Years not exceeding &4, at the Rates of 5, 
6, and. 7 per Ct. per Annum Compound Interest. 



Years. 


5 Per Cent. 


6 Per Cent. 


7 Per Cent. 


Years. 


5 Per Cent. 


6 Per Cent. 


7 Per Cent. 


1 


J. 05 


1.06 


1.07 


13 


1.8S564 


2.13292 


2.40985 


2 


1.1025 


1.1236 


1.1449 


14 


1.97993 


2.26090 


2.57853 


3 


1.15762 


1.19101 


1.22504 


15 


2.07892 


2.39655 


2.75903 


4 


1.2155 


1.26247 


1.3108 


16 


2.1S287 


2.54035 


2.95216 


5 


1.27638 


1.33822 


1.40255 


17 


2.29201 


2.69277 


3.15SS1 


6 


1.34 


1.41851 


1.50073 


18 


2.40661 


2.S5433 


3.37994 


7 


1.4071 


1.50363 


1.60578 


19 


2.52695 


3.02559 


3.61654 


8 


1.47745 


1.59384 


1.71S19 


20 


9.65329 


3.20713 


3.8697 


9 


1.55132 


1.68947 


1.83S46 


21 


2.7S596 


3.39956 


4.14057 


10 


1.628S9 


1.79084 


1.96715 


22 


2.92526 


3.60353 


4.43041 


11 


1.71033 


1.S9829 


2.104S5 


23 


3.07152 


3.81974 


4.74054 


12 


1.795S5 


2.01219 


2.25219 


24 


3.22509 


4.04873 


5.0723S 



REBATE. 

Rebate is a deduction or Discount upon money paid before it is due. 

To Comprite tlie Re"bate upon any Sum. 

Rule — Multiply the amount by the rate per cent, and by the time, and divide the 
product by the sum of the product of the rate per cent, and the time added to 100. 

Example. — What is the rebate upon $12,075 for 3 years, 5 months, and 15 days, at 
6 per cent. ? 

3 years 5 months and 15 days == 3.4574 years. 
12075X6X3.4574 _ 250488.63 _ 
100+(6X3.4574) ~ 120.7444 = 2 ° 74 ' 53 ~ $2 ° 74 53 * 



INTEREST AND DISCOUNT. ANNUITIES. 59 



INTEREST AND DISCOUNT. 

To Ascertain tlie Principal, the Time, Rate per Cent., and 
Interest "being given. 

Rule.— Divide the given interest by the interest of $l r etc., for the given rate 
and time. 
Example.— A Vhat sum of money at 6 per cent, will in 14 months produce $14? 
14-^ .07 = 200 dollars. 

To Ascertain the Itate per Cent., the Principal, Interest, 
and. Time "being given. 

Rule.— Divide the given interest by the interest of the given sum, for the time, at 
1 per cent. 

Example If $32.66 was the discount from a note of $400 for 14 months, what was 

that per cent. ? 

The interest on 400 for 14 months at 1 per cent.= 4.66. 
Then 32.66 -r- 4.66 = 7 per cent. 

To Ascertain the Time, the Principal, Rate per Cent., and 
Interest "being given. 

Rule.— Divide the given interest by the interest of the sum, at the rate per cent. 
for one year. 
Example— In what time will $10S produce $11.34, at T per cent. ? 
The interest on 108 for one year is 7.56. 

11.34 -=- 7.56 = 1.5 years. 



EQUATION OF PAYMENTS. 

Rule.— Multiply each sum by its time of payment in days, and divide the sum of 
the products by the sum of the payments. 

Example.— At owes 13 $300 in 15 days, $60 in 12 days, and $350 in 20 days; when 
ia the whole due ? 

300x15=4500 
60X12= 720 
350X20 = 7000 
710" ) 12220 (17 -\-days. 



ANNUITIES. 

To Ascertain the Amount of Annuity, the Time, and Hate 
of* Interest "being given. 

Rule. — Raise the ratio to a power denoted by the time, from which subtract 1 ; di- 
vide the remainder by the ratio less 1, and the quotient, multiplied by the annuity, 
will give the amount. 

Note.— $1 added to the given rate per cent, is the ratio, and the preceding table 
in Compound Interest is a table of ratios. 

Example. — What is the amouut of an annual pension of $100, interest 5 per cent., 
which has remained unpaid for four years ? 

1.05 ratio; then 1.05*— 1 = 1.21550625 — 1 = .21550625, and .21550C25-f- (1.05 — 1) 
.05 = 4.310125, which X 100= $431.0125. 

To Ascertain the Present 'Worth of an Annuity, the Time, 
and Hate heing given. 

Rule. — Ascertain the value of it for the whole time; and this amount divided by 
the ratio, involved to the time, will give the worth. 



60 



ANNUITIES. 



Example. — What is the present -worth of a pension or salary of $500, to continue 
10 veal's at 6 per cent, compound interest ? 

$500, by the last rule, is worth $6590.3975, which, divided by 1.06io (by table, 
page 53, is 1.79084) = $36S0.l5. 
Or, multiply the tabular amount in the following table, by the given annuity, and 
the product will be the present worth. 

Ta"ble showing the Present Worth of* an Annuity 
at 5, (3, and. 7 per Cent. Compound Interest for 
any Number of* Years under 34. 



Years. 


5 Per Cent. 


6 Per Cent. 


1 Per Cent. 


Years. 


5 Per Cent. 


6 Per Cent. 


7 Per Cent. 


1 


.95238 


.94339 


.f345 


18 


11.63958 


10.3276 


10.0591 


2 


1.S5941 


1.83339 


1.S08 


19 


12.03532 


11.15311 


10.3356 


3 


2.72325 


2.67301 


2.C243 


20 


12.46221 


11.46992 


10.594 


4 


3.54595 


3.4651 


3.3872 


21 


12.S2115 


11.76407 


10.S355 


5 


4.32948 


4.21236 


4.1001 


22 


13.163 


12.04158 


11.0612 


6 


5.07569 


4 91732 


4.7605 


23 


13.4S307 


12.3033S 


11.2722 


7 


5.73637 


5.58:33 


5.3S92 


24 


13.79864 


12.55035 


11.4693 


8 


6.46321 


6.20979 


5.9712 


25 


14.09394 


12.7S335 


11.6536 


9 


7.107S2 


6.S0169 


6.5152 


26 


14.37518 


13.00316 


11.3253 


10 


7.72173 


7.36003 


7.0235 


27 


14.64303 


13.21053 


11.9867 


11 


8.30641 


7.SS637 


7.4986 


28 


14.S9813 


13.40616 


12.1371 


12 


8.86325 


S. 33334 


7.9426 


29 


15.14107 


13.59072 


12.2777 


13 


9.39357 


8.S5263 


S.3576 


39 


15.37245 


13.764S3 


12.409 


14 


9.89S64 


9.2949S 


S.7454 


31 


15.59281 


13.92908 


125313 


15 


10.37966 


9.71225 


9.1079 


32 


15.S0268 


14.08398 


12.64G5 


16 


10.S3778 


10.10539 


9.4466 


33 


16.00255 


14.22917 


12.753S 


17 


11.27407 


10.47726 


9.7632 


34 


16.1929 


14.36613 


12.854 






Illustration.— As above; 10 years at 6 per cent. =7.36008, and 7.36003x500 = 
36S0.04 dollars. 

When Annuities do not commence till a certain period of time, they are said to be 
in Reversion. 



To Ascertain the Present Worth of an Annuity in Re- 
version. 

Rule Take two amounts under the rate in the above table, viz., that opposite 

the sum of the two given times and that of the time of reversion ; multiply their dif- 
ference by the annuity, and the product is the present worth. 

Example. — What is the present worth of a reversion of a lease of $40 per annum, 
to continue for 6 years, but not to commence until the end of 2 years, allowing 6 per 
cent, to the purchaser ? 

64-2 = 8 years= 6.20979 

2 " = 1.S3339 

4.37640X40= 175.05.6 dollars. 
For Half-yearly and Quarterly payments, multiply the annuity for the given time 
bv the amount in the following table : 



Rate per 
Cent. 


Half -yearly. 


Quarterly. 


Rate per 
Cent. 


Half-yearly. 


Quarterly. 


3 

t 


1.007445 
1.00S675 
1.009902 
1.011126 
1.012348 


1.011181 
1.013031 
1.014877 
1.016720 
1.018559 


$ 
* 


1.013567 
1.014781 
1.015993 
1,017204 


1.020395 
1.022257 
1.024055 
1.0258S0 



Example. — What will an annuity of $50, payable yearly, amount to in 4 years, at 
5 per cent., and what if payable half yearly ? 
By table, page 58, l.C5* = 1.2155. 

1.2155 — 1 -r- (1.05 — 1 ) = 4.31, and 4.31 X50 = 215.50 dollars for yearly payment, 
and 215.50x1,012343, as by above table = 218. 16 " " half-yearly" 



PERPETUITIES. — COMBINATION. 6 1 

PERPETUITIES. 

Perpetuities are such Annuities as continue forever. 

To Ascertain, tlie Value of a Perpetual Ann-aity. 

Rule.— Divide the annuity by the rate per cent., and multiply the quotient by the 
unit in the preceding table. 

Example. — What is the present worth of an annuity for $100, payable semi-an- 
nually, at 5 per cent. ? 
100 -T-.05 =2.000, and 2.000x1.012348, from preceding table = 2.024.70 dollars. 

For Perpetuities in Reversion, subtract the present worth of the annuity for the 
time of reversion from the worth of the annuity, to commence immediately. 

Example. — What is the present worth of an estate of $50 per annum, at 5 per 
cent., to commence in 4 years ? 

50 -f- . 05 . =1000 

$50, for 4 years, at 5 per cent. =3.54595 (from table) X50 = 177.2975 

822.7U25 dollars, 
which in 4 years, at 5 per cent, compound interest, would produce $1000. 



COMBINATION. 

Combination is a rule for ascertaining how often a less number of numbers or 
things can be chosen varied from a greater, or how many different collections may 
be formed without regard to the order of each collection. 

The combinations of any number of things signify the different collections which 
may be formed of their quantities, without regard to the order of their arrangement. 

Thus the 3 letters, cr, 6, c, taken all together, form but one combination, abc. 
Taken two and two, they form 3 combinations, as ab, ac, be. 

Rule.— Multiply together the natural series 1, 2, 3, etc., up to the number to be 
taken at a time. Take a series of as many terms, decreasing by 1, from the number 
out of which the combination is to be made, ascertain their continued product, and 
divide this last product by the former. 

Example. — How many combinations may be made of 7 letters out of 12 ? 
1X2X3X4X5X6X7 5040 j, 39916S0 i- 

n» and -RK77r- = ^2. 



12X11X10X9X8X7X6 3991660' 5040 

"When two !N*nm"bers or Tilings are Combined. 

Rule.— Multiply together the natural series 1, 2, 3, etc., to a less term than the 
number of the combinations ; ascertain their continued product, and proceed as before. 

Note The class of the combination is determined by the number of elements or 

things to be taken ; if two are taken the combination is of the 2d class, and so on. 

Example. — There are 3 cards in a box, out of which 2 are to be drawn in a re- 
quired order. Here there are 2 terms ; hence 2 — 1 = 1, and - — - = -■ = 6 -r- 1 = 0. 

Combination, without Repetitions. 

Rule. — From the number of terms of the series subtract the number of the class 
cf the combination, less 1 ; multiply this remainder by the successive increasing terme 
of the series, up to the last term of the series; then divide this product by the num- 
ber of permutations of the terms, denoted by the class of the combination. 

Example. — How many combinations can be made of 4 letters out of 10, excluding 
any repetition of them in any second combination ? 

10 — (4 — 1 ) = 7 = number of terms — number of class, less 1. 
7x8x9x10 = 5040 = prod, of remainder, and the successive terms up to the la>t term. 

1x2x3x4 = 24= per mutations of the class of the combination. Then -— -^ = 210. 

F 



62 PROBABILITY. 

Combinations witli Repetitions. 

In this case the repetition of a term is considered a new combination. Thus j, 35, 
admits of but one combination, if not repeated ; if repeated, however, it admits of 
three combinations, as 1, 1 ; 1, 2 ; 2, 2. 

Rule. — To the number of the terms of the series add the number of the class of 
the combination, less 1 ; multiply the sum by the successive decreasing terms of the 
series, down to the last term of the series ; then divide this product by the number 
of permutations of the terms, denoted by the class of the combination. 

Example. — How many different combinations of numbers of 6 figures can be made 
out of 11 ? 

11 -f- (6 — 1 ) = 1G = sum of number of terms, and the number of class, less 1. 
16x15x14x13x12x11 — 5705760 —prod, of sum, and successive terms to last term. 
1x2x3x4x5x6 = 120 = permutations of the class of the combination. Then 

"Variations -with. Repetitions. 

Every different arrangement of individual number or things, including repetitions, 
is termed a Variation. 

The Class of the Variation is denoted by the number of individual things taken at 
a time. 

Rule — Raise the number denoting the individual things to a power, the exponent 
of which is the number expressing the class of the variation. 

Examine — How many variations with 4 repetitions can be made out of 5 figures? 
5^ = 625. 






PROBABILITY. 

The Probability of any event is the ratio of the favorable cases to all the cases 
vhich are similarly circumstanced with regard to the occurrence. Thus, from a re- 
ceptacle containing 1 white and 2 black balls, the probability of drawing a white ball, 
by the abstraction of 1, is J ; the probability of throwing ace with a die is ^ : in 
other words, the odds are 2 to 1 against the first, and 5 to 1 against the second. 

If ra-f-n = the whole number of chances, m represents the number which are fa- 
vorable and n the unfavorable. — ■ — = the probability of the event, 
m-f-n 

Illustration. — If a cent is thrown twice into the air, the probability of its falling 
with its head up, twice in succession, is as 1 to 4. Thus, it may fall : 

1 . The head up twice in succession. 

2. The head up the 1st time and the wreath the 2d time. 

3. The wreath up the 1st time and the head the 2d time. 

4. The wreath up twice in succession. 

The^e are the only results possible, and being all similarly circumstanced as to 
probability, the probability of eacli case is as 1 to 4, or the odds are as 3 to 1. 

The probability of either the head or wreath being up twice in succession is as 1 to 
1, or the chances are even, because the 1st and 4th cases favor such a result; the 
probability of the head once and the wreath once in any order is as 1 to 2, because 
the 2d and 3d cases favor such a result; and the probability of the head or the wreath 
once is as 3 to 4, or the odds are as 3 to 1, because the 1st, 2d, and 3d, or the 2d, 3d, 
and 4th cases, favor such a result. 

Note. — 1 to 2 is an equal chance, for 1 out of 2 chances =1 to 1, being an equal 
chance ; again, 1 to 5 is 4 to 1, for 1 out of 5 chances is 1 to 4. 

Illu8. 2. — Suppose with two bags, one containing 5 white balls and 2 black, and 
the other 7 white and 3 black. The number of cases possible in one drawing from 
each bag is (5 -|- 2) X (7 + 3) = 7x10 =70, because every ball in one bag may be 
drawn alike to one in the other. 

The number of cases which favor the drawing of a white ball from both bags is 
5x7 = 35, for every one of the 5 white balls in one bag may be drawn in combina- 
tion with every one of the 7 in the other. For a like cause, the number of cases 
which form the drawing of a white ball from the 1st bag and a black one from the 
2d is 5X3 = 15 ; a black ball from the 1st bag and a white ball from the 2d is 7x2 
= 14 ; and a black ball from both is 3x2 = 5. 



PROBABILITY. 63 

Therefore the probability of drawing is as 

5x7 35 1 

-— - X =-= s = 1 to 1, a white ball from both bags. 
70 70 JL 

5x3 15 3 

— - = — = — = 11 to 3, a white ball from the 1st, and a black from the 2d. 

7x2 14 1 

— -= — - = - =4 to 1, a black ball from the 1st, and a white from the 2d. 

3x2 6 3 

-=rr- = — = — = 32 to 3, a black ball from both. 

5x3+2x7 29 

= = — = 41 to 29, a white ball from one, and a black from the other, 

1,3 29 
for both the 2d and third cases favor this result ; hence K + -7 = =r. 

5 14 70 

5x7 + 5x3 + 2x7 64 32 

Yo ~70~35~ 33 t0 32 ' at lcaSt ° ne whife * aM for the lst ' 2Jl 

13 1 32 
and 3d cases form this result ; hence 5 + -T7 + 5 = 35* 

Again, if the number of white and black balls in each bag are the same, say 5 
white and 2 black, 5+ 2x5 + 2 = 49, then the probability of drawing is as 
5x5 25 
—77—— -77c = 25 to 24, a white ball from both. 

49 49 ' 

5x2 10 

— — - = — =39 to 10, a white ball from the 1st, and a black from the 2d. 

49 49 
2x5 10 
—+ = — =39 to 10, a black ball from the 1st, and a white from the 2d. 

49 49 
2x2 4 
— — -= — = 45 to 4, a black ball from both. 

49 49 

Illus. 3. — When two dice are thrown, the probability that the sum of the num- 
bers on the upper sides is any given number, say 7, is as follows : 

As every one of the six numbers on one of the dice may come up alike to or in 
combination with the other, the number of throws is 6x6 = 36. 

(land 6] 
The number 7 may be a combination of < 2 u 5 > ; and as these numbers may 

(3 « 4) 
be upon either dice, there are 3x2 = 6 throws in favor of the combination of 7; 

6 1 
hence the probability of throwing 7 is -^-.= -, or as 5 to 1. 

Illus. 4.— The probability of a player's partner at Whist holding a given card is 
as follows : 
The number of cards held by the other 3 players is 3x13 = 39; the probability, 

therefore, that it is held "by the partner is — , but it may be one of the 13 cards which 

o9 

13 1 
he holds ; hence the probability is 1 Xl3 = — = -, or as 2 to 1. 

39 3 

Illus. 5. — The probability of a player's partner at Whist holding two given cards 
is as follows: 

1x2 2 
The number of combinations of 39 things, taken 2 and 2 together, is -- — -- ==7. 1 

therefore the probability that these 2 cards are any given 2 cards in the partner's 

ii2*_L. { 1 

hand is 39 x 38 = = — - = 740 to 1 ; but they may be any 2 cards in the part- 

- — - 39X19 741 

ner's hand ; therefore, since the number of combinations of 13 cards, taken 2 and 2 

1x2 2 78 2 

together, is — — — = — - = 78, the probability required is -^r— rr, or as 17 to 2. 

22 
Similarly, the probability that he holds any 3 given cards is as-^-, or4\s CS1 to 22. 

703 



64 PKOBABILITY. 

The probabilities at a game of Whist upon the following points are : 
7 to 5, that one hand has two honors, and two hands one ; 
13 to 2, that two hands have each two honors; 
17 to 2, that each hand holds an honor; 
5 to 1, that one hand has three honors, and one hand one; 
94 to 1, that the four ho?iors are held by one hand. 

Illus. 6 — If 5 half dollars are thrown into the air, the probability of any of the 
possible combinations of their falling is as follows : 

Hence the probabilities are : 

Q) =.03125=1 to 31 5 heads; 

5X Q *= .15625 = 5 to 27 4 heads and 1 tail ; 

10x U\ =.3125 =5 to 11 Z heads and 2 tails; 

10 X Q = - 3125 = 5 to 11 2 heads and 3 tails; 

5x (h 5 = .15625 = 5 to 27 1 head and 4 tails; 

= .03125 = 1 to 31 6 tails. 



All Wagers are founded upon the principle of the product of the event, and the 
contingent gain being equal to the amount at stake. 

Illustration — Suppose 3 horses, A, B, and C, are started for a race, and I have 
wagered 12 to 5 against A, 11 to 6 against B, and 10 to 7 against C. 

If A wins, I win 6 + 7 — 12 = 1. 

"B " I " 5 + 7-11 = 1. 

«C " I u 5 + 6-10 = 1. 
Hence I win 1, whichever horse wins, from having taken the field against each 
horse at the odds named. 

A are 5 to 12") ( ^f in favor of A, 

The odds given in \ „ ,, ~ u 1n f ; the corresponding ) 6 ., _, 

*- * i B b 11 T orobabilitv is \ rf B » 



favor of ) D _ 1L { probability is U? 



c " 7 " io; iJ* « c, 



and 4 + — + — = — = 1-06 = LOG to 1 in favor of the taker of the odds. 

The odds given upon the first seven favorite horses entered for the Oaks Stakes of 
1S2S were so great that the probability in favor of the taker of the odds when reduced 
was as follows : 
T . ,, an . OM / 1. 5 to 2 3. 4tol 5. 14tol 7. 15 to 1 

The odds* ere | 2 . 5 t0 2 4. 7 to 1 6. 14 to 1 

/4X3X16 = 192 

2,2,1.1.1.1,1 4,5.3 4,1,3 Ux7xl6 = 112 

= 7 + 7 + 5 + 8 + r5 + T5 + rG^7 + l^ + i6 = 7 + 3 + i6 = )^l!Xj = _5? 



S67 



7x3x16 33« 



== 5^-j = 1.092 = 1.092 to 1, in favor of the taker of the odds, yet neither of the horses 
upon which these odds were given won. 

Illus. 2.— If the odds are 3 to 1 against a horse running a race, and 6 to 1 against 
another horse winning a second race, the probability of the 1st horse winning is ^, 
and of the other i Therefore the probability of both the races being won is ^g, 
and the odds against it 27 to 1, or 1000 to 37.037. The odds upon such an event were 
given in 1S2S at 1000 to 60, or 16.67 to 1. 



CHRONOLOGY, 



65 



Tatole showing tlie 0<ids toetweeia Results or 
Chances, and. "between any Number and. the 
Whole Number, at the various Odds against each, 
also the Value of each Chance in parts of IOC 



Odds against 


Value of 


Odds against 


Value of 


Odds against 
each. 


Value of 


each. 


Chance. 


each. 




Chance. 


Chance. 


Even 


50. 


3^ to 


1 


22.22 i 


9>£ to 1 - 


9.52 


11 to 10 


47.62 i 


4 u 


1 i 


20. 


10 " 1 


9.09 


6 " 5 


45.45 


4^ " 


1 


1S.18 i 


12 " 1 


7.7 


5 " 4 


41.44 


5 " 


1 


10.66 


15 » I 


6.25 


5% " 4 


42.1 


5# " 


1 


15.38 


18 " 1 


5.26 


6 u 4 


40. 


6 " 


1 


14. 2S 


20 u 1 


4.76 


6X " 4 


38.1 


6^ " 


1 


13.33 i 


25 " 1 


3.S4 


7 " 4 


30.30 


7 u 


1 


12.5 l 


30 " 1 ■ 


3.22 


7X u 4 


34.78 ! 


ly 2 « 


1 


11.76 i 


40 " 1 


2.44 


2 " 1 


33.33 


8 " 


1 


11.11 


50 u 1 


1.96 


*% " 1 


28.57 


8% " 


1 


10.52 


60 " 1 


1.64 


3 " 1 


25. 


9 " 


1 


10. 


100 " 1 


.99 



Operation.— Divide 100, or the unit, as the case may be, by the sum of the odds, 
and multiply the quotient by the lesser chance or odds. 

100 
Illustration. — 6 to 4. 6 -f- 4 = 10, and -^r x4 = 40 = value of chance. 



CHRONOLOGY. 

A Solar day is measured by the rotation of the earth upon It3 axi3 with respect to 
the Sun. 

The motion of the Earth, on account -of the ellipticity of its orbit and of the pertur- 
bations produced by the planets, is subject to an acceleration .and retardation. To 
correct this fluctuation, time-pieces are adjusted to an average or mean solar day 
(mean time), which is divided into hours, minutes, and seconds. 

In Astronomical computation and in Nautical time the day commences at M., and 
in the former it is counted throughout the 24 hours. 

In Civil computation the day commences at midnight, or A.M., and is divided into 
two portions of 12 hours each. 

A Solar Year, termed also an Equinoctial, Tropical, Civil, or Calendar Year, is the 
time in which the Sun returns from one Vernal Equinox to another; and its average 
time, termed a Mean Solar Year, is 365.24224 solar days, or 365 days, 5 hours, 4S 
minutes, and 49.536 seconds. 

A Year is divided into 12 calendar months, or 365 days. 

A Calendar Month varies from 2S to 31 days. 

A Mean Lunar Month,* or lunation of the moon, is 29 days, 12 hour3, 44 minutes, 
2 seconds, and 5.24 thirds. 

A Bissextile or Leap Year consists of 3G6 days ; the correction of on3 year in four 
is termed the Julian; hence a mean Julian year is 3S5.25 days. 

In the year 1582 the error of the Julian computation of a year had amounted to a 
period of 10 days, which, by order of Pope Gregory VIII., was suppressed in the Cal- 
endar, and the 5th of October reckoned as the 15th. 

The error of the Julian computation, .00776 day?, is about 1 day in 12S.79 yea-s, 
and the adoption of this period as a basis of intercalation is termed the Gregorian 
Calendar, or New Style,* the Julian Calendar being termed the Old Style. 

The error of the Gregorian year (365.2425 days) amounts to 1 day in 3571.4283 
years. 

The New Style was adopted in England in 1752 by reckoning the 3d of September 
as the 14th. 

By an English law, the years 1800, 1900, 2100, 2200, 2300, and 2500, and any other 
100th year, excepting only every 400th year, commencing at 2000, are not to be reck, 
oned Bissextile years. 

* Ferguson. f Now adopted in every Christian country except Russia. 

0* 



66 CHRONOLOGY. 

The Dominical or Sunday Letter is one of the first seven letters of the alphabet 
and is used for the purpose of determining the day of the week correspondin/to any 
given date. In the Ecclesiastical Calendar the letter A is placed opposite to the 1st 
day of the year, January 1st ; B to the second; and so on through the seven letter* • 
then the letter which falls opposite to the first Sunday in the year will also tall on! 
posite to every following Sunday in that year. 

In the Ecclesiastical Year the intercalary day is reckoned upon the 24th of Feb- 
ruary ; hence the 24th and 25th days are denoted by the same letter, the dominical 
letter being set back one place. ' 

In the Civil Year the intercalary day is added at the end of February, the change 
of letter taking place at the 1st of March. "'»"ge 

To Compute tlie Dominical Letter. 
Rule — Divide the number of centuries and the years of the given century each 
by 4, and the years again by 7 ; multiply the remainders respectively bv 2, 2 and 4 • 
add together the three products, and increase their sum by 1 ; then divide the whole 
sum by 7, and the remainder will be the ordinal number of the dominical letter re- 
quired. 

Note — If remain, it will be the 7th, or G-. 

Example.— What will be the dominical letter for the year 1942 ? 

Centuries i = 19, which -^ 4 = 4, and 3 rem. ; years = 42, which -4- 4 = 10, and 2 
rem. ; and 42 again by t = 6, hence the remainders = 3.2.0. 

Then, 3x2 = 6; 2 X 2=4, and 0x4 = 0, and 6+4+0+1 = 11, and 11-^-7 = 4 re- 
mainder, the ordinal number of the required letter being D. 

Note.— In bissextile years two dominical letters are used, one before and the other 
after the intercalary day. 

A Dominical Cycle is a period of 400 years, when the same order of dominical let- 
ters and days of the week will return. 

X Cycle of the Sun, or the Sunday Cycle, is the 2S years before the same order of 
dominical letters return to the same days of the month, and it is considered as hav- 
ing commenced 9 years before the era of the Julian Calendar. 

To Compute the Cj-cle of tlie Sun. 

EuLE.-Add 9 to the given year; divide the sum bv 2S ; the quotient is the num- 
ber of cycles that have elapsed, and the remainder is the number or years of the 

Note.— Tlie use of this computation is the determination of the dominical letter 
for any given year of the Julian Calendar for each of the 2S years of a cycle. 

By the adoption of the Gregorian Calendar, the order of the letters is necessarily 
interrupted by the suppression of the century bissextile years in 1900, 2100. 2*00 
etc., and a table of dominical letters must necessarily be reconstructed for the fol- 
owing century. 

Tlie Lxinar Cycle, or Golden Xumber, is a period of 19 vears. after which the new 
□oons fall on the ?arae days of tlie month of the Julian year, within 1.5 hour?. 
The year of the birth of Jesus Christ is reckoned the first of the Lunar Cycle. 

To Compute tlie Lunar Cycle, or Golden ISTximl^er. 

Rule.— Add 1 to the given year ; divide the sum by 19, and the remainder is the 
golden number. 

Note. — If remain, it is 19. 

Example.— What is the golden number for 1SC6? 

1866 + 1 -^- 19 = 9S, and the remainder = 5= the golden number. 

Note.— There are two objections to the permanency of this lunar period. In the 
first place, the Julian year, which is the basis of the computation, does not correctly 
^T*/?^?^! the - mean S0lar or the Gre S° rian year. In the second place, a pe- 
riod of OQ.jO 75 days is nearly 1.5 hours in excess of 235 mean astronomical lnnationV; 
and hence the correct time of the new moon will in successive cycles occur so much 
2arher, and will retrogade a day in every 30S years. 



CHK0N0L0GT. 



67 



The Epact for any year is a number designed to represent the age of the moon on 
the 1st day of January of that year. 

To Compute th.e Epact. 

Rule Multiply the golden number less 1 by 11, divide the product by 80, and 

the remainder is the result required. 

Note If remain, it is 29. 

Example.— Required the epact for 1865? 
The golden number — 4. 

4 — 1 X 11 = 33, and 33 -i- 30 = 1 and 3 remainder ; hence 3 j= epact 

To Compute tlie Roman Indictioxi. 

Rule. — Add 3 to the given year ; divide the sum by 15, and the remainder is the 
indiction. 
Note. — If remain, the Indiction is 15. 

The Dionysian Period is a period of 532 years, the product of the lunar and solar 
cycles (19X2S), and it was designed for the purpose of including all the varieties of 
the new moons and dominical letters, so that after every 532 years they were ex- 
pected to recur in the same order. The measure of the lunar eycle, however, not be- 
ing exact, and the Sunday cycle being interrupted at the centenary years that are 
not bissextile, this period is altogether in disuse. * 

129 -f (given year — 1S00) =a year of the Dionysian, extending to 2203. 

The Number of Direction is the number of days that Easter-day occurs after the 
21st of March. 

Easter-day is the first Sunday after the first full moon which occurs upon or next 
after the 21st of March ; and if the full moon occurs upon a Sunday, then Easter-day 
is the Sunday after, and it is ascertained by adding the number of direction to the 
21st of March. It is therefore March N -f 21, or April N — 10. 

Illustration.— If the number of direction is 19, then for March, 19 -J- 21 = 40, and 
40 — 31 = 9 = 9th of April ; 
again for April, 19 — 10 = 9 = 9th of April 

Note. — The moon upon which Easter immediately depends is termed the Paschal 
Moon. 

Full Moon is the 14th day of the moon, that is, 13 days after the preceding day of 
-,he new moon. 



Perpetual Ta"ble for Ascertaining tlie Number of 
Direction, the Epact and. Dominical Letter "being 
given. 



o 




Dominical I 


ETTER. 




OS 

w 




Dominical I 


KTTER. 




ft. 
H 


A 

No. 


B 


c 

No. 


D 

No. 


E 

No. 


F 


G 


A 

No. 


B 

No. 


c 

No. 


D 


E 

No. 


F 

No. 


G 




No. 


No. 


No. 


No. 


No. 





26 


27 


2S 


29 


30 


24 


25 


15 


12 


15 


14 


15 


9 


10 




1 


26 


27 


28 


29 


23 


24 


25 


16 


12 


13 


14 


8 


9 


10 




2 


26 


27 


28 


22 


23 


24 


25 


17 


12 


13 


7 


8 


9 


10 




3 


26 


27 


21 


22 


23 


24 


25 


18 


12 


6 


7 


8 


9 


10 




4 


26 


20 


21 


22 


23 


24 


25 


19 


5 


6 


7 


8 


9 


10 




5 


19 


20 


21 


22 


23 


24 


25 


20 


5 


6 


7 


8 


9 


10 




o 


19 


20 


21 


22 


23 


24 


18 


21 


5 


6 


7 


8 


9 


3 


4 


7 


19 


20 


21 


22 


23 


17 


18 


22 


5 


6 


7 


8 


2 


3 


4 


8 


19 


20 


21 


22 


16 


17 


18 


23 


5 


6 


7 


1 


2 


3 


4 


9 


19 


20 


21 


15 


16 


17 


18 


24 


33 


34 


35 


29 


30 


31 


32 


10 


19 


20 


14 


15 


16 


17 


18 


25 


33 


34 


35 


29 


ao 


31 


32 


11 


19 


13 


14 


15 


16 


17 


18 


26 


33 


34 


28 


29 


30 


31 


32 


12 


12 


18 


14 


15 


16 


17 


18 


27 


33 


27 


28 


:;<> 


30 


31 


83 


13 


12 


13 


14 


15 


16 


17 


11 


28 


26 


27 


28 


29 


30 


31 


32 


14 


12 


13 


14 


15 


16 


10 


11 


29 


26 


27 


23 


29 


30 


31 


25 



68 



CHRONOLOGY. 



Perpetual Tal>le fbr Ascertaining Easter-day, th.e 
Epact and. Dominical Letter being given. 



m 






Dominical L 


ETTER 






s 


A 


B 


C 


D 


E 


F 


G 




Apr. 


Apr. 


Apr. 


Apr. 


Apr. 


Apr. 


Apr. 


u 


16 


17 


18 


19 


20 


14 


15 


1 


16 


17 


IS 


19 


13 


14 


15 


2 


16 


17 


18 


12 


13 


14 


15 


3 


16 


17 


11 


12 


13 


14 


15 


4 


16 


10 


11 


12 


13 


14 


15 


5 


9 


10 


11 


12 


13 


14 


15 


6 


9 


10 


11 


12 


13 


14 


8 


7 


9 


10 


11 


12 


13 


7 


8 


3 


9 


10 


11 


12 


6 


7 


8 


9 


9 


10 


11 


5 


6 


7 


8 


10 


9 


10 


4 


5 


6 


7 


8 


11 


9 


3 


4 


5 


6 


7 


8 


12 


2 


3 


4 


5 


6 


7 


8 


13 


2 


3 


4 


5 


6 


7 
Mar. 


1 


14 


2 


3 


4 


5 


6 

Mar. 


U 


1 


to 


2 


3 


4 


5 


30 


31 


1 



i 




Dominical Lettkb 






— 
& 


A 


B 


c 


D 


E 


F 


G 










Mar. 








1G 


2 


3 


4 

Mar. 


29 


3) 


31 


l 


17 


2 


3 
Mar. 


28 


29 


30 


31 


1 


IS 


2 
Mar. 


27 


2S 


29 


30 


31 


1 


19 


26 


27 


2S 


29 


30 


31 


1 
Mar. 


20 


26 


27 


28 


29 


30 


31 


25 


21 


26 


27 


2S 


29 


30 


24 


25 


22 


26 


27 


28 


29 


23 


24 


25 


23 


26 


27 


£8 


22 


23 


24 


25 




Apr. 


Apr. 


Apr. 


Apr. 


Apr. 


Apr. 


Apr. 


24 


2o 


U 


25 


19 


20 


21 


22 


25 


2^ 


24 


25 


19 


20 


21 


22 


20 


23 


24 


18 


19 


20 


21 


22 


27 


23 


17 


IS 


19 


2) 


21 


22 


28 


16 


17 


18 


19 


20 


21 


22 


29 


16 


17 


13 


19 


20 


21 


15 






The Roman lndiction is a period of 15 years, in use by the Romans. The precise 
time of its adoption is not known beyond the fact that the year 313 A.D. was a first 
year of a Cycle of lndiction. 

The Julian Period is a cycle of 79S0 years, the product of the Lunar and Solar 
Cycles and the lndiction (19X2SX15), andit commences at 4714 years B.C. 
6513 -j- (given year — ISOO) = year of the Julian Period, extending to 3267. 

Note.— If the year of the Julian Period is divided by 19, 2S, 15, or 32, the re- 
mainders will respectively give the Lunar and Solar CycUs, the lndiction, and the 
Year of the Dionysian. * 



Dates of the Day of the Week, corresponding to 
the Day determined by the preceding Table. 

Thus, if Monday is the day determined by the year given, the following dates are 
the Mondays in that year : 



February, 

March, 

November. 


February,* 
August. 


May. 


January, 
October. 


January,* 

April, 

July. 


September, 
December. 


June. 


1 


2 


3 


4 


5 





7 


8 


9 


10 


11 


12 


13 


14 


15 


16 


17 


18 


19 


20 


21 


22 


23 


24 


25 


26 


27 


2S 


29 


30 


31 











Noth.— In leap-year, January and February must be taken in the columns 
marked *. 

To Compute the Moon's Age. 

Rule.— To the day of the month add the epact and number of the month, then 
subtract 29 days 12 hours and 44 minutes (the period of a mean lunation) as often as 
the sum exceeds this period, and the result will give the moon's age. 

Note.— This is an approximate rule, serviceable only when the lunations of the 
moon are not known with precision. 



TABLE OF EPACTS, DOMINICAL LETTERS, ETC. 



69 



d. h. 



1 IT 

4 



CTuxribers of tlxe Month.. 

d h. 

July 4 2 

August 5 13 

September . . 7 



April 
May. 
June. 



d. h. 

1 16 

2 3 

3 14 



October 
November. . 
December . . 



January 

February 

March 

Example. — Required the age of the Moon on the 5th of February, 1S59. 

Given day 5 ) 

Epact 26 > 32d. 177*., from which subtract 29.12. — 2 days S 

Number of month . . 1.17 j 



Table of E pacts, 13oininical Letters, and. an. Alma- 
nac, from 1773 to 19G1. 







Dom. 


*j 






Dom. 


.j 






Dom. 


*j 


Years. 


Days. 


Let- 
ters. 


S 


Years. 


Days. 


Let- 
ters. 


05 


Years. 


Days. 


Let- 
ten. 


§ 

H 


1773 


Tuesday. 


c 


6 


1S16 


Friday.* 


GF 


1 


1859 


Tuesday. 


B 


26 


1774 


Wednesd. 


B 


17 


1817 


Saturday. 


E 


12 


1860. 


Thursday-* 


AG 


7 


1775 


Thursday. 


A 


2S 


1818 


Sunday. 


D 


23 


1861 


Friday. 


F 


18 


1776 


Friday.* 


GF 


9 


1819 


Monday. 


C 


4 


1862 


Saturday. 


E 


29 


1777 


Saturday. 


E 


20 


1820 


Wednesd.* 


BA 


15 


1863 


Sunday. 


D 


11 


1778 


Sunday. 


D 


1 


1821 


Thursday. 


G 


26 


1864 


Tuesday.* 


CB 


22 


1779 


Monday. 


C 


12 


1S22 


Friday. 


F 


7 


1865 


Wednesd. 


A 


3 


1780 


Wednesd.* 


BA 


23 


1823 


Saturday. 


E 


18 


1866 


Thursday. 


G 


14 


1781 


Thursday. 


G 


4 


1824 


Monday.* 


DC 


29 


1867 


Friday. 


F 


25 


1782 


Friday. 


F 


15 


1825 


Tuesday. 


B 


11 


1868 


Sunday.* 


ED 


6 


17S3 


Saturday. 


E 


26 


1S26 


Wednesd. 


A 


22' 


1S69 


Monday. 


C 


17 


1784 


Monday.* 


DC 


7 


1827 


Thursday. 


G 


3 


1870 


Tuesday. 


B 


28 


1785 


Tuesday. 


B 


18 


1828 


Saturday.* 


FE 


14 


1871 


Wednesd. 


A 


9 


17S6 


Wednesd. 


A 


29 


1829 


Sunday. 


D 


25 


1872 


Friday.* 


GF 


20 


1787 


Thursday. 


G 


11 


1830 


Monday. 


C 


6 


1873 


Saturday. 


E 


1 


178S 


Saturday.* 


FE 


22 


1831 


Tuesday. 


B 


17 


1874 


Sunday. 


D 


12 


17S9 


Sunday. 


D 


3 


1832 


Thursday.* 


AG 


28 


1875 


Monday. 


C 


23 


1790 


Monday. 


C 


14 


1833 


Friday. 


F 


9 


1876 


Wednesd.* 


BA 


4 


1791 


Tuesday. 


B 


25 


1834 


Saturday. 


E 


20 


1877 


Thursday. 


G 


15 


1702 


Thursday * 


AG 


6 


1835 


Sunday. 


D 


1 


1878 


Friday. 


F 


26 


1793 


Friday. 


F 


17 


1S36 


Tuesday.* 


CB 


12 


1879 


Saturday. 


E 


7 


1794 


Saturday. 


E 


2S 


1S37 


Wednesd. 


A 


23 


1880 


Monday.* 


DC 


18 


1795 


Sunday. 


D 


9 


1838 


Thursday. 


G 


4 


1881 


Tuesday. 


B 


29 


1796 


Tuesday.* 


CB 


20 


1839 


Friday. 


F 


15 


1S82 


Wednesd. 


A 


11 


1797 


Wednesd. 


A 


1 


1840 


Sunday.* 


ED 


26 


1883 


Thursday. 


G 


22 


1798 


Thursday. 


G 


12 


1841 


Monday. 


C 


7 


1884 


Saturday.* 


FE 


3 


1799 


Friday. 


F 


23 


1842 


Tuesday. 


B 


18 


1885 


Sunday. 


D 


14 


1S00 


Saturday. 


E 


4 


1843 


Wednesd. 


A 


29 


1886 


Monday. 


C 


25 


1801 


Sunday. 


D 


15 


1844 


Friday.* 


GF 


11 


1887 


Tuesday. 


B 


6 


1802 


Monday. 


C 


26 


1845 


Saturday. 


E 


22 


1888 


Thursday-* 


AG 


17 


18)3 


Tuesday. 


B 


7 


1846 


Sunday. 


D 


3 


1889 


Friday. 


F 


28 


1804 


Thursday* 


AG 


18 


1847 


Monday. 


C 


14 


1890 


Saturday. 


E 


9 


1805 


Friday. 


F 


20 


1848 


Wednesd.* 


BA 


25 


1891 


Sunday. 


D 


20 


1806 


Saturday. 


E 


11 


1849 


Thursday. 


G 


6 


1892 


Tuesday.* 


CB 


1 


1S07 


Sunday. 


D 


22 


1850 


Friday. 


F 


17 


1893 


Wednesd. 


A 


12 


1808 


Tuesday.* 


CB 


3 


1851 


Saturday. 


E 


28 


1894 


Thursday. 


G 


23 


1809 


Wednesd. 


A 


14 


1852 


Monday.* 


DC 


9 


1895 


Friday. 


F 


4 


1810 


Thursday. 


G 


25 


1853 


Tuesday. 


B 


20 


1896 


Sunday.* 


ED 


15 


1811 


Friday. 


F 


6 


1S54 


Wednesd. 


A 


1 


1897 


Monday. 


C 


26 


1S12 


Sunday.* 


ED 


17 


1855 


Thursday. 


G 


12 


1898 


Tuesday. 


B 


7 


1813 


Monday. 


C 


28 


1856 


Saturday.* 


FE 


23 


1899 


Wednesd. 


A 


IS 


1814 


Tuesday. 


B 


9 


1857 


Sunday. 


D 


4 


1900 


Thursday. 


G 


29 


1S15 


Wedne.sd. 


A 


20 


1858 


Monday. 


C 


15 


1901 


Friday. 


F 


10 



Use of the Taule. — To ascertain the day of the week on which any given day 
of the month falls In any year from 1773 to 1901. 

Illustration. — The great fire occurred in New York on the 16th of December, 
18. ,5 : what was the day of the week ? 

Opposite 1S35 is Sunday; and by the preceding table, under December, it is ascer- 
tained that the 13th was Sunday : consequently^ the IGth ivas Wednesday. 



70 



moon's age, new moon, etc. 



Tahle showing the Age of* the Moon on the Day 
preceding the first Day of a Month for the Years 

1878 to 1883, at 12 M., at New York. 

[By S. H. Weight, AM., Ph.D.] 

Month. 1878. | 1879. 1880. 



January ... 
February. . . 

March 

April 

May 

June 

July 

August 

September . 

October 

November. . 
December . . 



26 18 46 

25 2 32 

26 8 27 

27 21 24 
27 19 47 
29 3 57 

4 15 
1 19 12 

3 11 8 

4 3 2 

5 17 45 

6 7 35 



January, 1SS3, 21d. lh. 26m. 



D. H. M. 

7 19 17 
9 5 

7 12 5S 

8 19 56 

9 3 

10 10 44 
10 20 18 

12 7 45 

13 20 55 

14 11 9 
16 1 44 
16 17 41 



D. H. M. 

18 5 34 

19 18 13 

19 5 53 

20 16 16 

21 1 40 

22 10 19 
22 IS 48 

24 3 32 

25 13 16 

26 15 
26 12 4 
28 2S 



H. M. 

2 56 

16 13 

5 23 
IS 12 

6 10 

17 6 

3 14 
11 43 
20 7 

4 52 
14 8 

22 



D. H. M. 

10 11 49 
12 24 
10 14 
12 4 21 
12 19 6 
14 9 15 
14 22 26 

16 10 4 

17 19 48 

18 3 43 

19 10 44 
19 17 36 



Tahle showing the Times or Periods of New 
Moons at New York for the Years 1878 to 1882. 



Month. 




1878. 


1879. 


1S80. 




1881. 




1882. 




D. 


H. M. 


D. H. M. 


D. H. M. 


i>. 


H. M. 


H. M. 


Jan.... 


3 


9 23 a.m. 


22 7 a.m. 


11 5 47 p.m. 


29 


7 47 p.m. 19 11 36 a.m. 


Feb.... 


2 


3 33 " 


20 11 2 p.m. 


10 6 24 a.m. 


23 


6 37 a.m. J 17 10 p.m. 


March . 


3 


2 36 p.m. 


22 4 4 " 


10 7 44 p.m. 


29 


5 43 p.m. 19 


7 39 a.m. 


April . . 


2 


4 13 " 


21 9 A.M. 


9 10 20 a.m. 


2S 


5 50 a.m. 


17 


4 59 p.m. 


May . . . 


30 


8 3 A.M. 

9 4 r.M. 


21 1 16 " 


9 1 41 " 


27 


6 54 p.m. 


17 


2 45 a.m. 


June .. 


7 45 a.m. 


19 3 42 p.m. 


7 5 12 p.m. 


2G 


8 46 a.m. 


15 


1 34 p.m. 


July... 


29 


4 48 p.m. 


19 4 15 a.m. 


7 8 23 a.m. 


26 


12 " 


15 


1 56 A.M. 


August 


28 


52 a.m. 


17 3 5 p.m. 


5 10 44 p.m. 


24 


3 53 p.m. 13 


4 12 p.m. 


Sept... 


26 


8 5S " 


16 51 a.m. 


4 11 45 a.m. 


23 


7 S a.m. 12 


S 12 A.M. 


Oct 


25 


6 15 p.m. 


15 10 16 " 


3 11 56 p.m. 


22 


9 52 p m. 12 


1 16 " 


Nov . . . 


24 


4 25 a.m. 


13 6 19 p.m. 


2 11 32 a.m. 


21 11 3S a.m. 


10 


C 24 p.m. 


Dec. . . . 


23 


4 42 p.m. 


13 6 26 a.m. 


/ 1 10 13 p.m. 
\31 9 4 a.m. 


21 


11 " 


10 10 34 A.M. 



January, 1SS3, 9d. lh. 5m. A.M. 



To Compute the Age of the Moon at Meridian on 
any Day in the Years 1878 to 1882. 

Rule. — Look in the table for the year, and under it, in the line op- 
posite to the given month, is given a period which, when added to the 
day of the month, will give the age of the moon, or the period since it 
was new, on the meridian of New York. 

When, however, the sum of this period and the day of the month 
exceeds the period of the actual lunation, subtract the period of the lu- 
nation therefrom ; and if the remainder yet exceeds the lunation, sub- 
tract a like period, and the remainder will give the age in days, hours, 
and minutes. 

Note. — When the Period of an. actual Lunation can not be d-termined, subtract 
29 tkfl/s^ 12 hour*, 41 min. f which id the period of a mean lunation ; and proceed as 
for an actual lunation. 



MOOD'S AGE AND LUNATIONS. 71 

Example.— Required the age of the moon in New York on the 22d of May, 18T9, 
at 12 M. 

Under 1S79, and opposite to May, in the table preceding (p. 70), is 9d. 3/?,, to which 
add 22 days = 31 days 3 hours; from which subtract the period of a mean lunation 
= (31d. 3/i.) — (29c?. 12h. 44m.) = Id. 14/*. 16m., the age required. 

By Actual Lunation, as by the following Rule, the age is as follows: 

(Time of new moon for May, 1879, 2\d. lh. 16m. A.M.) + 30 days in the preceding 
month, April = (51c7. lh. lGm.); from which subtract the time of the preceding new 
moon (table, p. 70), (21cf. 9Jh. A.M.) = 29d. 16h. 16m,., the moon's actual age when new 
in May. Hence (bit/. 3/<) — (29d. 16h. 16m.) = Id. lOh. 44m., the moon's age at 12 M. 
on 22d May. 

To Compute the Period of Lunation. 
Rule. — From the sum of the time of the new moon at which a re- 
quired lunation ends, and the number of days in the preceding month, 
subtract the time of the preceding new moon, and the result will give 
the period required. 

Note If the minuend is in P.M., and subtrahend in A.M., add 12 hours to the 

result ; and if the minuend is in A.M., and the subtrahend in P.M., subtract 12 hours 
from the result. 

A Mean Lunation is 29.53053S7 days = 29 days 12 hours 44 min. 2 sec. and 5.24 
primes; but a True Lunation varies with every moon; hence, when the sum of the 
epact and day of the month exceeds 29c/. 12/?. 44m., the age of the moon will not be 
ascertained with precision unless the lunation under computation happens to be the 
exact length of the mean. But all epacts and days, the sum of which is less than 
29cL 12h. 44m., can be ascertained by the use of the table and rule with exactness. 

Thus, on the 9th of January, 1864, the moon's age at New York was 29ct 20h. 33m. 
— 29cf. 12/*. 44771. z=7h. 49m. ; but 9/?. 11m. was the exact age, and that lunation was 
29c/. lift. 22m. Again, on the 8th of January, the day preceding, the moon's age was 
28i. 20/i. 33m. at 12 M., the exact time given by a like table and rule. 

Example. — Required the period of lunation ending 20th February, 1S79. 

Time of nev moon 20cZ. llh. 2m. P.M. (table, p. 70) + 31 days in the preceding 
month, January = (51c/. 11/?. 2m.) — (22d. lh. A.M.), time of preceding new moon 
(table, p. 70) = 29c/. 4/?. 2m. ; to which add 12 hours for period from P.M. to A M. 
= 29t/. 16/?. 2m., the period required. 

To Compute trie -A.ge of trie Moon at Mean Noon 
at any otlier Location than at IN"ew York, or that 
given in the Ta"ble. 

Rule. — Ascertain the age as per preceding rule, and add or subtract 
the difference of longitude or time, according as the place may be west 
or east of it, to or from the time determined by the rule. 

Example. — What will be the age of the moon at Cincinnati, Ohio, on the 1st of 
January, 1S79 ? 

Difference of longitude = 41 min. 58 sec. ; hence, by table and rule page 70, age of 
the moon at New York on that day at 12 M. is 7 days 19 hours 17 min. ; to which 
add 41 min. 58 sec. —7 days 19 hours 58 min. 58 sec. 

Note — The time of the moon's Southing, or of its age, when ascertained for New 
York, will answer approximately for any part of the Atlantic coast, as it crosses the 
different meridians at the same relative time that it does at New York, and does not, 
therefore, need a correction for any difference of longitude when the longitude is not 
very remote. 

The moon's Southing, as usually given in the United States Almanacs, both Civil 
and Nautical, is computed for Washington ; but as the time-table (p. 90) is computed 
for the meridian of New York, this location is given in the preceding tables, for the 
purpose of maintaining a uniformity of expression. 

When the time of new moon is ascertained for a location, and it is 
required to ascertain it for any other, add the difference of longitude 
or time of the place, if east, and subtract it if it is west of it. 



72 TIDES. ,> 

To Compute tlie Time of Higri-water next after 
th.e Moon's Transit, or Southing, at Different Lo- 
cations, without the aid of a Xantical Almanac. 

Rule. — From the table (p. 70) note the period of time under the 
year, and opposite to the month for which the time of the high- water is 
required. 

To this number add the day of the month, and subtract the period 
of the actual lunation, if it can be determined, and if not, 29tf. 12A. 
44m. (a mean lunation) from the sum, when it exceeds the period of 
the lunation used. 

Opposite to this, the age of the moon, in the left-hand column of the 
following table, note the hours and minutes in the adjoining column, 
which add to the time of high-water of the given place, on the days 
of the full and change of the moon, or the establishment of the Port (for 
which see following table), and subtract \2h. 26m., or 2±h. 52m. (a lunar 
day), whenever the result exceeds either of these times, and the re- 
mainder will give the time of high-water required. 

Example. — Required the time of high-water at New York, January 30th, 1379, 
next after the moon's transit. 

In table, opposite to January and under :. 197». 7 m., which being added 

to 30. the day oi the month = 37d. ldh. 1m. — 291 I4h. 7m. (the actual lunation end- 
ing with the new moon in January, 1S79) = Sd. 5A. = the moon's age on that day at 
n M. 

Opposite to Si. in the following table (p. 73) is 61. 44m. P.M., and the difference 
between that and S.5i. or i'2h. — 25m. ; hence St 5A. — 3*1 = 5/?., and as 12h. : 25m. 
: : 5/». : 10.4m., which added. to Qh. 44m. =z&h. 54 + m. ; which added to SA. 13m. (the 
establishment of the Port, or the time of high-water at the full and change of the 
moon at Governor's Island, New York) = 15A. 7m., from which subtract 12A. 26m. 
= 2h.41m. P.M. 

When the moon souths at an hour ichich, when added to the establishment of the 
T . >crs after M,, then the first tide that succeeds this southing of the 

moon occurs in the A.M. of the ensuing day. 

When the time of the tide preceding this is required; from the sum of the moon's 
soothing, and establishment of the Port, subtract 12A. 26m., and the remainder will 
give the time of high-water on that day. 

lLLrsTRATio>r. — On the 9th of June, 1S64, the time of the moon's southing at x 
York, was 4A. 12m. P.M., to which add SA. 13m. for the establishment of the Port, 
and the sum is 12h. 25m., from which subtract 12A., and the difference of 25 is 25/m. 
. :' the next day, the 10th. 

When the time of high-icaier, folloiring the meridian passage of the moon, ex- 
ceeds 12 P.M., it is in the A.M. of the following day. 

Thus, if the moon souths at New York, 41. 12m. P.M. 
Establishment of the 1 . S 13 

IS 25 = 12 hour^ 25 min.from M. of one 
day*, or 25 min, A.M. of thefoUoicing dai'. 

When the time of high-icater preceding the moon % s southing is required, subtract 
from the time obtained as above V2h. 26m., the half of a lunar day, or contrariwise 
if the half of a lunar day exceeds the time. 
Thus, 12A. 26m. 

Time as above, 12 25 

— 1, or = 1 min. before M. of the day. 
Ttw. — The time for a tide being ascertained, that of the next succeeding is as 
certained by the addition of 12 hours 26 minutes. 

To Compute trie Time of High-water "by trie Aid 
of ttie American Xau.tical Almanac. 

Rule. — Ascertain the time of transit of the moon for Greenwich, 
preceding the time of the high-water required. 



TIDES. MOONS SOUTHING. 



73 



For any other location (west of Greenwich), multiply the time in 
the column " difY. for one hour" by the longitude of the location west 
of Greenwich, expressed in hours, and add the product to the time of 
transit. 

Note It is frequently necessary to take the transit for the preceding astronom- 
ical day, as the latter does not end until noon of the day under computation. 

Example. — Required the time of high-water at New York on the £5th of August, 
1864. • 

Longitude of New York from Greenwich = 4h. 56m. 3 sec, which ,- multiplied by 
2.17 min., the difference for 1 hour = 10.71 min. for the correction to be added to the 
time of transit, to obtain the time of transit at New York. 

Time of transit, 187*. 38.8m. ; then 187i. 38.8m. + 10.71m. = 18 hours 49.5 min. 

Time of transit at New York, 24tf. 187*. 50m. 

Establishment of the Port, 8 13 

25d. 37*. 3m. r= time of high-water. 

Note. — The time of the 25th at 37i. 3m. Astronomical computation == 25th at 37i. 
3m. P.M. Civil Time. 



To Compute tlie Time of High-water at tlie Full 
and. Ch.an.ge of* tlie Moon, tlie Time of High-wa- 
ter and. the Age of the Moon on any Day being 
given. 

Rule. — Note the age of the moon, and opposite to it, in the last col- 
umn of the following table, take the time, which subtract from the time 
of high- water at this age of the moon, added to 12h. 26m., or 24/i. 52m. 
as the case may require (when the sum to be subtracted is the great- 
est), and the remainder is the time required. 

Example. — What is the time of high-water at the full and change of the moon at 
New York? 

The time of high- water at Governor's Island on the 25th of Jan. , 1864, was 97i. 
20m. A.M. civil time. The age of the moon at 12 M. on that day was 16d. 87i. 59m. 

Opposite to 16 days, in the following table, is 137i. 28m., and the difference between 
16d. and 16d. 127i. (16.5) is 25m.; hence, if 127i.=25m., 16d. 8h. 59m.— 16d.=Sh. 
59m.=18.7 or 19m., which, added to 137i. 28m.=137i. 47m. 

Then 97i. 20ra.-f 127*. 26m. (as the sum to be subtracted is greater than the time) 
— 13/i. 4Tm.=217i. 46/n.— 13/i. 47m.=77i. 59m. 

This is a difference of but 14 minutes from the establishment of the Port. 

Table showing the Time after apparent N"oon be* 
fore the Moon next passes the Meridian, the -A.ge 
at ISToon "being given. (S.H. Wkigut, A.M., Ph.D.) 



Age of 
Moon. 


Moon at 


Age of 


Moon at 


Age of 
Moon. 


Moon at 


Age of 


Moon at 


Age of 


Moon at 


Merid'n. 


Moon. 


Merid'n. 


Meridian. 


Moon. 


Meridian. 


Moon. 


Meridian. 


Days. 


H. M. 


Days. 


H. M. 


Days. 


H. M. 


Days. 


H. M. 


Days. 


H. M. 




P.M. 




P.M. 




P.M. 




A.M. 




A.M. 


.0 





6 


5 03 


12 


10 06 


18 


15 OS 


24 


20 11 


.5 


25 


6.5 


5 28 


12.5 


10 31 


18.5 


15 34 


24.5 


20 37 


1 


50 


7 


5 53 


13 


10 56 


19 


15 59 


25 


21 02 


1.5 


1 16 


7.5 


6 19 


13.5 


11 21 


19.5 


16 24 


25.5 


21 27 


2 


1 41 


8 


6 44 


14 


11 47 
A.M. 


20 


16 49 


26 


21 52 


2.5 


2 06 


8.5 


7 09 


14.5 


12 12 


20.5 


17 15 


26.5 


22 17 


3 


2 31 


9 


7 34 


15 


12 37 


21 


17 40 


27 


22 43 


3.5 


2 57 


9.5 


7 59 


15.5 


13 02 


21.5 


18 05 


27.5 


23 OS 


4 


3 22 


10 


8 25 


16 


13 28 


22 


18 30 


2S 


23 33 


4.5 


3 47 


10.5 


8 50 


16.5 


13 53 


22.5 


IS 56 


28.5 


23 58 


5 


4 12 


11 


9 15 


17 


14 28 


23 


19 21 


29 


24 24 


5.5 


4 3S 


11.5 


9 40 


17.5 


14 43 


23.5 


19 46 


29.5 


24 48 



G 



TIDES. 



Tide-TaT^le for tlie Coast of the United States, 

Showing Time of High-water at the Full and Change of the Moon, termed 
the Establishment of the Port, being the Mean Interval between Time 
of Moon's Transit and Time of High-water, 

[By Prof. A. D. Bacbje, U. S. Coast Survey.] 







Rise and 


' 


• 


Rise 


and 




Fall. 


Places and Time. 




Fall. 


Places and Time. 


tio 




be 




G 


t* 






a 


£h 




a, 

B) 


es 

9 






B0 




COAST FROM PORTLAND 








CHESAPEAKE BAY AND 








TO NEW YORK. 


h. m. 


Feet. 


Feet. 


RIVERS. 


h. m. 


Feet. 


Feet. 


Portland Me. 


11 25 


9.9 


7.6 


OldPt. Comfort§. Va. 


8 17 


3.0 


2.0 


Portsmouth ... N. H. 


11 23 


9.9 


7.2 


Point Lookout . . Md. 


12 58 


1.9 


0.7 


Newburyport. . Mass. 


11 22 


9.1 


6.6 


Annapolis u 


17 4 


1.0 


0.8 


Salem " 


11 13 


10.6 


7.6 


Bodkin Light... " 


IS 8 


1.3 


0.8 


Boston Light. . M 


11 12 


10.9 


8.1 


Baltimore M 


IS 59 


1.5 


0.9 


Bo3tonf " 


11 27 


10.3 


8.5 


James R.(CityPt.)Va. 


14 37 


3.0 


2.5 


Nantucket .... u 


12 24 


3.6 


2.6 


Richmond u 


16 53 


3.4 


2.3 


Edgartown. ... 4 


12 16 


2.5 


1.6 










Holmes^ Hole. 4 


11 43 


1.8 


1.3 


COASTS OF N. AND S. 








Tarpaulin Cove " 


8 4 


2.8 


1.8 


CAROLINA, GEORGIA, 








Wood's Hole, n. side. 


7 53 


4.7 


3.1 


AND FLORIDA. 








Wood's Hole, s. side. 


8 34 


2.0 


1.2 


Hatteras Inlet . N. C. 


7 4 


2.2 


1.8 


Bird Isl'd Light, Mass. 


7 50 


5.3 


3.5 


Beaufort M 


7 2G 


3.3 


2.2 


X. Bedford Entrance ) 


7 57 


4.6 


2.8 


Smithv. (C. Fear) " 


7 19 


5.5 


3.8 


(Dumpling Rock) . \ 


Charleston 11 (C. II. ) 


7 26 


6.0 


4.1 


Newport R. I. 


7 45 


4.6 


3.1 


Wharf) . . . S. C. | 






Point Judith ... " 


7 32 


3.7 


2.6 


Fort Pulaski Ga. 


7 20 


S.O 


5.9 


Montauk Point. N.Y. 


8 20 


2.4 


1.8 


Savannah (DryDock ) 


S 13 


7.6 


5.5 


Sandy Hook N. J. 


7 29 


5.6 


4.0 


Wharf) Ga. \ 






New York* N.Y. 


8 13 


5.4 


3.4 


St. Augustine. . . Fla. 


8 21 


4.9 


3.6 










Cape Florida ... 4 


S SI 


l.S 


1.2 


LONG ISLAND SOUND. 








Sand Key " 


8 40 


2.0 


0.6 


Watch Hill .... R. I. 


9 


3.1 


2.4 


Key West " 


9 22 


1.6 


1.0 


Stonington Ct. 


9 7 


32 


2.2 


Tampa Bay 4 


11 21 


l.S 


1.0 


Little Gull Isl'd. N.Y. 


9 38 


2.9 


2.3 


Cedar Keys (Depot ) 


13 15 


3.2 


1.6 


New London Ct. 


9 28 


3.1 


2.1 


Key) Fla. ) 








New Haven u 


11 16 


6.2 


5.2 












11 11 


S.O 


4.7 


WESTERN COAST. 








Oyster Bay N.Y. 


11 7 


9.2 


5.4 


San Diego Gal. 


9 38 


5.0 


2.3 


Sands's Point . . u 


11 13 


8.9 


6.4 


San Pedro 4 


9 39 


4.7 


2.2 


New Rochelle . . u 


11 22 


8.6 


6.6 


Cuyler's Harbor. " 


9 25 


5.1 


2.8 


Throg's Neck... " 


11 20 


9.2 


6.1 


San Luis Obispo. u 


10 8 


4.S 


2.4 










! Monterey 4 


10 22 


4.3 


2.5 


COAST OF NEW JERSEY. 








1 South Farallone. u 


10 37 


4.4 


2.8 


Cold Spr'g Inlet. N. J. 


7 32 


5 4 


3.6 


San Francisco . . " 


12 6 


4.3 


2.8 


Cape May Landing " 


8 19 


6.0 


4.3 


Mare Island (San ) 
Francisco Bay) . . ) 


13 40 


5.2 


4.1 


DELAWARE DAY AND 








Benicia Cal. 


14 10 


5.1 


3.7 


RIVER. 








Ravenswood. ... ** 


12 36 


7.3 


4.9 


Delaware Breakwater 


8 


45 


3.0 


Bodega " 


11 17 


4.7 


2.7 


Higbee's (Cape May) . 


8 33 


6.2 


3.9 


Humboldt Bay . " 


12 2 


55 


3.5 


Egg Isl'd Light. N.J. 


9 4 


7.0 


5.1 


Astoria Or. 


12 42 


7.4 


46 


Mahon's River . . Del. 


9 52 


6.9 


5.0 


Nee-ah Harb'r,Wash. 


12 33 


7.4 


4.8 


Newcastle " 


11 53 


6.9 


6.6 


I Port Townshend " 


3 49 


5.5 


4.0 


Philadelpliia Pa. 


13 44 


6.8 


5.1 


Semi-ah-moo Bay u 


4 50 


6.6 


4.8 



Note.— The mean interval has been increased 12 hours 26 minutes (half a mean 
lunar day) for some of the ports in Delaware River and Chesapeake Bay, so as to 
give the succession of times from the mouth ; hence 12 hours 26 minutes is to be 
subtracted from the establishments which are greater than that time, in order to gir« 
the interval required. 



TIDES. 



75 



Bench. Marks referred, to in preceding Table. 

t Boston. — The top of the wall or quay, at the entrance to the diy-dock in the 
Charlestown navy-yard, 14.76 feet above mean low-water. 

t New Yokk. — The lower edge of a straight line, cut in a stone wall, at the head 
of the wooden wharf on Governor's Island, 14.56 feet above mean low-water. 

§ Old Point Comfort, Va. — A line cut in the wall of the light-house, one foot 
from the ground, on the southwest side, 11. feet above mean low-water. 

U Charleston, S. C The outer and lower edge of embrasure of gun No. 3, at 

Castle Pinckney, 10.13 feet above mean low-water. 

Establishment of the Port for several Places not 
in.clu.ded in th.e preceding Table. 





Rise 


Time. 


and 




Fall. 


H. M. 


Feet. 


3 30 


1 


8 15 


5 


12 


60 


11 


12 


11 


25 


11 30 


11 


11 30 


6 


9 1 


5 





Rise 


Time. 


and 




Fall 


H. M. 


Ft. 


7 51 


6 


11 30 


15 


9 34 


5 


7 30 


9 


9 35 


6 


2 30 


2 


8 25 


5 


12 


30 



Albany N. Y. 

Amboy N. J. 

Bay ofFundy N. S. 

Blue Hill Bay. 

Campo Bello Me. 

Cape Ann u 

Cape Cod Mass. 

Cape Hatteras N. C. 



Cape Henry Va. 

Eastport Me. 

Egg Harbor,. N. J. 

Halifax N. S. 

Hell Gate N. Y. 

Kingston Jam. 

Providence R. I. 

St. Johns N. S. 



Rise and Fall of Tides at several Places in th.e 
Grnlf of Mexico. 



Places. 


a 


a 


S3 


Places. 


S3 

05 


S" 


(5. 

OS 




1 


02 


fc 




"JS 


72 


5z 




Feet. 


Feet. 


Feet. 




Feet. 


Feet. 


Feet. 


St. George's Island. .Fla. 


1.1 


l.S 


.6 


Isle Derniere La. 


1.4 


1.2 


.7 


Pensacola " 


1.0 


1.5 


.4 


Entrance to Lake Cal- i 


1.5 


1.1 


.6 


Fort Morgan (Mobile ) 


1.0 


1.5 


.4 


casieu La. \ 


Bay) Ala. j 


Galveston Texas 


1.1 


1.6 


.8 


Cat Island Miss. 


1.3 


1.9 


.6 


Aransas Pass .... " 


1.1 


1.8 


.6 


Southwest Pass La. 


1.1 


1.4 


.5 


Brazos Santiago. . " 


.9 


1.2 


.5 



Establisliinent of the Port for several Places in 
Europe, etc. 

Port. | Country. [ Time. Port. Country. I Time. 



Amsterdam 

Antwerp 

Beachy Head 

Belfast 

Bordeaux 

Bologne 

Bremen 

Brest Harbor 

Bristol 

Cadiz 

Calais 

Calf of Man 

Cape St. Vincent . 

Chatham 

Cherbourg 

Clear Cape 

Cork Harbor 

Cowes 

Dover Pier 

Dublin Bar 



Netherlands . . 



England. 
Ireland. . 
France . . 



Netherlands . 

England .... 

Spain 

France 

St. Geo. Channel 

Spain 

England . . 
France . . . 
Ireland... 



Isle of Wight . . 

England 

Ireland 



Time. 


h. m. 


3 


4 25 


11 50 


10 43 


6 50 


11 25 


6 


3 47 


7 21 


1 40 


11 49 


11 5 


2 30 


1 2 


7 49 


4 


5 1 


10 46 


11 12 


11 12[ 



Fnnchal 

Gravesend 

Greenock 

Havre-de-Grace . . 

Holyhead 

Hull 

Lisbon 

Liverpool 

London Bridge . . . 

Nassau 

Newcastle 

Pembroke Dk.-y'd 

Quebec 

Portsmouth D.-y'd 

Rye Bay 

Sierra Leone 

Southampton .... 
Thames R., mouth 
Waterford Harbor 
Woolwich 



Madeira 

England 

W. C. of Scotl'd 

France 

Wales 

England 

Portugal 

England 

River Thames . 
New Providence 

England 

Wales 

Canada 

England 



Africa . . . 
England. 



Ireland. . 
England. 



h. m. 
11 30 

1 14 

8 
9 51 

10 11 

6 29 

2 GO 

11 16 
2 7 

7 30 

1 22 
6 12 
8 

11 41 
11 20 

8 15 
11 40 
12 

6 6 

2 15 



76 



TIDES. 



To Approximate to the Time ^*Khich. lias elapsed from 
Low or High. Water, by knowing the Rise or Fall of 
the Tide in the Interval. 

If the proportion of the rise and fall in a given time were the same in the differ- 
ent ports, this could easily be shown in a single table, giving the proportional rise 
and fall. The proportion, however, is not the same in different ports, nor in the 
same port for tides of different heights. 

The following table shows the relation between the heights above low-water for 

each half hour, for New York and for Old Point Comfort, and for spring and neap 

tides, at each place. Units express the total rise of high- water above low-water, and 

the figures opposite to each half hour denote the proportional fall of the tide from 

' high-water onward to low water. 

Table to Ascertain the Rise and. Fall of* a Tide for 
any Given Time from High or Low Water, 

Giving the Height of the Tide above Low -water for each Half Hour before 
or after High-wate?\ the Total Range being taken as Equal to 1. 



Time before 


New York 


Old Pt. Comfort 


Time before 


New York. 


Old Pt Comfort. 


or after 
Hieh-water. 






or after 
High water 






Spring. 


Neap. 


Spring. 


Neap. 


Spring 


Neap. 


Spring. 


Neap. 


h m 










h m 










_ 


1. 


1. 


1. 


1. 


3 30 


.49 


.31 


.49 


.44 


30 


.68 


.9S 


.98 


.98 


4 


.39 


.19 


.37 


.34 


1 


.94 


.93 


.95 


.94 


4 20 


.28 


.10 


.26 


.22 


1 30 


.89 


.86 


.88 


.87 


5 


.18 


.02 


.17 


.13 


2 


.SO 


.72 


.80 


.78 


5 SO 


.09 


_ 


.08 


.05 


2 30 


.72 


.59 


.70 


.68 


6 


.05 


_ 


.03 


.01 


3 


.60 


.45 


.59 


.57 


6 30 




- 


- 


- 



Spring tides occur about 2 days after the full and change of the moon, and Reaps 
two days after the first and last quarter. 

Illustration.— At New York, 3 hours after high-water, a spring tide has fallen .6 
(.60) of the whole fall. Suppose the whole rise and fall of that day to be 5.4 feet, 
then 3 hours after high- water the tide will have fallen 3.24 feet, or 3 feet 3 inches 
nearly. Conversely, if a spring tide has fallen 3 feet 3 inches, we know that high- 
water has passed about 3 hours. 

Tides oftlie G-ulf of Mexico. 

On the coast of Florida, from Cape Florida around to St. George's Island, near Cape 
San Bias, the tides are of the ordinary kind, but with a large daily inequality. From 
St. George's Island, Apalachicola entrance, to Derniere Isle, the tides are usually of 
the single-day class, ebbing and flowing but once in 24 (lunar) hours. At Calcasieu 
entrance the double tides reappear, and except for some days about the period of the 
moon's greatest declination, the tides are double at Galveston, Texas. At Aransas 
and Brazos Santiago the single-day tides are as perfectly well marked as at St 
George's, Pensacola, Fort Morgan, Cat Island, and the mouths of the Mississippi. 
For some 3 to 5 day?, however, about the time when the moon's declination is noth- 
ing, there are generally two tides at all these places in the 24 hours, the rise and 
fall being quite small. 

The highest high and lowest low waters occur when the greatest declination of the 
moon happens at full or change. The least tides when the moon's declination is 
nothing at the first or last quarter. 

Tides of tlie IPacific Coast. 

On the Pacific coast there is, as a general rule, one large and one small tide during 
each day, the heights of two successive high-waters occurring, one A.M., and the 
other P.M. of the same 24 hours, and the intervals from the next preceding transit 
of the moon are very different. These inequalities depend upon the moon's declina- 
tion. When the moon's declination is nothing, they disappear, and when it is the 



ELEMENTARY BODIES AND ANALYSIS OF SUBSTANCES. *7*i 



greatest, either north or south, they are the greatest. The inequalities for low-water 
are not the same as for high, though they disappear, and have the greatest value at 
nearly the same time. 

When the moon's declination is north, the highest of the two high tides of the 24 
hours occurs at San Francisco, about eleven and a half hours after the moon's south- 
ing (transit) ; and when the declination is south, the lowest of the two high tides oc- 
curs about this interval. 

The lowest of the two low-waters of the day is the one which follows next the 
highest high-water. 

To Convert Ch.exn.ical Formulae into a Mathematical 
Expression. 

Rule. — Multiply together the equivalent and the exponent of each substance, and 
the product will give the proportion in the compound by weight. Divide lUOO'by 
the sum of their products, and multiply this quotient by each of these products, and 
the products will give the respective proportion of each part by weight in 1000. 

Example.— The chemical formulae for alcohol is C±H§02* Required their pro- 
portional parts by weight in 1000? 

C± Carbon =6.1x4 = 24.4) (525.82) 

i/ 6 Hydrogens 1x6= 6 V x21.55^ 129.3 Vby weight. 
2 Oxygen = 8x2 = 16_) ( 344.8 ) 

1000-^46.4 =21.55 199.92 



Elementary Bodies, -with, their Synxbols and. 
Equivalents. 

Body. [Symb. Equiv. Body. Symb.[ Equiv. 



Body. 


Symb. 


Equiv. 


Aluminium 


Al 


13.T 


Antimony . 


Sb 


64.6 


Arsenic 


As 


37.7 


Barium . . . 


Ba 


68.6 


Bismuth. . . 


Bi 


71.5 


Boron 


B 


11. 


Bromine . . 


Br 


78.4 


Cadmium. . 


Cd 


55 8 


Calcium. . . 


Ca 


20.5 


Carbon . . . 


C 


6.1 


Cerium . . . 


Ce 


46. 


Chlorine . . 


CI 


35.5 


Chromium . 


Cr 


26.2 


Cobalt 


Co 


29.5 


Columbium 


Ta 


184.8 


Copper 


Cu 


31.7 


Didymium. 


D 


48. 


Erbium . . . 


E 


— 


Fluorine . . 


F 


18.7 


Glucinum . 


G 


6.9 


Gold 


Au 


196.6 



Symb. 


Equiv. 


H 


1. 


I 


126.5 


Ir 


C8.5 


Fe 


28. 


Ln 


48. 


Pb 


103.7 


L 


7. 


Mg 


12.7 


Mn 


26. 


Hg 


200. 


Mo 


47.9 


Ni 


29.5 


Nr 


— 


N 


14.2 


No 





Os 


99.7 


O 


8. 


Pd 


53.3 


Pe 





P 


15.9 


Pt 


98.8 



Hydrogen . . . 

Iodine 

Iridium 

Iron '. . 

Lantanium. . 

Lead 

Lithium 

Magnesium. . 
Manganese. . 

Mercury 

Molybdenum 

Nickel 

Niobium. . . . 

Nitrogen 

Norium 

Osmium 

Oxygen 

Palladium . . 
Pelopium . .. 
Phosphorus . 
Platinum . . . 



Potassium . . 
Rhodium . . . 
Ruthenium . 
Selenium . . . 

Silicon 

Silver » 

Sodium 

Strontium . . 
Sulphur 
Tellurium . . 
Terbium. . . . 
Thorium . . . 

Tin 

Titanium . . . 
Tungsten. . . 
Uranium . . . 
Vanadium . . 

Yttrium 

Zinc 

Zirconium .. 



K 

R 

Ru 

Se 

Si 

Ag 

Na 

Sr 

S 

Te 

Tb 

Th 

Sn 

Ti 

W 

U 

V 

Y 

Zn 

Zr 



39.2 
52.2 
52.1 
40. 
22. 
108.3 
23.5 
43.8 
16.1 
64.2 

60. 

58 9 

24 5 

92. 

60. 

68.5 

32. 

32.3 

34. 



Analysis of certain Organic Substances "by "Weight. 



Sugar 

Starch 

Gum 

Lignin 

Tannin 

Indigo 

Camphor . . . 
Caoutchouc. 
Albumen . . . 

Fibrin 

Casein 

Urea 

Gelatine . . . 



Car- Hydro- Oxy- 
bon. gen. gen. 



42.2 
44.2 
42.7 
52.5 
52.6 
73.3 
73.4 
87.2 
52.9 
53.4 
59.8 
18.9 
47.9 



51.2 
40.1 
50.9 
41.8 
43.6 
10.4 
14.6 

23.9 
19.7 
11.4 
26.2 
27.2 



Nitro- 
gen. 



13.8 
.3 

15.7 
19.9 
21.4 
45.2 
17. 



G* 



Hordein 

Veratrin 

Clnchonin . . . 

Quinine 

Brucine 

Strychnine 

Narcotine 

Morphine 

Oil, Spermaceti 

Castor 

Linseed . . . 

Alcohol 

Atmospheric air 



Car- 


Hydro- 


[ Oxy- 


bon. 


gen. 


gen. 


442 


6.4 


47.6 


66.7 


S.5 


'19.6 


77.8 


7.4 


5.9 


75.8 


7.5 


8.6 


70.9 


6.7 


17.4 


76.4 


6.7 


11.1 


65. 


5.5 


27. 


72.3 


6.4 


16.3 


78. 


11. S 


10.2 


74. 


10.3 


15.7 


76. 


11.3 


12.7 


£2.7 


12.9 


34.4 


— 


— 


77. 



Nitro- 
gen. 



1.8 

5. 

8.9 

8.1 

5. 

58 

2.5 

5. 



78 



FOOD. 



Food. 

HUMAN AND ANIMAL SUSTENANCE. 



Nitrogen. 
Gra. 
200 
180 



Least Quantity of Food required to sustain Life. 

Carbon. 
Grs. 

Adult Man, 4300 
Adult Woman, 3900 

Mean, 4100 190 

These quantities and proportions are contained in about 2 lbs. 2 oz. 
ordinary bakers' bread. 

A man, for his daily sustenance, requires about 1220 grs. nitrogenous 
matter, and bread contains 8.1 per cent, of it. 

Therefore 2 lbs. 2 oz. = 14875 grains X 8. 1 = 1205 grains. 



Nutritive Values of Food in Grains per Pound. 

Food. Carbon. Nitrogen. Food. Carbon. 



Beef. 

Barley Meal 

Bakers' Bread 

Buttermilk 

Bullock's Liver. . . . 
Beer and Porter. . . 

Carrots 

Cheddar Cheese. . . 

Cocoa 

Dry Bacon 

Fat Pork 

Flour, Seconds 

Fresh Butter 

Green Vegetables. . 

Green Bacon 

Indian Meal 

Lard 

Molasses 



1.854 
2.563 
1.975 

387 

934 

274 

. 508 

3.344 

3.934 

5.987 

4.113 

2.700 

6.456 

420 

5.426 

3.016 

4.819 

2.395 



184 

68 

88 

44 

204 

1 

14 

306 

140 

95 

106 

116 

14 

76 
120 



Mutton 

New Milk 

Oat Meal 

Pearl Barley. 

Potatoes 

Parsnips 

Rye Meal , 

Rice 

Red Herrings 

Split Peas , 

Sugar , 

Skimmed Milk... 
Skim Cheese.... 

Suet , 

Salt Butter , 

Turnips 

Whev 

Whitefish 



1.900 

599 

2.831 

2.660 

769 

554 

2.693 

2.732 

1.435 

2.698 

2.955 

438 

1.947 

4.710 

4.585 

263 

154 

871 



Nitrogen. 



189 
44 

136 
91 
22 
12 
86 
68 

217 

248 

43 

483 



13 

13 

195 



Nutritive Equivalents. Computed from the Amount of Nitrogen in the 
Substances when Dried. Human Milk at 1. 

Rice 

Potatoes 

Corn 

Rye 

Radish 

Wheat 

Barley 

Oats." 

Bread, Black 

Bread, White 

Peas 

Lentils 

Haricots 

Beans 

Milk, Cows' 

Egg, Yolk 

Oysters 



.81 

.84 


Cheese 

Eel 


3.31 
4.34 


1 


Mussel 


5.28 


1.06 


Liver, Ox 


5.70 


1.06 


Pigeon 


7.56 


1.19 


Mutton 


7.73 


1.25 


Salmon 


7.76 


1.38 
1.66 


Lamb 

Eirs, White 


8.33 

8.45 


1.42 


Lobster 


8.59 


2.39 


Skate 


8.59 


2.76 


Veal 


8.73 


2.83 


Beef 


8.80 


3.20 


Pork 


8.93 


2.37 


Turbot 


8.98 


3.05 


Ham 


9.10 


3.05 


Herring 


9.14 



FOOD. 



79 



Quantities of Different Foods required to furnish 1220 Grains of 
Nitrogenous Matter. 



Cheese 

Pease 

Meat, lean . 
Fish, White 
Meat, fat... 
Oatmeal... 



Pounds. 

.4 
.7 
.9 
1 

1.3 
1.5 





Pounds. 


Corn Meal 


1.6 


Wheat Flour. . . 


1.7 


Bacon, fat 


1.8 


Bread 


2.1 

2.3 


Rye Meal 


Rice 


2.8 



Barle}- Meal.. . 

Milk 

Potatoes 

Parsnips 

Turnips 

Beer or Porter 



Pounds. 

2.9 

4.2 

8.3 

15.9 

15.9 

158.6 



DIGESTION. 

Time required for Digestion of several Articles of Food. 

(Beaumont, M.D.) 



Apples, sweet and mellow, 
sour and mellow . . 

sour and hard 

Barley, boiled 

Beans, boiled 

Beans and Green Corn, boiled 
Beef, roasted rare 

roasted dry 

Steak, broiled 

boiled 

boil'd, with mustard, etc. 

tendon, boiled 

tendon, fried 

old salted, boiled 

Beets, boiled 

Bread, Corn, baked 

Wheat, baked, fresh . . 

Butter, melted 

Cabbage, crude 

crude, vinegar 

crude, vin'r, boil'd < 

Carrots, boiled 

Cartilage, boiled 

Cheese, old and strong 

Chickens, fricasseed 

Custard, baked 

Duck, roasted < 

Dumplings, Apple, boiled. . . . 
Eggs, boiled hard 

boiled soft 

fried 

uncooked 

whipped, raw 

Fish, Cod or Flounder, fried. . 

Cod, cured, boiled 

Salmon, salt'd and boil'd 
Trout, boiled or fried. . . 

Fowls, boiled or roasted 

Goose, roasted 

Gelatine, boiled 



30 



30 



1 

2 

2 

2 

2 

3 

3 

3 

3 

2 

3 

5 

4 

4 

3 

3 

3 

3 

2 

2 

4 

4 

3 

4 

3 

2 

2 

4 

4 

3 

3 

3 

3 

2 

1 30 

3 30 

2 

4 

1 

4 

3 

2 



Heart, Animal, fried 

Lamb, boiled 

Liver, Beef's, boiled. 

Meat and Vegetables, hashed. 

Milk, boiled or fresh < 

Mutton, roasted 

broiled or boiled. . . . 

Oysters, raw 

roasted 

stewed 

Parsnips, boiled 

Pigs, Sucking, roasted 

Feet, soured, boiled 

Pork, fat and lean, roasted. . . 

recently salted, boiled. 

" " fried . . 

" broiled 

" " raw. 

Potatoes, boiled 

baked 

roasted 

Rice, boiled 

Sago, boiled 

Sausage, Pork, broiled .... 

Soup, Barley 

Beef and Vegetables. 

Chicken 

Mutton or Oyster 

Sponge-cake, baked 

Suet, Beef, boiled 

Mutton, boiled 

Tapioca, boiled 

Tripe, soured 



Turkey, roasted { ^:. 

boiled , 

Turnips, boiled , 

Veal, roasted 

fried 

Brains, boiled 

Venison Steak, broiled , 



80 



FOOD. 



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FOOD. 



81 



Alimentary ^Principles. 

The primary division of Food is into Organic and Inorganic. 

The Organic is sub-divided into Nitrogenous and Non-Nitrogenous ; 
the Inorganic is composed of water and various saline principles. The 
former elements are destined for the growth and maintenance of the 
body, and are termed the "plastic elements of nutrition." The latter 
are designed for undergoing oxidation, and thus become the source of 
heat, and are termed ** elements of respiration," or " Calorifaciant. " 

Although Fat is non-nitrogenous, it is so mixed with nitrogenous matter that ifc 
becomes a nutrient as well as a calorifaciant. 



Note. — 
in height. 



Calorific Powers of* Different Foods. 
Every pound of water raised 1° F. is equivalent to 772 lbs. lifted 1 foot 



Calorific Power and Mechanical Energy of \0 Grains of the following 
Foods, in their Normal Condition, when completely Oxidized in the 
Animal Body. 



Food. 


Water raised 
1° in lbs. 


Pounds raised 
1 foot in height. 


Food. 


"Water raised 
1° in lbs. 


Pounds raised 
1 foot in height. 


Dry Flesh 

Albumen 

Lump Sugar . . 


13.12 

12.85 

8.61 


10.130 
9.920 
6.649 


Arrow Root. . 

Butter 

Beef Fat 


10.06 
18.68 
20.91 


7.776 
14.421 
16.140 



Sugar in "Various Products. 



Sugar, crude 95 

Molasses 77 

Buttermilk 6.4 

Carrots 6.1 

Parsnips 5.8 



(Per Cent.) 

Oatmeal 5.4 

Milk 5.2 

Barley Meal 4.9 

Rye Flour 3.7 

Wheat Bread 3.6 



Potatoes 3.2 

Turnips 3.1 

Peas 2 

Corn Meal 4 

Rice 4 



Volume of Oxygen required to Oxidize 100 parts of the following 
Foods as consumed in the Body : 

Grape Sugar 106 

Starch 120 

Hence, assuming capacity for oxidation as a measure, albumen has 
half the value of fat as a food-producing element, and a greater value 
than either starch or sugar. 



Albumen 150 

Fat • 293 



Relative Value of Various Foods as Productive of* 
Force. 



When Oxidized in the Body. 



Cabbage 1 

Carrots 1.2 

E^g, White 1.4 

Milk 1.5 

Apples 1.5 

Ale 1.8 

Fish 1.9 

Potatoes 2.4 

Porter 2.6 



Veal, lean 2.8 

Mackerel 3.8 

Hani, lean 4 

Bread, crumb 5.1 

Egg, hard boiled.. 5.4 

Egg, Yolk 7.9 

Sugar 8 

Tsinglass 8.7 

Rice 8.9 



Pea Meal 9 

Wheat Flour 9.1 

Arrowroot 9.3 

Oatmeal 9.3 

Cheese 10.4 

Cocoa 16.3 

Butter 17.3 

Fat of Beef. 21.6 

Cod-liver Oil 21.7 



82 



FOOD.— MISCELLANEOUS NOTES. 



Analysis of Various Fruits. 



Fbuit. 


Water. 


Sugar. 


Acid. 


Albumen. 


Pectoug 
Substances. 


Seeds, 
Skin, etc. 


Ash. 


Apple . . 
Pear . . . 
Grape . . 


85 

84 
80 


7.6 

7.4 

13.7 


1 

.1 
1 


.2 
.3 

.8 


2.8 

3.3 

.6 


2.9 
4.6 
3.5 


.5 
.3 

.4 



■ 



MISCELLANEOUS NOTES. 
SHOT. 

Diameter and iNTnnVber of Pellets in an Ounce of 
Lead Snot, American Standard.— [Le Roy & Tatham.] 



TT 

T 

BBB 

BB 

B 



Diameter 
in Inches. 



.21 
.20 
.19 
.18 
.IT 



32 

38 
44 

49 
53 



Diameter 
in Inches. 



.16 
.15 
.14 
.13 
.12 
.11 



82 

98 

121 

149 

209 



No. 


Diameter 
in Inches. 


Pellets. 


7 


.10 


278 


8 


.09 


3T5 


9 


.OS 


560 


10 


.0T 


822 


11 


.06 


982 


12 


.05 


1778 



"Weather-foretelling [Plants [Han^eman.] 

If Rain is imminent. — duckweed * Stellaria media; its flowers 
droop and do not open. Crowfoot anemone, Anemone ranunculoides; 
its blossoms close. Bladder Ketmia, Hibiscus trionum ; its blossoms do 
not. open. Thistle, Carduus acaulis; its flowers close. Clover, Tri- 
folium pratense, and its allied kinds, and Whitlow grass, Draba verna; 
they droop their leaves. Nipple- wort, Lampsana communis; its blos- 
soms will not close for the night. Yellow Bedstraw, Galium verum; 
it swells, and exhales strongly ; and Birch, Betula alba, exhales and 
scents the air. 

Indications of Rain. —Marigold, Calendula pluvialis ; when its flow- 
ers do not open by 7 A.M. Hog Thistle, Sonchus arvensis and olera- 
ceus; when its blossoms open. 

Rain of short duration. — Chickweed, Stellaria media; if its leaves 
open but partially. 

. ^ If cloudy. — Wind flower, or Wood Anemone, Anemone memorasa; 
its flowers droop. 

Termination of Rain. — Clover, Trifolium pratense ; if it contracts its 
leaves. Birdweed and Pimpernel, Convolvulus and Anagallis arvensis; 
if they spread their leaves. 

Uniform Weather. — Marigold, Calendida pluvialis; if its flowers 
open early in the A.M. and remain open until 4 P.M. 

Clear Weather. — Wind-flower, or Wood Anemone, Anemone memo- 
rasa; if it bears its flowers erect. Hog Thistle, Sonchus a?*vcnsis and 
oleraceus; if the heads of its blossoms close at and remain closed dur- 
ing the night. 

* The Chickweed spreads its leaves at 9 A.M., and they remain open until noon. 



MISCELLANEOUS NOTES. 83 

Locomotive Traction and. DR.esistan.ce. 
Formula to ascertain the Traction of a Locomotive, 

d representing diameter of cylinder in inches ; I, length of stroke of piston ; and 
D, diameter of ivheel in feet or inches ; p, the mean pressure in pounds per square 
inch ; and T, the traction in pounds at rails. 

Formula to ascertain the Resistance of a Locomotive and Train upon 
a Level Railway. 

S 2 

S representing speed of train in miles per hour; and R, resistance of each ton 
(2240) of the gross weight in pounds. 

Estimated Consumption of Bituminous Coal per actual 
Horses' Power. 



Type of Engine. 



Improved Compound , 

Ordinary, with surface condenser and superheater., 

Ordinary, Injection 

Non-condensing 



Coal in lbs. 
per hour. 



3.5 

4.5 



Expansion and. Contraction of Building Stones. 

[Lieut. W. H. C. Bartlett, U. S. E.] 

Expansion or Contraction for each Degree of Temperature. 

For One Inch. 

Sandstone 000009532 

Whitepine 00000255 



For One Inch. 

Granite 000004825 

Marble 000005668 



Resistance of* Stones, etc., to the Effects of Freezing. 

Various experiments show that the power of stones, etc., to resist 
the effects of freezing is a fair exponent of that to resist compression. 



To Preserve HVTeat. 

Meat of any kind may be preserved in a temperature of from 80° to 
100°, for a period of ten days, after it has been soaked in a solution of 
1 pint of salt dissolved in 4 gallons of cold water and X gallon of a 
solution of bisulphate of calcium. 

By repeating this process the preservation may be extended by the 
addition of a solution of gelatin or the white of an egg to the salt and 
water. 

Silk. — A thread of silk is the 2500th of an inch in diameter. 
Spider s Thread. — Four miles of a spider's thread weighs one grain. 
Soap Bubble.— The film of a soap bubble is the 2500000th of an 
inch in thickness. 

Gold Leaf is the 280000th part of an inch in thickness. 



84 



MISCELLANEOUS NOTES. 



-A.ir and "Ventilation. 

^An average-sized man will exhale from his lungs and body from .6 to 
. / of a cubic foot of carbonic acid per hour. A lighted oil lamp or two 
candles will furnish the same volume. 

Assuming, then, that there are 4 volumes of carbonic acid in 10 000 
volumes of air, and that a man in a room with a lighted lamp or two 
candles furnishes from 1.2 to 1.4 cubic feet of acid per hour, there will 
be required to maintain the air at the required condition for health for 
SnA rt man ' the allowable pollution of it being 6 volumes in 10 000, fully 
3000 cubic feet of fresh air. By experiments made in Paris it was shown 
that there was required from 2400 to 3120 cubic feet per hour. 

Result of Observations of tlie "Vitiation, of tlie Air. 

[Angus Smith, M.D.] 



Atmosphere 3.2 to 3 4 

City Parks 3.2 to 3.8 

Streets 3.8 to 4.4 

" " in a fog 6. to 6.8 



Theatres, average . 
Offices " 

Workshops " 
Mines " 



8. to 32 
17. to 22 
20. to 30 

78. to 250 



[See continuation, p. 629.] 



Latitude and Longitude of Principal Places and 
Pu/blic Observatories. 

Compiled from the Records of the U. S. Coast Swxey and Topographical 

Engineer Corps, the Imperial Gazetteer, and BowditcKs Navigator. 

Longitude com pitted from the Meridian of Greenwich. 

L. represents Light-house; Ch., Church; S. H., State-Jiouse ; C. H., Custom-house; 

C. S., Coast Survey; and Obs., Observatory. 
N. and S., the divisiom of the Latitude; and E. and \V\, the courses East and West 

of Greenidch. 






Place. 



] Latitude ! Longitude. 



NORTH AND SOUTH 
AMERICA. 

Acapulco Mex. 

Albany N. Y. 

Annapolis Md. 

Ann Arbor Mich. 

Antigua, E.Pt...W.L 

Auburn N.Y. 

Augusta Ga. 

Augusta Me. 

Austin Tex. 

Baltimore, Mon't. . Md. 

Bangor, M. Ch Me. 

Baton Kouge La. 

Benicia Cal. 

Beaufort, Arsen'l, S. C. 
Bellevue, Am. Fur Co. 

Post 

Boston, S.H Mass. 

Boston, L. u 

Balize La. 

Brazos Santiago. .Tex. 

Bridgeport Conn. 

Bristol R.I. 

Brooklyn, X. Yd.. N.Y. 

Brunswick Me. 

Buffalo, L N.Y. 

Burlington N. J. 

Burlington Vt. 



N. 

16 50 : 
42 39 ! 

38 5S 
42 16 

17 05 
42 55 
,33 28 
!44 IS 
30 13 

39 17 
44 4S 
30 26 
38 03 
32 25 



48 



38 OS 24 
42 21 30 

42 19 36 
29 OS 05 
26 06 
41 10 30 

41 40 11 
40 42 

43 54 29 

42 50 

40 04 52 

44 27 



W. 



49 09 
44 40 
29 0C 
43 03 
45 
2S 
54 
50 



97 39 



36 39 
45 42 
IS 

07 13 
41 23 



95 47 46 
71 03 30 

70 53 06 
89 01 04 
97 12 

73 11 04 

71 16 05 

73 58 30 
69 57 24 
78 50 

74 52 37 
78 10 



Place. 



NORTH AND SOUTH 

AMERICA. 

Bath,W. S.Ch....Me. 



43 



Barnegat, L. N. J. 39 



Beaufort N. C. 

Barbadoes, S. Pt.. W. I. 



Buenos Ayres. .Brazil 34 

Cambridge, Obs.. Mass. 42 
Calais, C. S. Sta'n, Me. 45 

Camden S. Q. 34 

Canandaigua N. Y. 42 

Cape Ann, S. L. . Mass.* 42 
CapeCod, L. P. L. . " 42 
Cape Flat'ry, L..W. T. |48' 
Cape Hancock, Colo. R. 46 
Cape Hatteras, L . N. C. 35 

Cape May, L N. J. 38 

Cape Race .'N. S. 46 

Cape Henlop3n,L. Del. 3S 

Cape Fear N. C. 33 

Cape Car n a vera], Fla. 28 
Cape Florida, L. . . " 25 
Caraccas . . Maracaibo 10 



Cape St. Roque, Brazil 
Cape Horn, S. Pt. Her- 
mit's Island 



N. 

54 55 
46 
43 05 
03 
S. 
36 OS 
N. 
22 48 
11 05 
17 

54 09 

38 11 
2 

>23 15 
16 35 
15 02 

55 48 

39 24 
46 06 
48 

27 30 
39 54 
30 



5 28 
55 50 



Longitude. 



w. 

9 48 40 
4 06 
6 39 2S 
9 37 



5S 22 



07 40 
16 30 
33 
17 

34 10 
09 48 
43 54 
01 45 
30 54 
57 IS 
04 3 
04 07 
57 
33 

09 02 
01 30 



35 17 
67 16 



LATITUDE AND LONGITUDE. 



85 



Talkie of Latitude and. Longitude- (Continued). 



Place. | Latitude. 


Longitude. | 


Place. 


Latitude, j 


Longitude 


NORTH AND SOUTH S. 


W. 


NORTH AND SOUTH 


N. 


w. 


AMERICA. o / // 


O y // 


AMERICA. 


, // 


/ // 


Callao, Flag Staff, Peru 


12 4 


77 13 


Harrisburg Penn. 


40 16 


76 50 




N. 




Hartford, S. H..Conn. 


41 45 59 


72 40 45 


Cape Sable N. S. 


43 24 


65 36 


Holmes Hole, Ch., 






Cape Sable, C. S..Flo. 


25 6 53 


81 15 


Mass. 


41 27 13 


70 35 59 


Cape Charles Va. 


37 7 18 


75 57 54 


Huntsville Ala. 


34 36 


S6 57 


Cape Henry, L.... u . 


36 55 30 


76 2 


Hudson N. Y. 


42 14 


73 46 


Cape Breton u 


45 57 


59 4S 5 


Indianapolis Ind. 


39 55 


S6 5 


Castine Me. 


44 22 30 

29 7 27 


68 45 
82 56 12 


Jackson Miss. 

Jalapa Mex. 


32 23 

19 30 8 


90 8 


Cedar Keys, Depot Isl. 


96 54 30 


Charleston, C.Ch. .S.C. 


32 40 44 


79 55 39 


Jefferson City Mo. 


£8 36 


92 8 


Chagres, Centre of Pla- 






Key West, L Flo. 


24 33 


81 47 13 


teau N. G. 


9 20 


80 1 21 


Kingston, C.H..C.W. 


44 8 


76 28 37 


Cheboygan, L. . . Mich. 


45 40 9 


84 24 37 


Kingston Jamaica 


17 5S 


76 46 


Chicago, R. C. Ch..Ill. 


41 53 48 


87 37 47 


Knoxville Tenn. 


35 59 


S3 54 


Cincinnati, Obs. . Ohio 


39 5 54 


84 29 31 


Laguayra . .Maracaibo 


10 36 


67 2 


Charlestown, B. Hill 








S. 




Monument .'. .Mass. 


42 22 36 
10 26 


71 3 18 
75 38 


Lima Peru 


12 3 

N. 


77 6 


Carthagena N. G. 






Cleveland, L Ohio 


41 31 


81 51 


Lancaster Penn. 


40 2 36 


76 20 33 


Columbia, S. H... S.C. 


33 59 57 


SI 1 54 


Lexington Ky. 


38 6 


84 18 


Columbus Ohio 


39 57 


83 3 


Little Rock Ark. 


34 40 


92 12 


Concord, S. H...N. 11. 


43 12 29 


71 29 


Lockport N.Y. 


43 11 


78 46 


Corpus Christi. . .Tex. 


27 47 18 


97 27 2 


Los Angelos Cal. 


34 3 15 


118 10 44 


Council Bluffs, Xeb. T. 


41 30 


95 48 


Louisville Kv. 


38 3 


S5 30 




41 44 34 


124 11 22 


Lowell... St. A.'s Ch., 






Campeachy . .Yucatan 


19 49 


90 33 


Mass. 


42 38 46 


71 19 2 


Dayton Ohio 


39 44 


84 11 


Matamoras Tex 


25 52 50 


97 27 50 


Des Moines Iowa 


41 35 


93 40 


Machias Bay Me. 


44 33 


67 22 


Detroit, St. P. Ch., 






Madison, Dome. .Wis. 


43 4 31 


89 23 ?6 


Mich. 


42 19 46 


83 2 23 


Marblehead, L.. . Mass. 


42 30 14 


70 50 39 


Dover -...Del. 


39 10 


75 30 


Matagorda, C. S. Sta- 






Dover N. H. 


43 13 

15 38 


70 54 
61 26 


tion Tex. 


28 41 29 

19 25 45 


95 57 29 


Dominica,N. P'r, W.I. 


Mexico .Mex. 


99 5 6 


Dubuque Iowa 


42 29 55 


90 39 57 


Macon, Arsn'l Ga. 


32 50 24 


83 37 39 


Eastport, Un. Ch. .Me. 


44 54 10 


00 5S 59 


Milwaukee Mich. 


43 2 24 


87 54 4 


Edenton, C. 11. . . N. C. 


36 3 27 


76 85 4S 


Montgomery, S. II . Ala. 


32 22 46 


86 17 4S 


Erie, L Penn. 


42 8 43 


80 4 12 


Mobile, E. Ch u 


30 41 26 


88 1 29 


Fredericksb'g, E. Ch., 






Montreal C. E. 


45 31 


73 32 56 


Va. 


38 18 6 


77 27 17 


Monterey, C. S. Sta- 






Falls St. Anth'y, Minn. 
Fire Island, L. . . N. Y. 


44 5S 40 
40 37 54 


93 10 30 
73 12 48 


tion Cal. 


36 37 £6 


122 49 31 


Martinico, S. Point, 




Fort Gibson, Ind. Ter. 


£5 47 35 


95 15 10 


W.I. 


14 27 


60 55 


Fort Laramie. . Neb. T. 


42 12 10 


104 47 43 


Montserrat,W. E. Pi'r, 






Fort Leavenworth, Ks. 


39 21 14 


94 44 


W.I. 


16 48 


62 12 


Frankfort Ky. 


3S 14 


84 40 


Maracaibo, Maracaibo 


10 39 


71 45 


Frederick Md. 


3J 24 


77 18 


Monte Video, Ratlsl'd, 


S. 




Frederickton N. B. 


46 3 


66 3S 15 


Brazil 


34 53 


56 13 


Galveston, Oath' 1, Tex. 


29 18 17 


94 46 59 


Mona Island, E. Pier, 


• N. 




Gloucester, E. P. L., 






W.I. 


18 7 


67 47 


Mass. 


42 34 47 


70 39 33 


Matanzas Cuba 


23 3 


81 40 


Guadaloupe,S.W.Pt., 






Nantucket, S. Tower, 






W.I. 


15 57 


61 44 


Mass. 


41 16 54 


70 5 36 


Georgetown Ber. 


32 22 2 


64 37 6 


Nashville, U. . . Tenn. 


36 9 33 


86 49 3 




S. 




Nassau, L N. P. 


25 5 2 


77 21 2 


Guayaquil Quito 


2 13 


79 53 


Natchez Miss. 


31 34 


91 24 42 


Grand Cayman, E. 


N. 




Nebraska, Junction of 






Pier W.I. 


19 20 


81 10 


Forks of Platte Riv. 


41 5 5 


101 21 24 


Havana, Moro. . . Cuba 


23 9 


82 21 23 


New Bedford Mass. 


41 38 10 


70 55 10 


Hole in the Wall, L, 






Newbern N. C. 


35 20 


77 5 


Bahamas. 


25 51 5 


77 10 6 


Newburgh N. J. 


41 31 


74 1 


Halifax, Obs., D.Y'rd, 






Newburyport, E. L., 






N.S. 


44 39 4 


63 35 


Mass. 


42 43 25 


70 48 40 



II 



LATITUDE AND LONGITUDE. 



Table of Latitude and. Longitude— (Continued). 



| Latitude. | Longitude. 



NORTH AND SOUTH 


N. 


W. 


AMERICA. 


O / // 


O y /. 


Newcastle, K.Cli. . Del. 


39 39 36 


75 33 27 


New Haven, Col, Conn. 


41 IS 26 


72 55 25 


New London, P. Cb., 






Conn. 


41 21 16 


72 5 29 


New Orleans, M't. .La. 


29 57 46 


90 2 30 


Newport, L R. I. 


41 29 12 


71 IS 29 


New York, C. II . N. Y. 


40 42 43 


74 3 


Norfolk, F. Bank. .Va. 


36 50 53 


76 18 47 


Norwich Conn. 


41 33 


72 7 


Nantucket, L Mass. 


41 23 24 


70 2 24 


Ocracoke, L N. 0. 


35 6 28 


75 5S 51 


Ogdensburg, L. . .N. Y. 


44 45 


75 30 


Olympia Wash. T. 


47 3 


122 55 


Ottawa C.W. 


45 23 


75 42 


Old Point Comfort, L. . 






Va. 


37 2 


70 IS 6 


Panama, Cath'l, N. G. 


8 57 9 


79 27 17 


Pensacola, Sq're. .Flo. 


30 24 33 


S7 12 4 


Perote Mex. 


19 2S 57 


97 8 15 


Philadelphia, S. H., 






Penn. 


39 56 53 


75 8 42 


Pittsburg Penn. 


40 32 


80 2 


Petersburg, C. H. . Va. 


37 13 47 


77 23 55 


Plattsburg N.Y. 


44 42 


73 26 


Plymouth, C. S. Stat'n, 






Mass. 


41 57 23 


70 39 47 


Point Hudson. .W. T. 


48 T 3 


122 44 3.3 


Portland, C. H....Me. 


43 39 2S 


70 14 5S 


Providence, U. Ch. R. I. 


41 49 26 


71 23 59 


Portsmouth, n.l.N. H. 


43 4 14 


70 42 12 


Puebla de los Angelos, 






Mex. 


19 15 


98 2 21 


Porto Rico, N. E. Pier, 






W. I. 


18 24 


65 39 


Port au Prince. . W. I. 


IS 33 


72 16 3 


Porto Cabello, M'caibd 


10 28 


63 7 


Porto Bello N. G. 


9 34 


79 40 


Prairie du Chien,Wis. 


43 2 


91 8 35 


Quebec, Citadel. .C. E. 


46 49 12 


71 12 15 


Raleigh, Square. . N. C. 


35 46 50 


7S 37 50 


Richmond, Cap. ..Va. 


37 32 16 


77 25 43 


Rochester, Rochr. H., 






N.Y. 


43 S 17 


77 51 


Rio Janeiro, Sugar L'f 


22 56 

N. 
43 55 


43 9 


Sackett's Harb'r, N.Y. 


75 57 


Savannah, Exch. . .Ga. 


32 4 52 


81 5 15 


Sacramento Cal. 


38 34 41 


121 27 44 


St. Augustine Flo. 


29 4S 30 


81 T5 


St. Louis Mo. 


3S 37 2S 


90 15 16 


St. Paul Minn. 


44 52 46 


95 4 54 


Salem, Spire Mass. 


42 31 12 


70 53 36 


Saltillo Mex. 


25 26 22 


101 1 45 


Salt Lnke City... Utah 


40 45 S 


112 6 8 


San Antonio Tex. 


29 25 22 


98 29 15 


San Diego, C.S.O., Cal. 


32 41 58 


117 13 22 


Sanduskv, L Ohio 


41 3> 30 


S2 42 15 


Sandy Hook, L... N.J. 


40 27 42 


73 59 48 


San Francisco, Presi- 






dio Cal. 


37 47 36 


122 26 4S 



Place. 



| Latitude. Longitude. 



NORTH AND SOUTH 
AMERICA. 

San Francisco, C. S. 

Station Cal. 37 

San Louis Obispo. " |35 

San Pedro " 33 

Santa Fe N. Mex. ! 35 

Schenectady N.Y. 1 42 

Syracuse u 143 

Springfield, L 111. 39 

Stoniugton, L. . . Conn. 41 
Sweet Water River, i 

Mouth of. ..Neb. T. 42 

St. Christopher, N. Pt., 

W.I. 

St. Eustatia, Town, 

W.I. 

St. Josephs Mo. 

St. Bartholomew, S. 

Point W.I. 

St. Martins, Fort. " 

St. Croix, Obs " 

St. John's u 

St. Thomas, Fort Ch' n, 
W.I. 

St. Domingo W. I. 

St. Jago de Cuba, En- 
trance W.I. 

St. Vincent's, S. Point, 

W.I. 

Turk's Island, N. Pt. 

G.Turk W.I. 

Tobago, N.E.Pr. " 
Trinidad, Fort . . " 
Tampa Bay,E.Key.Flo. 

Tallahassee u 

Tampico, Bar Mex. 

Taunton, T. C. Ch., 
Mass. 

Toronto C.W. 

Trenton, P. Ch... N.J. 
Troy . ...Un'y. ..N.Y. 

Tuscaloosa . . Ala. 

Utica, Dut.Ch... N.Y. 

Vandalia 111. 

Vera Cruz Mex. 

Victoria Tex. 

Vincennes Ind. 

Valparaiso, Fort, Chili 33 

Washington, Capitol 

West Point, Obs. M. A., 
N.Y. 

Wheeling Va. 

Wilmington, C. II., 
N.C. 

Wilmington, T. H., 
Del. 

Worcester, Ant. II., 
Mass. 

York Penn. 

Yorktown Va 



N. 



W. 



48 122 23 19 
10 38 120 43 31 
43 20 US 16 3 
41 6 100 1 22 



48 
3 
48 
19 36 



42 27 18 


17 24 


17 29 


23 3 13 


17 53 30 


IS 5 


17 44 30 


IS 18 


IS 21 


IS 29 


19 58 


13 9 


21 32 


11 20 


10 39 


27 36 


30 2S 


22 15 30 


41 51 11 


43 33 35 


40 13 10 


42 43 44 


33 12 


43 6 49 


3S 50 


19 11 52 


2S 46 57 


3S 43 


S. 


33 2 


N. 


3S 53 20 


41 23 26 


40 7 


34 14 3 


39 44 27 


42 16 17 


39 5S 


37 13 



73 55 
76 9 16 

59 33 
71 54 

107 45 27 

C2 50 

63 
109 40 44 

62 56 54 

63 3 

64 40 42 
64 42 

64 55 IS 
69 52 

75 52 

61 14 

71 10 

60 27 

61 32 

82 45 15 
84 36 
97 51 51 

71 5 55 

79 23 21 

74 45 30 
73 49 41 

S7 42 

75 13 

S9 



36 



71 41 

77 15 

73 57 1 
SO 42 

77 56 47 

75 32 42 

71 48 13 

76 40 
76 34 



LATITUDE AND LONGITUDE. 



87 



Ta"ble of Latitude and Longitude— (Continued). 



Place. 



I Latitude.] Longitude. 



EC ROPE, ASIA, AFRICA, 
AND THE OCEANS. 

Antwerp 

Alexandria, L 

Archangel 

Athens 

Aleppo 

Algiers, L 

A msterdam 

Borneo, Koads 



Batavia, Obs. 



Bussorah 

Botany Bay, Cape 
Koads 



Barcelona 

Bombay, Flag Staff. . 

Bristol 



Bremen 

Berlin, Obs 

Brussels, Obs 

Benconlen, Fort, Su- 
matra 



Cape Clear , 



St. 



Calais 

Constantinople, 
Sophia 

Cape St. Mary, Mada- 
gascar 



Canton 

Cronstadt 

Copenhagen . 



Cape of G. Hope, Obs. 
Cadiz 



Calcutta 

Christiana 

Corinth 

Cairo. 

Candia 

Ceylon, Pt. Pedro . 



Congo River . 

Dublin 

Dover 



Edinburgh 

Falkland Islands, St. 
Helena, Ob.-= 



Fayal, S. E. Point . . . 

Feejee Group, Ovo 

lau, Obs 



N. 

51 13 • 
31 12 
64 32 
37 53 
36-11 
36 47 

52 22 
5 

S. 
6 8 

N. 
30 30 

S. 
34 2 

N. 
41 23 
IS 56 

51 27 

53 5 

52 30 

50 51 

S. 

3 43 

N. 

51 26 

50 58 

41 1 

S. 
!5 39 

N. 
23 7 
59 59 
55 41 



33 56 

N. 

36 32 



22 34 
59 55 

37 54 
30 3 
35 31 

9 49 

S. 

6 8 

N. 

53 23 

51 8 

55 57 
S. 

15 55 
N. 

38 30 
S. 

17 41 



12 



E. 

O / 

4 24 
29 53 
40 33 
23 44 
37 10 

3 4 

4 53 
115 

106 50 

48 

151 13 

2 11 

72 54 

W. 

2 35 
E. 

5 49 

13 23 45 

4 22 

102 19 

W. 

9 29. 

E. 

1 51 

2S 59 

45 7 

113 14 
29 47 
12 34 

IS 28 45 
W. 

6 IS 
E. 

SS 20 
10 43 
22 52 
31 18 
25 8 
SO 23 

12 9 
W. 
6 20 30 

E. 

1 19 

W. 

3 12 

5 45 

28 42 
K. 

173 53 



Place. 



I Latitude, j Longitude, 



EUROPE, ASIA, AFRICA, 
AMD THE OCEANS. 

Florence |43 46 

Funchal, Madeira . . . 
Greenwich 



Geneva. , 



Gallego Island . 

Glasgow 

Gibraltar 



Genoa . . . 
Honolulu . 



Hood's Island, Marq's 

Hamburg 

Havre 

Jeddo 

Jerusalem ..... 



Liverpool, Obs. 



Leyden 

Leghorn, L. . 



Lisbon . 



Leipsic 

Moscow 

Malta, Valetta. . 
Messina, L. . . . 



Madrid . 
Malaga. 



Mocha 

Muscat 

Marseilles 

Majorca, Castle . 

Manilla 

Madras 



New Zealand, N. Cape 
New Hebrides, Table 

Island 

Niphon, Cape Idron, 

Japan 

Naples, L 

Navigators' Islands, 

Opoun, E. Pier. . . . 
Owhyhee 

Odessa 

Pekin 

Palermo, L 

Paris, Obs 

Prince of Wales Isl'd, 

Torres Strnit 
Porto Praya, Cape 

Verd Islands. 



N. 


O / // 


43 46 


32 OS 


51 2S 3S 


46 11 59 


1 42 


55 52 


36 7 


44 24 


21 19 


S. 


9 26 


N. 


53 33 


49 29 


35 40 


31 48 


53 24 47 


52 9 28 


43 32 


38 42 


51 20 20 


55 40 


35 54 


38 12 


40 25 


36 43 


13 20 


£3 37 


43 IS 


30 34 


14 36 


14 4 9 


S. 


34 24 


15 28 


N. 


34 36 3 


40 50 


S. 


14 9 


20 23 


N. 


46 28 


39 54 


38 S 


48 50 13 


S. 


10 46 


N. 


14 54 



E. 

O / . 

11 16 

W. 

16 56 



E. 

9 15 
W. 

104 5 

4 16 

5 22 
E. 

8 53 

157 52 

158 57 

.9 5S 

6 

140 

•37 20 

W. 

3 

E. 
4 29 15 
10 18 
W. 

9 9 
E. 

12 22 
35 33 

14 30 

15 35 
W. 

3 42 

4 26 
E. 

43 12 
58 35 

5 22 
2 23 

121 2 
80 15 45 

173 1 

167 7 

138 50 35 

14 16 

W. 

169 2 

155 54 

E. 

30 44 

116 2S 

13 22 
2 20 

142 12 

W. 

23 3 



88 



LATITUDE AND LONGITUDE. 



Table of Latitude and. Longitude— {Continued). 

Place. Latitude. Longitude. Place. | Latitude. Longitude. 



Latitude. 


Longitude. 


s. 


E. 


O / // 


O y /, 


3S 51 32 


151 IS 


N. 




41 54 


12 27 


5 54 


4 29 


S. 


W. 


16 30 


155 10 


N. 


E. 


44 37 


33 30 


33 26 


27 7 


14 55 


100 


21 11 


72 47 




W. 


2S 2S 


16 16 


1 17 


103 50 


S. 




33 52 42 


151 23 


N. 


W. 


36 50 


5 53 



EUROPE, ASIA, AFRICA. 
AND TlIE OCEANS. 

Port Jackson 



Rome, St. Peter's. 
Rotterdam 



Scilly Islands . 

Sevastopol 

Smyrna 

Siam 

Surat, Castle.. 



Santa Cruz Ten. 

Singapore 

Sydney 

Seville 



EUROPE, ASIA, AFRICA. 
ANT) THE OCKANB. 

Senegal, Fort 

Sierra Leone 



Suez 

St. Helena . 



Stockholm, Obs. . . 

St. Petersburg 

Toulon 

Tripoli 

Tunis, City 

Tangier 35 

Venice 40 

Vienna I4S 

Warsaw, Obs !52 



N. 

1 " 
S. 
30 
N. 
59 
S. 
55 
X. 

20 31 
56 



07 

54 
47 
47 
50 
13 
13 
S. 
Zanzibar Island.. Sp. | 6 2S 



W. 

16 32 

13 IS 

E. 

32 34 

5 45 

15 6 
30 19 

5 22 

13 11 
10 6 

5 54 

14 26 

16 23 
21 2 

39 33 



Pul>lio and Private Observatories. 

Longitude given in Time. 



Latitude. ] Longitude. 



N. 



W. 



Albany, Dudley. 142 33 49.55 4 54 59.52 

Berlin 5J 

Birr Castle, Earl 
of Rosse 



Brussels 

Cambridge, U. S. 



53 5 47 

50 51 10.7 
42 22 49 



Cambridge 52 12 51.6 

S. 
Cape ofG. Hope. |33 56 3 
Copenhagen, Uni- N. 

verity 55 40 53 



50 19. S 

W. 
25 22 
12 43.6 

E. 

45 3.6 

24 37.7 

W. 

Georgetown, U.S. 3S 54 26.1 5 S 13.15 
Greenwich 51 2S 33 



Dublin 53 23 13 

Edinburgh 55 57 23.2 

Florence 43 40 41.4 

Geneva 46 11 59.4 



53 35.5 

Vv\ 
31 40.9 

E. 

17 23. 9 

W. 

4 44 32 

E. 

22.75 

1 13 55 



Hamburg 53 33 5 

Leipsic 51 20 20.1 

Leyden 52 9 28.2 



E. 

39 54 1 
49 23.5 
17 57.5 



j Latitude. | Longitude. 



Liverpool 



N. 

.53 24 47. S 



Madras 13 4 S.l 

Marseilles 43 17 50 

Moscow 55 45 19. S 

Munich, Bogen 

hausen 4S $i 45 

Naples, Capo di 

Monte 

Palermo 

Paris .- 



Portsmouth. 
Quebec . 



40 51 46.6 
3S 6 41 
43 59 13 



50 48 3 



W. 

h. m. s. 

12 .11 
E. 
5 20 57.3 

21 29 
2 30 16.96 

46 26.5 



56 5S.S6 
53 24.17 
9 20.63 
W. 

4 23.9 
E. 
40 4S 30 4 44 49.02 
Home, College... '41 53 52.2 i 49 54.7 

Stockholm 59 20 31 ,1 12 14. S 

St. Petersburg 

Academy 59 50 29.7 2 1 13.5 
I W. 
Santiago de Chili 33 26 24. S 4 4? IS. 9 

Washington 38 53 39 5 S 12 

Inkivchtsbei-g, E. 

Olmutz 49 35 49 19 .1 

L.M. Kutherfurcl, W. 

New York 40 43 IS. 53 4 55 55.73 
K. 
Sydney 33 51 41.1 10 4 59.S6 



DIFFERENCE IN TIME. 



89 



Table showing the Difference in. the Time at the 
following Places. 



Loncjitude, computed both from New York and Greenwich. 



I New York. \ Greenwich. 



Place. 



| New York. 



Greenwich. 



Acapulco 

Albany 

Alexandria, Fgy't 

Algiers 

Amsterdam 

Antwerp 

Auburn 

Augusta Ga. 

Austin 

Baltimore 

Bangor. 

Barbadoes 

Bath 

Baton Rouge 

Berlin 

Beaufort.. ..N. C. 

Boston S. II. 

Bombay 

Bremen 

Bridgeport 

Brunswick 

Buffalo, L 

Burlington.. N. J. 

Buenos Ay res 

Brooklyn, N. Yard 

Cadiz 

Callao 

Calais, Me. 

Calcutta 

Canton 

Cape Race 

Cairo Egypt 

Cape May 

Cape Horn 

Chicago 

Cincinnati. . . . 

Charleston 

Cleveland .... 
Columbus. . .Ohio 

Concord N. H. 

Charlestown 

Columbia S. C. 

Corpus Christi . . . 
Cape of GoodHope 
Constantinople . . 

Copenhagen 

Dayton 

Detroit 

Dubuque 

Dublin 

Dover N. II. 

Dover Del. 

Eastport 

Edinburgh 

Erie 

Florence 

Fort Leavenworth 
Frederic ksb'g, Va. 
Frederickt'n,N.P>. 
Frankfort Ky. 



h. m. s. 

1 43 ITS. 
1 IF. 

55 34 
5 8 10 
5 15 32 
5 13 36 

9 52 S. 
31 3G 

1 34 36 
10 2T 
S3 57 F. 
57 32 
16 45 

1 9 12S. 
5 49 35F. 

10 38 S. 

11 46F. 
9 47 36 

5 3L 16 

3 16 

16 10 

19 56 S. 

3 30 

1 2 32F. 



4 30 48 


12 52 S. 


26 53F. 


10 49 20 


12 2S 56 


1 23 44 


7 1 12 


3 54S. 


36 56F. 


51 31S. 


41 58 


23 43 


31 24 


36 12 


10 4F. 


11 47 


28 8S. 


1 33 48 


6 9 55F. 


6 51 56 


5 46 16 


40 44 S. 


36 10 


1 6 40 


4 30 38F. 


12 24 


6 S. 


28 4F. 


4 43 12 


24 ITS. 


5 41 4F. 


1 22 56 S. 


13 49 


29 27F. 


42 40 S. 



7i. m. 
6 39 

4 54 
1 59 

12 
19 
17 

5 5 
5 27 
30 

5 6 
4 35 

3 53 

4 3.) 

6 5 
53 

5 6 
4 
4 



44 
51 
35 
4 52 

4 39 

5 15 
4 59 

3 53 

4 55 
25 

5 8 
■4 29 
5 53 
T 32 

3 32 
2 5 

4 59 

4 19 

5 50 
5 37 
5 19 
5 27 
5 32 
4 45 

4 44 

5 24 

6 29 
1 13 
1 55 

50 
5 36 

5 32 

6 2 
25 

4 43 

5 2 

4 W 
12 

5 20 
45 

6 18 
5 9 

4 20 

5 38 



17 S. 

59 

32F. 

16 

32 

36 

52 S. 

36 

36 

27 

3 

2S 
15 
12 

35F. 
3SS. 
14 
36F. 
16 
44S. 
50 
56 
30 
28 
54 
12 
52 

T 
20F. 
56 
16 S. 
12F. 
54S. 

4 
31 
5S 
43 
24 
12 
56 
13 

8 
43 
55F. 
56 
16 
41S. 
10 
40 
22 
36 

56 
48 
17 
4F. 

56 S. 
49 
33 
40 

u 



Ftinchal 

Galveston , 

Genoa 

Geneva 

Georgetown, Ber. 

Gibraltar 

Glasgow 

Greenwich 



Halifax N. B. 

Harrisburg 

Hamburg 

Hartford 

Havana 

Havre 

Hudson 

Huntsville 

Indianapolis 

Jackson 

Jeddo 

Jerusalem 

Jefferson City 

Kingston. . .C.W. 
Kingston .. .Jam. 

Knoxville 

Leghorn 

Lima 

Lisbon 

Liverpool 

Little Rock 

Lexington 

Louisville 

Lowell 

Macon 

Madrid 

Malaga 

Malta 

Manilla 

Marseilles 

Matanzas 

Matagorda 

Matamoras 

Mexico 

Milwaukee 

Mocha 

Mobile 

Montreal 

Monterey. . . .Cal. 

Montgomery 

Moscow 

Monte Video 

Naples 

Natchez 

Nassau, L 

Nantucket, S. Ch. 

Nashville 

New Orleans 

New London 

Newport 

New Bedford 



h. m. s. 

3 48 20F. 
1 23 8S. 
5 31 32F. 
5 20 37- 

37 32 

4 34 32 
4 3S 56 

4 56 



h. m. s. 

1 T 40 
6 19 8 
35 32F. 

24 37 

4 IS 2SS. 

21 28 

IT 4 



41 40 


4 14 20 


11 20S. 


5 7 20 


5 35 52F. 


39 52F. 


5 IT 


4 50 43 S. 


33 253. 


5 29 25 


4 55 36F. 


24 


56 


4 55 4 


51 4SS. 


5 47 48 


4S 20 


5 44 20 


1 4 32 


6 32 


14 16 F. 


9 20 F. 


7 25 20 


2 29 20 


1 12 32S. 


6 8 32S. 


9 54 S. 


5 5 54 


11 4 


5 7 4 


39 36 


5 35 36 


5 37 12F. 


41 12F. 


12 24S. 


5 8 24S. 


4 19 24F. 


30 36 


4 44 


12 


1 12 43S. 


6 8 48 


41 12 


5 37 12 


46 


5 42 


10 44F. 


4 45 16 


33 80S. 


5 34 30 


4 41 12F. 


14 4S 


4 3S 16 


}7 44 


5 54 


5S F. 


13 8 


8 4 8 


5 17 23 


21 23 


30 40S. 


5 26 40 S. 


1 27 49 


6 23 49 


1 33 51 


6 29 51 


1 40 20 


6 36 20 


55 36 


5 51 36 


7 48 48 F. 


2 52 4SF. 


56 68. 


5 52 6S. 


1 49 F. 


4 54 11 


3 11 188. 


8 7 IS 


49 11 


5 45 11 


7 IS 12F. 


2 22 12F. 


1 11 S 


3 44 52 S. 


5 53 4 


57 4F. 


1 9 38 S. 


6 5 3SS, 


13 24 


5 9 24 


15 PSF. 


4 4 ) 22 


51 16S. 


5 47 16 


1 4 10 


6 10 


7 3SF. 


4 43 22 


10 58 


4 45 H 


12 19 


4 43 41 



90 



DIFFERENCE IX TIME. 



Table— (Continued). 



New York, j Greenwich. 



Place. 



New York. Greenwich. 



New York 

Newburg 

Newcastle 

New Haven. . . 

Norfolk 

Norwich , 

Odessa 

Odgensburg . . . 
Old PointComfort 

Ottawa 

Owhyhee 

Paris 

Panama 

Pekin 

Petersburg 

Palermo 

Pittsburg 

Philadelphia 

Plattsburg 

Portland 

Porto Rico 

Portsmouth, N. H. 

Pensacola 

Providence 

Porto Praya 

Prairie du Chien. 

Quebec 

Raleigh 

Rio Janeiro 

Richmond 

Rotterdam 

Rome 

Rochester 

Sackett's Harbor. 

Sacramento 

Santa Cruz, Ten' fe 

Savannah 

San Diego 

Sandusky 

San Antonio 

Salt Lake City . . 

Salem 

San Francisco... 

Santa Fe 

Singapore 

Sydney 

Seville 

Sierra Leone 



6 

4 

9 

7 

6 58 

C 

9 

6 

5 27 

5 5 

21 

12 41 

13 

5 49 

24 

4 

2 

15 
33 
13 
52 
10 
3 23 

1 S 
11 
18 

2 3 
13 

5 13 

5 45 

15 

7 

3 9 

3 50 
28 

2 52 
34 

1 37 

2 32 
12 

8 13 

2 8 

11 51 

5 9 

4 32 
4 2 



4S. 
14 
1SF. 
15S. 
32 F. 
56 

S. 
12 
48 
30 
20F. 
4SS. 
52F. 
36 S. 
28F. 

8S. 
34 
16F. 

1 
24 
12 
4SS. 
24F. 
45 
34 S. 
11F. 
31 S. 
24F. 
43 S. 
56F. 
48 
24 S. 
48 
51 

56F. 
21 S. 
53 
49 
57 
24 
25F. 
47 S. 

5 
20F. 
32 S. 

8F. 
48 



4 56 S. 



4 

14 



4 56 

5 2 

4 51 42 

5 5 15 

4 4S 2S 

2 2 56F. 

5 2 S. 
5 5 12 

5 2 4S 
10 23 36 

9 20F. 
5 17 4SS. 

7 45 52F. 
5 9 36S. 

53 2SF. 
5 20 8S. 
5 34 
4 53 44 
4 40 59 
4 22 36 

4 42 49 

5 4S 4S 
4 45 36 

1 32 12 

6 4 34 

4 44 49 

5 14 31 

2 52 36 
5 9 43 

17 56F. 

49 4S 
5 11 24S. 
5 3 43 

8 5 51 
15 4 
5 24 21 

7 4S 53 

5 30 49 

6 33 57 

7 28 24 

4 43 35 

5 9 47 
7 4 5 

6 55 20F. 
10 5 32 S. 

23 52 



Siam 

Schenectady 

Springfield 111. 

Sandy Hook 

Stonington 

St. Croix 

St. Thomas 

St. John's... W.I. 

St. Domingo 

St. Jago de Cuba . 

Smyrna 

Suez 

St. Petersburg . . . 

Stockholm 

St. Helena 

St. Augustine 

St. Louis 

St. Paul 

St. Joseph Mo. 

Syracuse 

Tampa Bay 

Tallahassee 

Tampico Car 

Taunton 

Toronto 

Toulon 

Tripoli 

Troy 

Trenton 

Tunis 

Tuscaloosa 

Turk's Island 

Utica 

Valparaiso 

Vandalia 

Vera Cruz 

Vincennes 

Venice 

Vienna 

Victoria Tex. 

Warsaw 

Washington, Cap. 
Wilmington, Del. 
Wilmington, N. C. 

Wheeling 

West Joint 

Worcester 

York 



53 12 [ Yorktown. 



.1 G6 F. 

20 
1 2 12S. 
IF. 

8 24 
37. 17 

36 19 

37 12 

16 32 

7 2SS. 

6 44 2SF. 

7 6 16 
6 57 16 
6 S 24 

4 33 

30 20S. 
1 5 1 

1 12 20 

2 22 43 

8 37 
35 1 
42 24 

1 35 27 

ii stf; 

21 33S. 

5 17 28F. 
5 4S 44 

1 17 

3 2S. 
5 36 24F. 

54 4SS. 

11 S0F. 

4 52 S. 

9 16F, 
1 SS. 
1 2S G4 

53 40 

5 53 44F. 

6 1 32 

1 32 4S. 

6 20 9F. 

12 IS. 
6 11 

15 47 
26 48 
12F. 

5 47 
10 40 S. 
10 16 



A. m. ». 
6 40 F. 

4 55 40 S. 

5 58 12 
4 55 59 
4 47 3G 
4 18 43 
4 19 41 
4 18 48 

4 39 £8 

5 3 28 

1 48 2SF. 

2 10 16 
2 1 16 
1 12 24 

23 S. 

5 26 20 

6 11 

6 16 20 

7 18 43 
5 4 37 
5 31 1 

5 3S 24 

6 31 27 

4 44 23 

5 17 33 
21 2SF. 
52 44 

4 54 43S. 

4 59 2 
40 24F. 

5 50 4SS. 

4 44 40 

5 52 

4 46 44 

5 56 S 

24 34 

5 49 40 
57 44F. 

1 5 32 

6 2S 4S. 
1 24 9F. 
5 S 12S. 
5 2 11 

5 11 47 
5 22 4S 
4 55 4S 

4 47 IS 

5 6 40 
5 6 16 



F representing Fast, and S Slow. 



To Ascertain the Difference of* Time between 
New York and. Greenwich and. any Place not 
given in the Table. 

Reduce the longitude of the place to time, and if it is W. of the as- 
sumed meridian it is Slow ; if E., it is Fast. 

If the difference for New York is required, and it exeeds 4k. 56w., 
subtract this sum, and the remainder will give the difference of time, S. ; 
and if it does not exceed it, subtract the difference from it, and the re- 
mainder will give the difference of time, F. 



TRAVELING DISTANCES. 



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a 



SAILING AND RAILROAD DISTANCES. 



95 



Sailing Distances between various Ports of En- 
gland, trie Oanadas, the United. States, etc., etc. 





Miles 




Miles. 




2,463 

383 

2,850 


New York to Panama, v ia C. Horn 

New Orleans to Minatitlan 

44 M Puerto Cabello, 


11 329 


Boston to Halifax 


816 


44 u Liverpool (via Halifax) 




Philadelphia to Liverpool 


3,147 


Honduras 


945 


Cape Bonavista to Cape Spear. . . 
Cape Spear to Cape Race 


76 


Bermudas to Nassau 


S04 


55 


Panama to David Chiriyui 


276 


Cape Race to Liverpool 


1,992 

457 
835 


44 44 San Juan del Sud. . . . 


570 


44 u Halifax 


44 u Gulf of Fonseca 

44 u Acapulco 


739 


11 u Boston 


1,416 
1,724 

2,S97 


" u New York 


1,004 
1,155 


44 u Manzanilla 


" "Philadelphia 

44 u Cape Pine 

St. Johns, N. F., to Quebec 


44 44 San Diego 


19 


44 44 Monterey 


3,19S 


891 


44 u San Francisco 


3,240 


44 "Boston 


890 


San Francisco to San Juan del Sud 


2,6S5 


u u Liverpool 


1,956 


44 u Gulf of Fonseca. 


2,591 


44 u Galway 


1,677 


44 44 Acapulco 


l,S4l 


14 "Bristol . 


1,936 


44 4t Manzanilla 


1,543 


44 u Greenock 


1,848 


" " San Diego 


474 


u " St. Peter's L't. 


183 


44 44 Monterey 


105 


u u Cape Spear . . 


5 


44 44 Humboldt 


200 


" u Cape Race ... 


60 


* " Columbia R.Bar 


530 


44 u C. Bonavista. 


72 


44 44 Vancouver 


638 


New York to Minatitlan 


1,962 


" u Portland 


650 


14 u Puerto Cabello, Hon- 




U U Port Townshend 


732 


duras) 


2,114 


44 " Victoria 


715 



Sailing Distances between various Ports and New 
York and London. 



Not included in the preceding Talk. 





Alexandria . . 
Amsterdam.. 
Barbadoes . . . 

Batavia 

Bermudas . . . 

Bombay 

Boston 

Bremen 

Bristol 

Buenos Ayres 

Cadiz 

Calcutta 

Cape Race. . . 



Miles 


Miles. J 


4893 


2,980 1 


3291 


262 


1S55 


3,812 


S972 


11,492 


6S2 


3,142 


8522 


10,703 


340 


3,084 


3428 


408 


2979 


501 


6010 


6,162 


3125 


1.115 


9350 


11,531 


1004 


2,249 



Cork 

Cowes 

Funchal 

Galway 

Glasgow 

Greenock 

Halifax 

Havana 

HobartTown. 
Kingst'n,Jam 

Lima , 

Madras 

Norfolk 



Miles. 


Miles. 


2,782 


560 


3,092 


200 


2,760 


1,303 


2,720 


721 


2,913 


807 


2,895 


789 


560 


2,706 


1,161 


4,197 


9,187 


11,368 


1,456 


4,305 


10,050 


10,149 


8,707 


10,SS8 


390 


3,447 





Miles 


Miles. 


Pensacola .... 


1623 


4,654 


Philadelphia . . 


227 


3,404 


Quebec 


1360 


3,0S0 


Queenstown.. . 


2780 


551 


Rio Janeiro . . . 


4970 


5,076 


St. Helena 


5096 


3,433 


St. Johns 


1064 


2,214 


Southampton . 


3103 


211 


Swan River.. . 


8480 


10,661 


Teneriffe 


2909 


1,522 


Tortugas 


1151 


4,182 


Venice 


4953 


2,950 


Washington . . 


530 


3,612 



Distances "between several Cities of tne TJ. S. 9 
Not included in Table on page 91. 







Miles. 






Miles. 


New Orleans to Cairo 


548 
183 
100 
134 
147 
479 


St. Louis 
Louisville 

« 

Montgomery 
St. Louis 


to Nashville 

" Montgomery 


319 




44 Jackson 


490 


Memphis 
St. Louis 


" Grenada 


377 


" Little Rock 

" Cairo t 

" Omaha 


" Nashville 


1S5 
163 




" Denison 


5S5 



96 DEPTHS OF WATER ON BARS AND IN CHANNELS. 



Tatole showing tlie least Water in. the Channels of 
certain Harbors, Rivers, and. Anchorages on the 
Coast of the United. States. [U. S. Coast Subyey.] 

ATLANTIC COAST. 



Harbors, ete 



Locations. 



Feet. 



Kennebec River 

Portland Me. 

Portsmouth N. H. 

Newburyport 

Ipswich 

Annisquam . . . 

Gloucester 



Salem , . Mass, 

Boston Mass, 

Plymouth 



Barnstable Harbor. . . 
Newport R. I. 



New York . 



Arthur's Kill. 



Kill von Kull . 



Newark Bay . . 
Hudson River. 



Delaware Bay. 



Delaware River . 



Up to Hanniwell's Point 

Breakwater to anchorage 

Channel off town 

Narrows to the city 

Over Bar 

Over Bar 

Over Bar 

Channel to S. E. Harbor 

Up into inner Harbor 

Southern Ship Channel 

Inside of Salem Neck 

Channel, Lovell's and Gallop's Islands 

Channel, Governor's and Castle Islands 

Up to anchorage 

Anchorage in the Cow Yard 

Over Bar 

Anchorage S. and W. of Goat Island 

Wharves inside of Goat Island 

Newport to Prudence Island 

Mount Hope Bay 

Gedney's Channel 

Swash Channel 

South Channel 

Main Channel 

Ship Channel, after passing S.W. Spit buoy. 

Anchorage Perth Amboy 

* Woodbridge to Kossville 

fRossville to Chelsea 

t Chelsea, Western Channel, to Elizabethport. 

Elizabeth port to Shooter's Island 

Shooter's Island to Bergen Point L. H 

Bergen Point L. H. to New Brighton 

§ Bergen Pt. L. H. to mouth of Hickensack R. 

Castle Garden to Manhattanville 

Manhattanville to Yonkers 

Yonkers to Piermont 

HPiermont Ferry to Sing Sing 

Sing Sing to Haverstraw 

Haverstraw to Peekskill 

IT Main Channel, passing Delaw're Breakwat'r 

Off Brandywine L. II 

Main Channel to Bombay Hook L. 

Main Channel, Liston's Point 

Main Channel to Reedy Island 

Main Channel, Reedy Island L. H 

Opposite Delaware City 

Tp to Marcus Hook 

Opposite Chester 

Bar off Hog Island 

Greenwich Point to Philadelphia 

Capes at entrance to Hampton Roads 



25.5 
16 
27 
45 

7 

7.5 

6.5 
30 
24 
2S 
19 
28.5 
IS 
14 
24 

7.7 
33 
21 
31 
42 
23 
17 
21 
31 
23 
22 
13.5 
14 
13 

6.5 
10 
27 

7 

32 
27 
39 
24.5 
26 
27 
61 
43 
27.5 
20 
20 
24.5 
30 
20.5 
24 5 
1S.5 
21.5 
30 



Chesapeake Bay . 

* Two bars, each a quarter of a mile, have a less depth than 18 feet. 

-f- A small shoal, with 12 feet, lies in the middle of tlie Kill, opposite the wharf at Blazing Star: 
and another, with 10 feet, a quarter of a mile to the northward; but deeper water is found on the 
east siile of both. 

X A shoal, with 4 feet, obstructs the eastern channel, half way between Chelsea and its junction 
ftith the main channel. 

§ From Bergen Point Light, halfway to Newark Bay Light-house, 17 feet may be carried. 

i| A shoal of 12.5 feet occurs about a mile below Sing Sing. 

% Soundings varying between 10 and 15 fnthomi. 



PEPTHS OF WATER ON BARS AND IN CHANNELS. 97 



Harbora, etc. 



Table— (Continued). 

ATLANTIC COAST. 

Locations. 



Chesapeake Bay . . . 



Potomac River . 



York River Va. 

James River .... . Va. 



Elizabeth River. .Va 
Hatteras Inlet. .N. C. 



Ocracoke Inlet 

Albemarle Sound 

North River N. C 

Beaufort N. C 



Cape Fear 

Georgetown . . . .S. C. 



Bull's Bay 

Charleston S. C. 



Stono Inlet 

North Edisto 

St. Helena Sound. 



Port Royal. 
Tybee 



Savannah 

Ossabaw Sound 



Sapelo Sound 

Doboy Bar (Inlet) . 

St. Simon's Sound . 



Anchorage, Hampton Roads 

Hampton Roads to Sewall's Point 

S. of Sewall's Point (one mile and a half) 

Up to Norfolk 

Hampton Roads to. James River 



Tail of York Spit to Yorktown 

White Shoal Bar 

Up to Jamestown Island Bar 

Channel to one mile above Deep Water L. H. 

Jamestown Island Bar 

Harrison's Bar 

* Trent's Reach 

* Warwick Bar 

* Richmond Bar 

Norfolk to Navy-yard 

Over Bar 

t Over Bulkhead into Pamlico Sound 

Over Bar 

Anchorage, Wallico's Channel 

t Light-boat off Caroon's Point 

t Up the Sound to Martin's Point 

JAt entrance, and seven miles up from Albe- 
marle Sound 

Main Channel 

Through the Slue 

New Inlet Bar , 

Entrance to Winyah Bay 

Anchorage inside of North Island 

Up to Georgetown , 

Over Bar 

Anchorage * 

Main Bar 

North Channel 

Maffit's Channel 

Over Bar 

S. or Main Channel 

S. Channel 

S. Edisto 

S. E. Channel 

Bar near Tybee Island 

Tybee Roads 

Channel up to city 

S. Channel to Vernon River 

S. Channel to Ogeechee River 

Over Bar 

Entrance over Bar 

Anchorage in Sound 

Over Bar 

Entrance to Sound 

Turtle River to Blythe Island 

To Brunswick over Bar 

To Brunswick, Channel 

Over Bar 

Over Bar 



59 
25 

21 
23 

27 



33 

16 
19 
23 
15 
13.5 

8.5 
12.5. 

7 
25.5 
14 

7 

10 
19 

7 

5.5 

6.7 

15 

7 

S 

7 
27 

9 

13 
21 
11 
10 
11 
'6.5 
12 
17 
14 
10.5 
19 
31 
11 
12 
13 
IS 
15.5 
24 
15 
3S 
21 

9 
13 
11 

7 



St. Mary's 

St. Johns River. . Flo. 

* The effect of neap and spring tides is very small. The depth is affected much more sensibly by 
&he stage of the river above. 

+ The tide diminishes rapidly after entering the Inlet. 

X There are no lunar tides in Albemarle, Currituck, and Pamlico Sounds. 



98 



DEPTHS OF WATER OX BARS AND IX CHANNELS. 



Table— (Continued). 
ATLANTIC 



Harbors, etc. 



COAST. 

Locations. 



St. Johns River . .Flo. Channel up to Jacksonville 
St. Augustine | Over Bar. . 



Key West . 



Tortugas . . 
Tampa Bay 



Waccasassa Bay 

Cedar Keys 

St. Mark's 



St. Gecrge's Sound... 
Apalachicola 



St. Andrew's Bay 

Pensacola 



Florida Reef | Cape Florida L. H. W.' S. W. '% \y 

Turtle Harbor entrance *. .' 

Inside the Reefs, Hawk Channel 

Key Sambo Channel 

Main Channel to middle buoy on Shoals" ' 

Shoals to anchorage 

Sand Key Channel 

W. Channel 

N.W.Channel 

S. W. Channel 

Over Bar 

Channel Egmont and Passnge Keys' .. 

Channel to anchorage 

Main Channel over Bar ...... 

Over Bar " # 

Up to Fort St. Mark's.' 

E. entrance over Bar .............. 

Anchorage 

*Over Bar ...... • ' ' ' '. *...'..'. 

Up to anchorage .'.'.".".'.! 

*Main Channel, over Bar ....... 

* Over Bar 

Bar to Navy-yard 

Wharf at Pensacola 

*Over outer Bar 

Main Channel to Fort Morgan 

To Upper Fleet ...'.'.'.'!'. 

*Grant's Pass to Pascagoula Wharf 

Horn Island Pass, over Bar 

Anchorage, Horn Island ....... 

Up to Pascagoula Wharf. .......'...'" 

*Ch:innel 

N. W. Channel '.'.'.'.'.'.'.'.'.'.'. 

Anchorage 

*Ship Channel 

S. Pass .'.'.'.'.'.' 

Shell Bank Channel ...... 

*Pass a 1' Outre, N. Channel.* .'.'.[ 

S. Channel 

*Over Bar, N. entrance ..'.'.'.'..'. 

* Entering 

^Channel 

Channel 

*Bar of Grand Pass .' . . . . . . . . ' " ' ' ." .' [ ) 

Grand Passage to Independence Island 

*CIian'l inside, and N. of Ship Isl'd Shoal L.' S 

Channel N. of Ship Island Shoal 

Entrance to Cut-off Channel buov 

On Bulkhead 

Mouth of Atchafalaya River .....'. 
Mid-channel off L. H 

* En trance over Bar 

*Across the Bar 

* En trance over Bar 

*Over Bar 



Mobile Bay and River 



Mississippi Sound , 



Ship Island Harbor . 



Cat Island Harbor. . 



Mississippi Delta . 



Northeast Pass, 
Southeast Pass 

South Pass 

Southwest Pass 
Barrataria Bay. 



Derniere, or Last Isl'd 
Atchafalaya Bay 



Vermilion Bay . 
Calcasieu River 

Sabine Pass 

Galveston Bay . 
San Luis Pass 



Brazos River | *Over Bar . 

Matagorda Bay i * Entrance over Bar 

Aransas Pass \ *Aransas Pass 

Rio Grande j Channel. ... 



* The highest tides occur ut the moon's greatest declination. 



23 
7 
20 
26 
11 
34 
27 
30 
27 
30 
45 
54 
19 
17 



9 

7 

15.5 

19 

13 

10 

13 

22.5 

27 

21 

21 

36 

12 
7.5 

15 

19 
8 

19 

19.5 

IS 

16 

14 

15.2 

9.5 

12 

9.5 
10 

S 
155 

7.5 
15 
27 
14 

S 

6.5 
4S 
42 

5.5 

7.5 
12 



8 



25.1 
11.2 
21.5 
27.5 
12.5 
35.3 
2S.3 
31.3 
2S.3 
31.3 
46.2 
55.2 
20.4 
IS 4 
10.6 
11.5 
11.5 
95 
17.1 
20.6 
14.1 
11.1 
14 
23.5 
28 
22 
22 
37 
13 
8.7 
16.2 
20.2 
9.2 
20.3 
20.8 
19.3 
17.3 
15.3 
16.5 
10.6 
13.1 
10.6 
11.1 
9.1 
16.6 
S.7 
16.2 
28.4 
15.4 
9.6 
SI 
49.6 
43.6 
7.4 
9 

13.1 
9.1 
91 
in.1 
10.1 
4.9 



DEPTHS OF WATER ON BARS AND IN CHANNELS. 



99 



Harbors, etc. 



Ta"ble— {Continued). 
PACIFIC COAST. 
Locations. 



San Diego Bay 

San Pedro 

Point Duma 

San Buenaventura. . 
Santa Cruz Island . . 

Santa Barbara 

San Miguel Island. . 

Coxo Harbor 

San Luis Obispo 

San Simeon 

Monterey Harbor. . . 

Santa Cruz Harbor . 
San Francisco Bay . 



San Francisco Harbor 
Mare Island Straits . . 

Ballenas Bay 

Sir Frs. Drake's Bay. 

Tomales Bay 

Bodega Bay 

Coast 

Albion River 

Mendocino City 

Shelter Cove 

Humboldt Bay 

Crescent City Harbor 

Ewing Harbor 

Koos Bay 

Umpiua River 

Columbia River 



Shoalwater Bay 

Grenville Harbor. . . 

Nee-ah Harbor 

False Dungeness . . . 
New Dungeness .... 
Smith's Isl'd, N.side 
Bellingham Bay.. . . 

Port Townshend 

Port Ludlow 

Port Gamble 

Seattle 

Blakely Harbor 

Steilacoom Harbor. . 
Olympia Harbor . . . 



Entrance 

Abreast of La Plaza 

Point Pedro and Dead Man's Island . 

Anchorage 

Anchorage 

Anchorage, Prisoner's Harbor 

Anchorage inside of Kelp 

Cuyler's Harbor 

Anchorage 

Anchorage k 

Harbor anchorage 

Anchorage 

Near shore 

Anchorage 

From Four-fathom Bank to S. shore . 

Rincon Point 

Market Street Wharf 

Cunningham's Wharf 

On the Bar 

Mid-channel 

Mid-channel, Navy-yard, and Vallejo 
Inside of Breakers, Duxbury Reef. . . . 

Inside the Point 

Over Bar 

Inside of Reef, off Point 

Haven's anchorage 

Anchorage 

Anchorage 

Anchorage 

Channel 

Anchorage off city 

Anchorage 

Over Bar 

On Bar, opposite Mid-channel 

N. Channel to Baker's Bay 

*Entrance into S. Channel 

On Bar of S. Channel 

On Bar 

S. Channel 

Anchorage 

Anchorage 

Anchorage 

Anchorage 

Anchorage near Kelp 

Anchorage 

Anchorage 

Anchorage 

Anchorage 

Anchorage 

Anchorage 

Anchorage 

Mid-channel 



27.4 

18 

18 

54 

36 

75 

18 

37 

30 

33 

24 

42 

30 

27 

28 

66 

54 

36 

33 

25 

25 

24 

17 

10 

36 

48 

48 

SO 

22 

20 

21 

46 

11 

13 

24 

19 

16 

18 

25 

22 

36 

54 

45 

25 

18 

48 

36 

18 

20 

46 

18 

11 



LOfC 



* 21 feet may be carried in at mean low water. 



100 ALCOHOL. ACIDS. PROPERTIES OF VEGETABLES. 

Proportion of Alcohol in lOO Parts of the fbllo^ving 

Liquor s.-(Beande.) 



Small Beer... 1. and 1.0S 

Cider 5. 2 and 9.8 

Porter 3. 5 and 5. 20 

Brown Stout. 5.5 and 6.S 

Ale 6. 87 and 10. 

Perry 7.26 

Rhenish 7.5S 

Moselle 8.7 

Johannisberger 8.71 

Elder Wine.. 8.79 

Claret ordinaire .... 8.99 

Tokay 9.33 

Rudesheimer 10 . 72 

Marcobrunner 11 .G 

Gooseberry Wine.. 11. S4 

Frontignac 12 . 89 

Hockheimer 12.03 



Vin de Grave 

Champagne 

" Burgundy 
Hermitage, red 

u white . . 

Amontillado 

Barsac 

Sauterne 

White Port 

Bordeaux 

Shiraz 

Malmsey 

Sherry 

" old 

Alba Flora 

Constantia, red 

Port 

9 



12. OS 

12.01 

14.57 

12.32 

17.43 

12.63 

13. SO 

14.22 

15. 

15.1 

15.52 

10.4 

17.17 

23.86 

17.20 

IS. 92 

23. 



Colares 19 . 75 

Lisbon 18.94 

Malaga 17.2 

Cape Muscat 18.25 

Teneriffe 19.79 

Lachryma 19 .7 

Currant Wine 20.55 

Madeira 22.27 

14 Sercial 27.4 

Marsala 25.09 

Raisin Wine 25.12 

Cape Madeira 29 . 51 

Gin 51.6 

Brandy 53.39 

Rum 53. 6S 

Irish Whisky 53.9 

i Scotch Whisky 54.32 



TABLE SHOWING THE DILUTION PER CENT. NECESSARY TO REDUCE 
SPIRITUOUS LIQUORS. 
Water to be added to 100 volumes of spirit when of the following strength : 



Strength 
Required. 


90 


85 


80 


75 


70 


65 


00 


55 


50 


Per cent. 


Percent. 


Per cent. 


Per cent. 


Per cent. 


Percent. 


Per cent. 


Per cent. 


Percent. 


Per cent. 


85 


5.9 


















SO 


12.5 


6.3 
















75 


20 


13.3 


6.7 














70 


2S.6 


21.4 


14.3 


7.1 












65 


3S.5 


30. S 


23.1 


15.4 


7.7 










60 


50 


41.7 


33.3 


25 


10.7 


8.3 








55 


, 03.6 


54.5 


45 5 


30.4 


27.4 


IS. 2 


9.1 






50 


SO 


70 


60 


50 


40 


30 


20 


10 




40 


125 


112 5 


100 


S7.5 


75 


62.5 


50 


37.5 


C5 


30 


200 


1S3.3 


100.7 


150 


133.3 


110.7 


100 


83.3 


66.7 



Illustration. — 100 volumes of spirituous liquor having 90 per cent, of spirit 
contains : alcohol, 90, water, 10=100. 
To reduce it to 30 per cent, there is required 200 volumes of water. 

Hence 200 + 10 = 210, and — = — = 30 spmt ' or 30 per cent 
210 70 70 water, 

Acids. 

Acetic Acid (Vinegar), the acid of Malt beer, etc., etc. 
Lactic Acid, the acid of Millet beer and Cider. 
Tartaric Acid, the acid of Grape wine. 

Order of Acidity of several Wines. 



1. Moselle. 


3. Burgundy. 


5. Claret. 


7. Port. 


2. Rhine. 


4. Madeira. 


6. Champagne. 


8. Sherry. 



Nutritions Properties of* different "Vegetables and. 
cake, compared -with each other in Qxxantities. 



Oil- 



Oil-cake 1. 

Peas and Beans 1.5 

Rice 1.6 

Wheat, grain 2.5 

" flour 2. 

Oats 2.5 

Rye 2.5 



Bran, wheat. .2.75 and 3. 

Corn 3. 

Barley 3. 

Pea straw 3. 

Clover hay 4. 

Hay 5. 

Potatoes 14. 



Old Potatoes 20. 

Carrots 17.5 

Cabbages 18. 

Wheat straw 26. 

Barley " 26. 

Oat " 27.5 

Turnips 30. 



Illustration. —1 lb. of oil-cake is equal to IS lbs. of cabbage. 



SUBSTANCES OF VEGETABLE CROPS. TANNIN. 



101 



^Divisions of tne Compound Substances of tlie cultivated 
Crops of ^Vegetables. 



Vegetable. 



Saccharine 
or Sugary. 



Oleaginous 
or Oily. 



Albuminous 
or Fleshy. 



Sulphur, Phos- 
phorus, Iron, etc. 



Wheat Flour*. 

Barley 

Oats 

Rye 

Corn 

Rice 

Beans 

Peas 

Potatoes 

Mangel Wurzel 

Turnips 

Bran 

Wheat Straw . 
Oat " . 

Clover Hay . . . 



Per Cent. 

55 
60 
CO 
60 
TO 
75 
40 
50 
18 
11 



Per Cent. 
2.4 

V4 

2.4 

2.4 



4 
3.4 



PerCent. 
10 to 19 
12 to 15 
14 to 19 

12 

7 

24 to 28 

24 

2 

2 

m 

16 



Per Cent. 
20 to 30 

30 

40 

15 

30 

30 
8 to 15 

5 to 8 

50 
60 
90 



IVXineral Constituents absorbed or removed from an Acre 
of Soil by several Crops.— (Johnson.) 



Potassa 

Soda 

Lime 

Magnesia 

Oxide of Iron 

Phosphoric Acid 

Sulphuric Acid 

Chlorine 

Silica 

Alumina , 

Total 




Yield of Oil from Several Seeds. 





Per Cent. 




Per Cent. 


Poppy. . . 


. . 56 to 63 


Castor 


25 


Cress . . . 


• . . 56 to 5S 


Sunflower. . 


15 



Per Cent. 

Hemp 14 to 25 

Linseed 11 to 22 



Average Quantity of Tannin in several Substances. 

— (MOKFIT.) 



Per Cent. 

Catechu, Bombay 55. 

44 Bengal 44. 

Kino 75. 

Nutgalls, Aleppo 65. 

* 4 Chinese 69. 

Oak, old, inner bark 21. and 14.2 

" young, " 15.2 

11 " entire bark 6. 

" " spring-cut bark 22. 

44 44 root bark 8.9 

Chestnut, American rose, bark ... 8. 

44 horse, 44 2. 



Per Cent, 

root bark 5S. 

Sumac, Sicily and Malaga 16. 

44 Virginia 10. 

44 Carolina 5. 

Willow, inner bark 16. 

44 weeping 16. 

Sycamore bark 16. 

Tan shrub 44 13. 

Cherry-tree 24. 

Tormentil root <16. 

Corneus Sanguinea 44. 

Alder bark 36. 



100 lbs. of wheat flour contains : Fine flour ... 40 I Blue flour 

Common flour . . 80 Bran, etc. 

Middlings ... 10 | 

I* 



102 



FOODS, MANURES, ZINC, GLASS, BARTCEL. 



Analysis of different Articles of* Food, -with Reference 
only to their Properties for giving Heat and Strength. 

In 100 Parts. 

W£H| Substances. p r " Nitr °- 

gen. || bon. gen. 



Substances. 



Beef, meat 

Liver, Calf 8.. . 
Cod-fish, salted 

Sardines 

Mackerel 

Eels 

Egga 

Milk, Cow's 

Oysters 

Lobster 



Car- 


Nitro-I 


bon. 


gen. 


11 


3 


15.68 


3.93 


16 


5.02 


29 


6 


19.26 


3. 74 


30.05 


2 


13.5 


1.9 


8 


.60 


T.18 


2.13 


10.06 


2.93 



Substances. 



Cheese, Chest' r 

Beans 

Peas 

Wheat 

Flour. . . 

Rye Flour 

Barley 

Corn 

Buckwheat 



Car- 
bon. 



41.04 

42 

44 

41 

3S.5 

41 

40 

44 

42.5 

41 



4.13 Oatmeal 

4.5 Bread, stale . . . 

3. G6 Potatoes 

3 I Carrots 

1.64 Wine 

1.75 Alcohol 

Beer, strong. . . 

Oil, Olive 

Butter 183 

Coffee I 9 



44 
2S 
11 
5.5 



4.5 

98 



Note. — Multiply the figures representing the nitrogen by i 
enous matter is obtained. 



.64 
1.1 

5, and the equivalent amount of nitrog- 



1.95 
107 



.31 

.015 



.08 



Wheat 60* 



Standard Weights of G-rain per Bushel. 

Lbs. I Lbs. | Lbs. 

Corn and Rye. . 56 | Oats 32 | Barley... 48 



Relative "Valne of Foods compared -with lOO Ids 

Hay. 

Lbs. 
Cornstalks, dried . . . 400 

Carrots 276 

Rve 54 

Wheat 45 

Barley 54 



Lbs. 

Clover, green 400 

Corn, green 275 

Wheat straw 374 

Rye straw 442 

Oat straw 195 



ofOood 



Lbs. 



Oats 57 

Corn 59 

Sunflower seeds 62 

Linseed cake 63 

Wheat bran 105 






HVTanxxres. 

Relative Fertilizing Properties of various Manures. 
Peruvian Guano... 1. | Horse 048 | Farm-yard... .0298 



Human, mixed 



Swine 044 Cow . 



.0259 



Or, 1. lb. guano : 
aow. 



zl4X human, 21 horse, 22^ swine, 33^" farm-yard, and ! 



ZINC — SHEETS. 
Thickness and "Weight per Square Foot. 



Inch. 
.0311=10 OZ. 

.0457 = 12 oz. 



Inch. 

.0534 = 14 oz. 

.0611 = 16 oz. 



Inch. 

.0686 = 18 oz. 

. 0761 = 20 oz. 



WINDOW GLASS. 
Thickness and Weight per Square Foot. 



No. 


Thickness. 


Weight. 


No. 


Thickness. 


Weight. 


No. 


Thickness. 


Weight. 






Oz. 




Inch. 


Oz. 




Inch. 


Oz. 


12 


.059 


12 


17 


.083 


17 


26 


.125 


26 


13 


.063 


13 


19 


.091 


19 


32 


.154 


32 


15 


.071 


15 


21 


.1 


21 


36 


.167 


36 


16 


.077 


16 


24 


.111 


24 


42 


.2 


42 



Dimensions of* a Barrel. 

Diameter of head, 17 ins. ; bung, 19 ins. ; length, 2S ins. ; volume, 7689 cub. ins. 

Pyramid of Egypt. 

Base, square T45 feet. I Height 450 75 feet. 

Inclined height 56S.25 feet. | Weight 6 848 000 tons. 



CHURCHES, OPERA HOUSES. WORKS OF MAGNITUDE. 103 



Capacity of the Principal Churches and. Opera Houses. 

Estimating a person to occupy an area of 19.7 inches square. 
• Churches. 



St. Peter's 54000 

Milan Cathedral 37 000 

St. Paul's, Rome 32 000 

St. Paul's, London 25 600 

St. Petronio, Bologna 24400 

Florence Cathedral 24 300 

Antwerp Cathedral 24 000 

St. Sophia's, Constantinople 23 000 



St. John, Lateran 22 900 

Notre Dame, Paris 21 000 

Pisa Cathedral 13 000 

St. Stephen, Vienna 12 400 

St. Dominic s, Bologna 12 000 

St. Peter's, Bologna 11 400 

Cathedral of Sienna U ()00 

St. Mark's, Venice 7 000 



Opera Houses and Theatres. 



Carlo Felice, Genoa 2560 

Opera House, Munich 2370 

Alexander, St. Petersburg 2332 

San Carlos, Naples 2240 

Imperial, St. Petersburg 2160 

La Scala, Milan 2113 



Academy of Paris 2002 

Teatro del Liceo, Barcelona 4000 

Covent Garden, London 26S4 

Opera House, Berlin 163ft 

New York Academy 2526 

'f " Stadt 3000 



Area and. ^Population of the Earth. 



Divisions. 


Area. 


Population. 


Pop. to Sq. Mile. 




Square Miles. 






America 


14 491000 


706T7000 


5 


Europe 


3 760 000 


2S5954000 


79 


Asia 


16 313 000 


558562 000 


35 


Africa 


10036000 


71604000 


6 


Oceanica 


4500 000 


23 261 000 


5 


Total 


50 000 000 


1020 058 000 


20.4 



About j^jjth of the whole population are born every year, and nearly an equal 
number die in the same time; making about one birth and one death per second. 

The latest, and apparently the fairest, estimate of the world's population, makes 
it 1 150 000 000, divided as follows : 



Mnlattoes 315 000 000 

Blacks 173 000000 

Mohammedans 140 000 000 

Jews 14000000 



Whites 640000000 

Copper-colored 22 000 000 

Pagans. . .• 676 000 000 

Christians 320 000 000 

The 320 000 000 Christians are divided as follows : 

Church of Rome 170 000 000 | Greek and East Churches 60 000 000 

Protestants 90 000 000 

American Works of* Magnitude. 

Croton Aqueduct, N. Y.— Has a capacity of 100 000000 to 118000000 gallons per 
day, and from Dam to Receiving Reservoir is 38.134 miles in length. 

Illinois Central Railroad. — Length, Chicago to Cairo 365 miles, Centralia to Dun- 
leith 344 miles, total 709 miles. 

National Road. — Over the Cumberland Mountains to Illinois Town, 650% miles 
in length, and 80 feet in width. Macadamized for a width of 30 feet. 

Suspension Bridge, Niagara River. — Wire, span 1042 feet, 10 ins. 

Stones. — Of the U. S. Treasury, Washington, are heavier than any in the Pyra- 
mids of Egypt. 

Washington Aqueduct, D. C. (Captain M. C. Meigs, U. S. Engineers).— Conduit, 
cylinder of masonry 9 feet in diameter. Stone arch over Cabin John's Creek, 220 
feet span, 57^ feet rise. 

Iron pipe bridge over Reck Creek, 200 feet span, 20 feet rise. Arch of 2 lateral 
courses of cast-iron pipe, 4 feet internal diameter, and \% ins. thick. These pipes 
conveying the water not only sustain themselves over the great span, but support a 
street road and railway. 



104 



MISCELLANEOUS. 



Length. ofG-un Barrels (C. T. Coathufe.) 

The length of the barrel of a gun, to shoot well, measured from the vent- 
than 4- n0t lesS than 43 times the diameter of its bore, nor more 

M ason and Dixon's Line. 

39° 43' 26.3" N. mean latitude. 69.043 miles. 

Descent of Western Rivers. 

The slope of rivers flowing into the Mississippi from the East is about 
3 ms. per mile ; and from the West, 6 ins. 
The mean descent of the Ohio River from Pittsburg to the Mississinni 

E5f m «ft&£ %^2&1W£ that of the Mississi ^ t0 L 

Consumption of Atmospheric Air.-(Co A THVPE.) 

*£VZ^U& V °» UI?ie ° f carbonic acid gas given off bv the respira- 
Tn 94 hn, r f V Um ^ ^'"^ a , Inounts to *•<* P« cent, of the air respired. 
In 24 hours, the respiration of one healthy adult produces 10.7 cubic feet 

&oS 8 ' a removes from atmos P here exact1 ^ *S— ! 

m,?oh e ovtl Candl l (three l l a P ° U . nd ^ dest ™^S during its combustion, as 
much oxygen per hour as the respiration of one adultf 

adult\nmin7 ?me -° f oift that Can b V' e( * uired *>r the respiration of an 
adult human being in 24 hours, even if no portion of that which has been 
once respired were to be inspired again, does not exceed 266.7 cubic 

*ir A ^ h d * aper ' [SS . co u nf ; ned within a g^en volume of atmospheric 
air, tv ill become extinguished as soon as it has converted 3 per cent, of 
the given volume of air into carbonic acid. 

Days of tlie Roman Calendar. 

The Calends were the first 6 days of a month, the Nones the following 
v days, and the Ides the remaining davs. 

In March, May, July, and October, "the Ides fell upon the 15th and the 
A ones began upon the 7th. In the other months the Ides commenced 
upon the 13th and the Nones upon the 5th. 

^Periods of GJ-estation. 
Elephant, 1.9 years; buffalo and camel, 1 year ; horse and ass, 11 months • 
cow 9 months ; sheep, 5 months ; lion, 5 months ; dog, 9 weeks • cat 8 
weeks ; sow, 16 weeks ; guinea-pig, 3 weeks. 

Periods of Incubation. 
Swan, 42 days ; parrot, 40 days ; goose and pheasant, 35 davs ; hens 
ot all gallinaceous birds, 21 davs ; pigeon, 14 days ; canary 14 davs • 
duck, turkey, and peafowl, 28 days. The temperature of hatching is 104°.' 

Alimentary Principles. 

- 1 ;i W f t 7 ; o 2 '^i Ugar ^ 3 ' G ?, m; 4 - Starch; 5. Pectine; 6. Acetic Acid; 
/. Alcohol; 8. Oil or Fat,— \ egetable and Animal : 9. Albumen- 10 Fie- 
ri ne ; 11. Caseine; 12. Gluten ; 13. Gelatine; 14. Chloride of Sodium 

These alimentary principles, by their mixture or union, form our ordi- 
nary foods, which, by way of distinction, may be denominated compound 
aliments; thus, meat is composed of fibrine, albumen, gelatine fat etc • 
wheat consists of starch, gluten, sugar, gum, etc. ' ' ' ' 



WEIGHT OF SQUARE ROLLED IRON. 



105 



Comparison of Tonnages under Old and 'New Laws. 





Old 
Law. 




New Law. 


Diff. 

+ or- 

of old 

Meas't. 


Descriptioa of Vessel. 


Flush 
Deck. 


Houses, 
etc. 


Total. 


Full-built Ship . . 187 X 42 x 20 . 7 ft. 

Clipper 220. 5x42. 5x 17.7 tk 

Half Clipper.... 213 x 42 .5x28 " 
pit. Sea Steamer ....337 x41. 3x26.1 u 
sw. " " ....280.6x46 X32.8" 
sw. River Steamboat 329.9x35.4x10.4 " 
sw. f* " ..393 xol xlO.2 " 

pr. Steam-tug 81. 6x17. 4x 7.8" 

Coast'gSchooner.127 x31 xlO.6 " 

Yacht.... 72 x20 x 7.6 " 

Fishing Smack . . 36. 6x13. 6x 3.3" 
Caual-boat 94. 6x16. 6x 8 " 


Tons. 

1353 

1768 

1831 

2803 

2818 

1185 

2118 

102 

363 

96 

12.2 

118 


Tons. 

1518 
1280 
1687 
2554 
2419 

675 
1383 
55.4 

256 
42 

109 


Tons. 

107 
45 
100 
355 
225 
590 
1262 

93 
3 


Tons. 

1625 

1325 

1787 

2909 

2644 

1265 

2645 

55.4 

349 

45 

7.6 
109. 


PerCL 

20 + 

25 - 

2K- 

4 + 

6 - 

7 + 
20 -f 
46 - 

4 - 
53 - 
40 - 

7M~ 



Weight of Square Rolled Iron., 

From Xa Inch to 9)4 Inches. 



ONE FOOT IN LENGTH. 



Fide. | Weight. | 



Ins 






■% 

.X4 



■% 



Lbs. 

.013 
.053 
. .118 
.211 
.475 
.845 
1.32 
1.901 
2.588 
3.38 
4.278 
5.28 
6.39 
7.604 
8.926 
10.352 



Ins. 



n* 

•X 
■% 



Weight. 



Lbs. 

11.883 

13.52 

15.263 

17.112 

19.066 

21.12 

23.292 

25.56 

27.939 

30.416 

33.01 

35.704 

38.503 

41.408 

44.418 

47.534 



Side. | Weight. 



Ins. 

% 

u 

% 



% 



Lbs. 

50.756 
54.084 
57.517 
61.055 
64.7 
68.148 
72.305 
76. -1CA 
80.333 
84.48 
88.784 
93.168 
97.657 
102.24 
106.953 
111.756 



Side. 



Weight. 



Ins. 

6. 



>% 



m 

9. 



Lbs. 
116.671 
121.664 
132.04 
142.816 
154.012 
165.632 
177.672 
190. 13!) 
203.024 
216.336 
230.068 
244.22 
258.8 
273.792 
289.22 
305.056 



iLLrsTKA/rioN — What is the weight of a bar \% ins. by 12 inches in length? 
In column 1st, find \% ; opposite to it is 7.604 lbs , which is 7 lbs. and .604 of a lb. 
If the lesser denomination of ounces is required, the result is obtained as follows : 

Multiply the remainder by 16, point off the decimals, and the figures 
remaining on the left of the point give the number of ounces. 

Thus, .004 of a lb. =.604 X 16 =.9.604 = 7 lbs. 9.C64 ounces. 
To Ascertain, the Weight for less than a Foot in Length, 

Operation.— What is the weight of a bar 6^ inches square and 9# inches long ? 
In column 6th, opposite to 6^, is 132.04, which is the weight for a foot in length. 
6.25x12 inches =132.04 



.25 ' 
9.25 



is .5 

is .5 



of 6r 



66.02 
33.01 



of .3= 2.7503 



:101.7S0S^ouwte. 



106 WEIGHT OP ROUND AND FLAT ROLLED IRON. 



Weight of Round. Rolled Iron, 

From % 6 Inch to 12 Inches in Diameter. 



ONE FOOT IN LENGTH. 



E 


iam. 


Weight. Dia 


meter 


Weignt 


Diaemter. 


Weight 


j Diameter 


Weight. 


Ins. 


Lbs 


ns. 


Lbs. 


Ins. 


Lbs, 


Ins 


Lbs. 


% 


.01 


% 


13.44 


'% 


56.788 


■y 


149.328 




# 


.041 


% 


14.975 


•M 


59.9 


159. 45C 




£ 


.093 


K 


16.588 


•% 


63.094 


1G9.85C 




¥> 


.165 


% 


18.293 


5. 


66.35 


1 


180. 69C 




% 


.373 


4 


20.076 


-J6 


69.731 


191.808 




4 


.663 


% 


21.944 


-X 


73.172 


203.26 




°4 


1.043 3 




23.888 


'-% 


76.7 


215.04 




4 


1.493 


% 


25.926 


• X A 


80.304 




227.152 


1 


% 


2.032 
2.654 


% 


28.04 
30.24 




84.001 
87.776 


239.6 
252. 37G 




% 


3.359 


Y 


32.512 


>% 


91.634 


*io/ 4 


265.4 




H 


4.147 


H 


34.886 


6. 


95.552 


'/I 


278.924 




% 


5.019 


4 


37.332 


*% 


103.704 


• K 


292.688 




M 


5.972 


% 


39.864 


•% 


107.86 


•% 


306.8 




% 


7.01 4 




42.464 


4 


112.16 


n. 


321.21(1 




% 


8.128 


% 


45.174 


-% 


116.484 


J* 

12. 


336.004 




% 


9.333 


u 


47.952 


j% 


120.96 


351.104 


2 




10.616 


% 


50.815 


7. 


130.048 


366 536 




Vs 


11.988 


Y 


53.76 


■Y 


139.544 


382.208 



Weight of Flat Rolled Iron, 

From %X}i Inch to 5%X6 Inches. 









ONE FOOT 


IN LENGTH. 






Thickn 


Weight. 


Thickn 


Weight 


[Thickness 


1 "Weight 


] Thickness 


] Weight. 


Ins. 


Lbs. 


Ins. 


Lbs. 


Ins. 


Lbs. 


i Ins 


Lbs. 


a 




1. 




m 




UA 




■H 


.211 


■% 


.422 


■% 


2.64 


>A 


2.535 


■U 


.422 


■Y 


.845 


•i 


3.168 


-% 


3.168 


■ % 


.634 


■% 


1.267 


m 


3.696 


M 


3.802 


■% 
■ Y 




■Y 


1.69 


i. 


4.224 


>% 


4.435 


.264 


■ b 4 


2.112 


i-H 


4.752 


i. 


5.069 


.528 

.792 

1.056 


■ % 
1.X 


2.534 
2.956 


■Y 
■Y 


.58 
1.161 


1& 

l.V 


5.703 
6.337 
6.97 


.% 




■Y 


.475 


■% 


1.742 


i% 




•% 


.316 


■Y 


.95 


■Y 


2.325 




■Y, 


.633 


■M 


1.425 


■% 


2.904 


>K 


.686 


■% 


.95 


■Y 


1.901 


■% 


3.484 


>M 


1.372 


■Y 


1.265 


■% 


2.375 


■% 


4.065 


-% 


2.059 


■% 


1.584 


■% 


2.85 


l. 


4.646 


M 


2.746 




■% 


3.326 


l-M 


5.227 


-°d 


3.432 




l. 


3.802 


I.V 


5.808 


-% 


4.119 


■Y 


.369 


ik 




i-M 


6.389 


.% 


4.805 


•Y 


.738 








i. 


5.492 


■% 


1.108 


■Y 


.528 


134 




i-H 


6.178 


■Y 


1.477 


■ Y 


1.056 


■Y 


.633 


i-% 


6.864 


■% 


1.846 


■% 


1.584 
2.112 


■ Y 


1.266 


i>% 


7.551 


■% 


2.217 


■Y 


■% 


1.9 


i.X 


8.237 



WEIGHT OF FLAT IiOLLED IRON. 



107 



T a"ble— {Continued) . 



Thickn. 


| Weight. 


Thickn. 


| Weight. 


Thickn. 


I Weight. 


Thickn. 


I Weight 


Ins. 


Lbs. 


Ins. 


Lbs. 


Ins. 


Lbs. 


Ins. 


Lbs. 


1.% 




2M 




2.y 2 




2% 




■M 


.739 


i. 


7.181 


M 


6.336 


%.% 


20.91 


■M 


1.479 


hH 


8.079 


.% 


7.392 




22.07 


■% 


2.218 


KM 


8.977 


l. 


8.448 


23.22 


M 


2.957 


i,% 


9.874 


t*M 


9.504 


*r% 


24. 3£ 


■% 


3.696 


1..M 


10.772 


%,u 


10.56 




■M 


4.435 


l,H 


11.67 


i.8 


11.616 


2.% 




.,« 


5.178 


1.% 


12.567 


*<M 


12.672 


•y* 


1.21 


l. 


5.914 


1-.3S 


13.465 


1 -% 


13.728 


•K 


2.42 


l.H 


6.653 


2. 


14.362 


i.% 


14.784 


>% 


3.64 


1-H 


7.393 


2.% 




i.X 


15.84 


-% 


4.85 


i-% 


8.132 




2. 


16.896 


•% 


6.07 


bd 


8.871 


•% 


.95 


2 -K 


17.952 


•M 


7.28 


i'.% 

m 


9.61 




1.9 

2.851 

3.802 


2.^ 


19.008 
20.064 


4 

i.y* 


8.50 

9.71 

10.93 


■% 


.792 


•ff 


4.752 


2.% 




12.14 


■u 


1.584 


.% 


5.703 


M 


1.109 


13.36 


■% 


2.376 


•K 


6.653 


m 


2.218 


14.57 


■% 


3.168 
3.96 


l. 


7.604 
8.554 




3.327 
4.436 


i'Jk 


15.78 
17.00 


■% 


4.752 


IM 


9.505 


•% 


5.545 


i-M 


18.21 


■% 


5.544 


\\ 


10.455 


4i 


6.654 


2/ 8 


19.43 


i. 


6.336 


J.ff 


11.406 


-% 


7.763 




20.64 


l-H 


7.129 


1-8 


12.356 


l. 


•8.872 


21.86 


l\ 


7.921 


i-M 


13.307 


\M 


9.981 


23.07 


l-% 


8.713 


i.% 


14.257 


l,H 


11.09 


24.29 


l-k 


9.505 


2. 


15.208 


IM 


12.199 


2-K 


25.50 


\-\ 


10.297 


*# 


16.158 


i-X 


13.308 


2-M 


26.71 


i.% 


11.089 


2.K 




i.« 


14.417 






r> 






i.% 


15.526 


3. 




4. 




:& 


1.003 


i-% 


16.635 


•}£ 


1.26 


-Vs 


.845 


■X 


2.006 


2/ 8 


17.744 


•if 


2.53 


: U 


1.689 


•ft 


3.009 




18.853 


•38 


3.80 


'% 


2.534 


•i 


4.013 


19.962 


& 


5.06 


•i 


3.379 


-"» 


5.016 


2.% 


21.071 


•H 


6.33 


•°/s 


4.224 


•i 


6.019 


2.% 


22.18 


•'M 


7.60 


•H 


5.069 


.% 


7.022 


2% 




•% 


8.87 


.% 


5.914 


i. 


8.025 




i. 


10.13 


% 


6.758 


i;# 


9.028 


-Vs 


1.162 


!•% 


11.40 


7.604 


t.k 


10.032 


- x 6 


2.323 


12.67 


i.g 


8.448 


l.M 


11.035 


>% 


3.485 


J.X 


13.94 


!■% 


9.294 


i-K 


12.038 


M 


4.647 


15.20 


!••» 


10.138 


hk 


13.042 


•li 


5.808 


16.47 


1-M 


10.983 


t.-g 


14.045 


M 


6.97 


17.74 


i-X 


11.828 


i-K 


15.048 


.% 


8.132 


S* 


19.01 


!•% 


12.673 


2. 


16.051 


l. 


9.294 


20.27' 


■Vb 


.898 
1.795 


2->£ 


17.054 

18.057 


l.V 

i-X 


10.455 
11.617 
12.779 
13.94 


2-M 


22-81] 
25.34( 

27.*^] 


■% 


2.693 


•Vs 


1.056 


i.X 


15.102 


3.M 




■6 


3.591 


-X 


2.112 


i-H 


16.264 


•j£ 


1.37.' 


■% 


4.488 


-X 


3.168 


!•% 


17.425 


•K 


2.741 


■u 


5.386 


•^ 


4.224 


2. 


18.587 


•M 


4.1H 


■% 


6.283 


•X 


5.28 


2K 


19.749 


W 


5.491 



108 



WEIGHT OF FLAT ROLLED IKON. 



Tat>le— (Continued). 



Thickn. • 


Weight. 


Thickn. 


Weight. | 


Thickn. 


Weight 


Thickn. 


Weight. 


Ins. 


Lbs. 


Ins. 


Lbs. 


Ins 


Lds. 


Ins. 


Lbs. 


8.J4 




3.% 




4.M 




&y* 




-28 


6.865 


i.% 


22.178 


2. 


30.415 


-Vi 


8.871 


% 


8.237 


l-x 


23.762 


2-X 


34.217 


.% 


13.307 


•% 


9.61 


2. 


25.346 


2-X 


38.019 


i. 


17.742 


1. 


10.983 


2.X 


28.514 


2-M 


41.82 


i-x 


22.178 


i-^ 


12.356 


2.X 


31.682 


3. 


45.623 


?--x 


26.613 


i-X 


13.73 


2-M 


34.851 


3.X 


49.425 


ix 


31.049 


i-K 


15.102 


3. 


38.019 


3-X 


53.226 


2. 


35.484 


i-X 


16.475 


3.X 


41.187 


s.% 


57.028 


2-X 


39.92 


i-% 


17.848 


3-X 


44.355 


4. 


60.83 


2-X 


44.355 


i-M 


19.221 


4. 
'ft 




4.X 


64.632 


2.% 


48.791 


fc* 


20.594 
21.967 


1.69 


4.% 




3. 

3.X 


53.226 
57.662 


2.X 


24.712 


-U 


3.38 
6.759 


:X 


4.013 


3-X" 


62.097 


2-K 


27.458 


-H 


•K 


8.026 


3-X 


66.533 


2.JJ 


30.204 


•% 


10.138 


•ft 


12.036 


4. 


70.968 


3. 


32.95 


l. 


13.518 


i. 


16.052 


4.X 


75.404 


3.K 




l.X 


16.897 
20.277 


I'M 

l.X 


20.066 
24.079 


4.X 


79.839 
84.275 


•X 


1.479 


V A 


23.656 


i.% 


28.092 


5. 


88.71 




2.957 
4.436 


2. 


27.036 
30.415 


2. 
2.X 


32.105 
36.118 


5.M 




•K 


5.914 


t-M 


33.795 


2-X 


40.131 


• U 


4.647 


•M 


7.393 


2.% 


37.174 


2-M 


44.144 


•M 


9.294 


• M 


8.871 


3. 


40.554 


3. 


48.157 


-% 


13.94 


•X 


10.35 


3-& 


43.933 


3.X 


52.17 


l. 


18.587 


.1 


11.828 


3-X 


47.313 


3-X 


56184 


l.X 


23.234 


•IX 


13.307 


3-M 


50.692 


*'-X 


60.197 


l.X 


27.881 


•« 


14.785 


4.Vi 




4. 


64.21 


IH 


32.527 


■IX 


16.264 


■H 

% 

l. 


1.795 


4.X 


68.223 


2. 


37.174 


.IX 


17.742 


3.591 


4.X 


72.235 


2.X 


41.821 


-iM 


19.221 


7.181 


5. 




2-X 


46.468 


• i% 


20.699 


10.772 




2.% 


51.114 


• IX 


22.178 


14.364 


M 


4.224 


3. 


55.761 


.2 


23.656 




1 7 . 953 


•ft 


8.449 


3.X 


60.408 


•2X 


26.613 


21.544 


•X 


12.673 


3-X 


65.055 


.*K 


29.57 


25.135 


i. 


16.897 


3-X 


69.701 


■*K 


32.527 


2. 


28 . 725 


l.X 


21.122 


4. 


74.348 


.3 


35.485 


2.V 
2.H 

2.% 
3 


32.316 


l.X 


25.346 


4-X 


78.995 


,3# 


38.441 


35.907 


l-H 


29.57 


fX 


83.642 


3.% 




39.497 
43.088 


2. 

2.X 


33.795 
38.019 


4-M 
5. 


88.288 
92.935 


•X 


1.584 


3-X 
3.X 
3-K 
4. 


46.679 


2-X 


42.243 


5.X 


97.582 


•3^ 


3.168 
4.752 


50.269 
53.86 


2-M 
3. 


46.468 
50.692 


5.% 




.& 


6.336 


57 .45 


3.X 


54.916 


•X 


9.716 


•% 


7.921 






3-X 


59.14 


X 


14.574 


•M 


9.505 


134 




3-X 


63.365 


l. 


19.432 


•X 


11.089 


• X 


3.802 


4. 


67.589 


k.X 


24.29 


l. 


12.673 


•X 


7.604 


4-X 


71.813 


l.X 


29.148 


i-X 


14.257 


X 


11.406 


4.X 


76.038 


*X 


34.006 


1..& 


15.841 


l. 


15.208 


4.% 


80.262 


2. 


38.864 


I'M 


17.425 


l.X 


19.01 


5.K 




2-X 


43.722 


l.X 


19.009 


l.X 


22.812 




2-X 


48.58 


*.% 


20.594 


t.X 


26.614 


•X 


4. 43 J 


2-% 


53.437 



WEIGHT OF CAST IKON AND LEAD BALLS. 



109 



Thickn. 


Weight. 


Ins. 


Lbs. 


5. 3 A 




3. 


58.296 


3.H 


63.154 


•-H 


68.012 



Table— (Continued). 

Thickn. Weight. 



Ins. 

5.% 

3.% 

4. 

4.K 



Lbs. 

72.87 

77.728 

82.585 



Thickn. 


Weight 


Thickn. 


Weight. 


Ins. 

5.% 
*■% 

5, 


Lbe. 
87.443 

92.301 
97.159 


Ins. 

5.% 

5.% 
6.* 


Lbs. 

102.017 

106.876 
116.592 



Examples —What is the weight of a bar of iron 5.V ins. in breadth by .% in. 
in thickness? J * 

In column 7, page 103, find 5.^; and below it, in column 5.% ; and opposite, to 
that is 13.307, which is 13 lbs. and .307 of a pound. 

For parts of a pound and of a foot, operate according to the rule laid down for ta- 
uiGj pag6 105. 



Weight and. Volume off Cast Iron and Lead Balls* 

From 1 to 20 Inches in Diameter. 



Diameter. 


Volume. 


Cast Iron. 


Lead. 


Ins. 


Cubic Ins. 


Lbs 


Lbs. 


1. 


.5235 


.1365 


.2147 


1-X 


1.7671 


.4607 


.7248 


2. 


4.1887 


1.092 


1.718 


2-X 


8.1812 


2.1328 


3.3554 


3. 


14.1371 


3.6855 


5.7982 


3.X 


22.4492 


5.8525 


9.2073 


4. 


33.5103 


8.7361 


13.744 


4-X 


47.7129 


12.4387 


19.569 


5. 


65.4498 


17.0628 


26.843 


5-X 


87.1137 


22.7206 


35.729 


6. 


113.0973 


29.4845 


46.385 


6.X 


143.7932 


37.4528 


58.976 


7. 


179.5943 


46.8203 


73.659 


7-X 


220.8932 


57.587 


90.598 


8. 


268.0825 


69.8892 


109.952 


8.X 


321.555 


83.8396 


131.883 


9. 


381.7034 


99.5103 


156.553 


9-X 


448.9204 


117.0338 


184.121 


10. 


523.5987 


136.5025 


214.749 


11. 


696.9098 


181.7648 


285.832 


12. 


904.7784 


235.8763 


371.096 


13. 


1150.346 


299.623 


471.806 


14. 


1436.754 


374.5629 


589.273 


15. 


1767.145 


460.6959 


724.781 


16. 


2144.66 


559.1142 


879.616 


17. 


2572.44 


670.7168 


1055.066 


18. 


3053.627 


796.0825 


1252.422 


19. 


3591.363 


936.2708 


1472.97 


20. 


4188.79 


1092.02 


1717,995 



110 



WEIGHT OF CAST IRON PIPES. 



Weight of Cast Iron Pipes of different Thick- 
nesses, 

From 1 Inch to 36 Inches in Diameter. 









ONE 


FOOT IN LENGTH. 








Diam Thickn. 


Weight 


Diam. | Thickn. | 


Weight. 


Diam. 1 Thickn. 


Weight. 


Ins. i J 


ns. 


Lbs. 


Ins. 


Ins. 


Lbs. 


Ins. J 


ns. 


Lbs. 


1. 


3^ 


3.06 


6. 


•% 


49.6 


11. y 


■A 


58.82 




K 


5.05 






•Vs 


58.96 




■% 


74.28 


l.U 


H 


3.67 


<$.y 




•y 


34.32 




■% 


90.06 


t 


% 


6. 






-% 


43.68 




■% 


106.14 


l-K 


-% 


6.89 






.% 


53.3 


i 




122.62 




K 


9.8 






-% 


63.18 


12. 


'■% 


61.26 


1.% 


% 


7.8 


7. 




-y 


36.66 




■ '% 


77.36 




% 


11.04 

8.74 






•% 


46.8 
56.96 




■% 


93.7 


2. 






& 


110.48 




K 


12.23 






•% 


67.6 


1 




127.42 


•Sfc 


% 


9.65 




1 




78.39 


te.& 


'■ l A 


63.7 




y 


13.48 


?*& 




\u 


39.22 




■% 


80.4 


2.& 


% 


10.57 






•% 


49.92 




■% 


97.4 




"4 


14.66 






% 


60.48 
71.76 


i 


■% 


114.72 




% 


19.05 


132.35 


2.% 


% 


11.54 




1 




83.28 


13. 


'•k 


66.14 




% 


15.91 


8. 




'-y 


41.64 




■% 


83.46 




% 


20.59 






•% 


52.68 




.% 


101.08 


3. 


% 


12.28 






.% 


64.27 




■% 


118.97 




% 


17.15 






>% 


76.12 


1 




137.28 




% 


22.15 




1 




88.2 


13.^ 


'■% 


68364 




% 


27.56 


8-A 




'.y 


44.11 




■% 


86.55 


3-X 


| 


18.4 






% 


56.16 




% 


104.76 




% 


23.72 






% 


68. 




% 


123.3 




% 


29.64 






% 


80.5 


1 




142.16 


3.K ■ 


H 


19.66 




1 




93.28 


14. 


'■% 


71.07 




% 


25.27 


9. 




'-y 


46.5 




■% 


89.61 




% 


31.2 






% 


58.92 




% 


108.46 


S.H ■ 


y 


20.9 






% 


71.7 




% 


127.6 




% 


26.83 






% 


84.7 


1 




147.03 




% 


33.07 




1 




97.98 


14. y 


X 


73.72 


4. 


A 


22.05 


9.>£ 




H 


48.98 




% 


92.66 




% 


28.28 






% 


62.02 




% 


112.1 




% 


34.94 






% 


75.32 




% 


131.86 


4.^ 


y 


23.35 






% 


88.98 


i 




151.92 




% 


29.85 




1 


: 


L02.9 


15. 


X 


75.96 




% 


36.73 


10. 




y 


51.46 




% 


95.72 


*.« 


A 


24.49 






% 


65.08 




S A 


115.78 


i 


% 


31.4 






% 


78.99 




% 


136.15 




% 


38.58 






% 


93.24 


1 




156.82 


4.^ | 


y 


25.7 




1 


■: 


L08.84 


15.^ 


H 


78.4 




% 


32.91 


10.^ 




y 


53.88 




% 


98.78 




% 


40.43 






% 


68.14 




% 


119.48 


5. 


y 


26.94 






% 


82.68 




% 


140.4 




% 


34.34 






% 


97.44 


1. 




161.82 




% 


42.28 




l' 


\ 


12.68 


16. 


l A 


80.87 


5-K 


y 


29.4 


11. 




y 


56.34 




% 


101.82 




i 


37.44 






% 


71.19 




% 


123.14 




45.94 






% 


86.4 




% 


144.76 


6. 


% 


31.82 






% i 


01.83 


i 




166.6 




% 


40.56 




1. 


i 


17.6 


™.y • 


K) 


83.3 



WEIGHT OF CAST IRON PIPES. — CAST IRON, ETC. Ill 
Ta"ble— {Continued). 



Diam. | Thickn. 


Weight. 


Diam. Th 


ickn. 


\ Weight. 


Diam. 


Thickn. 


j Weight. 


IllS. ] 


ns. 


Lbs. 


Ins. I 


ns. 


Lbs. 


Ins. 




ns. 


Lbs. 


16.K 


-% 


104.82 


22. 


'% 


138.6 


30. 


1 




303.86 




.% 


126.79 




% 


167.24 




1 


M 


343.2 




>K 


149.02 




% 


196.46 


31. 




M 


233.4 


1 




171.6 


1 




225.38 






% 


273.4 


17. 


'> X A 


85.73 


23. 


% 


144.77 




1 




313.68 




% 


107.96 




% 


174.62 




1 


'-y 8 


354.24 




% 


130.48 




% 


204.78 


32. 




>K 


240.76 




-X 


153.3 


1 




235.28 






•% 


281.94 


1 




176.58 


24. 


% 


150.85 




1 




323.49 


17.J^ 


K 


88.23 




% 


181.92 




1 


'h 


365.29 




% 


111.06 




% 


213.28 


33. 




% 


248.1 




% 


134.16 


1 




245.08 






% 


290.5 




% 


157.59 


25. 


% 


156.97 




1 




333.24 


1 




181.33 




% 


189.28 




1 


% 


376.26 


18. 


% 


114.1 




% 


221.94 




1 


% 


420.77 




% 


137.84 


1. 




254.86 


34. 




% 


255.45 




% 


161.9 


26. 


% 


196.62 






% 


298.88 


1 




186.24 




% 


230.56 




1 




342.88 


19. 


% 


120.24 


1. 




264.66 




1 


Vs 


387.13 




% 


145.2 


27. 


% 


204.04 




1 


H 


431.76 




% 


170.47 




% 


239.08 


35. 




% 


262.7 


1 




195.92 


1. 




274.56 






% 


307.62 


20. 


% 


126.33 


28. 


% 


211.32 




1 




352.86 




% 


152.53 




% 


247.62 




1 


% 


398.1 




% 


179.02 


1. 




284.28 




1 


% 


443.96 


1. 




205.8 


29- 


% 


218.7 


36. 




% 


270.18 


21. 


% 


132.5 




% 


256.2 






% 


316.36 




% 


159.84 


l' 




294.02 




1 




362.86 




% 


187.6 


30. 


% 


226.2 




1 


% 


409.34 


i! 




215.52 




% 


264.79 




1 


}£ 


456.46 



Note. — These weights do not include any allowance for spigot and faucet ends. 



CAST IRON. 

To Compute tlie Weight of a Cast Iron Bar or Rod. 

Find the weight of a wrought iron bar or rod of the same dimen- 
sions in the preceding tables or by computation, and from the weight 
deduct the ^th P art '•> or > 

As .1000 : .9257 : : the weight of a wrought bar or rod : to the 
weight required. Thus, what is the weight of a piece of cast iron 
4x3.%xl2 inches? 

In table, page 108, the weight of a piece of wrought iron of these di- 
mensions is 50.692 lbs. 

Then 1000 : .9257 : : 50.692 : 46.93 lbs. 

To Compute the Weight of a piece of Cast or 
Wrought Iron of any Dimension or Form. 

By the rules given in Mensuration of Solids (page 270), ascertain tho 
number of cubic inches in the piece, then multiply by the weight of a 
eubic inch, and the product will give the weight in pounds. 



112 COPPER, LEAD, AND BRASS. 

Example. — What is the weight of a cube of wrought iron 10 inches square by IE 
inches in length ? 

10X 10X15 = 1500 cubic inches. 

.2816 weight of a cubic inch.* 
422.4 pound s. 
2. What is the weight of a cast iron ball 15 inches in diameter? 
By table, page 109, 15 ins. = 176.7149 cubic inches. 

.2607 weight of a cubic inch.* 
46 J. 6957 pounds. 



COPPER. 

To Compute tlie "Weight of Copper. 

Rule. — Ascertain the nurabsr of cubic inches in the piece; multiply them by 
.3211S,* and the product will give the weight in pounds. 

Example.— What is the weight of a copper plate % an inch thick by 16 inches 
aquare ? 

162 — 256 

.5 for % an inch. 

12S7x .3241S = 41.495 pounds. 

Brazier's Sheets arc 30x60 inches, and from 12 to 300 lbs. per 
square foot. 

Sheathing Copper is 14x48 inches, and from 14 to 34 oz. per 
square foot. 



LEAD. 

To Compute the Weight of Lead. 

Rule. — Ascertain the number of cubic inches in the piece ; multiply the sum by 
.41015,* and the product will give the weight in pounds. 

Example — What is the weight of a leaden pipe 12 feet long, 3.75 inches in diame- 
ter, and 1 inch thick ? 

By Rule in Mensuration of Surfaces, to ascertain the Area of Cylindrical Rings. 
Area of (3. 75 + 1 + 1) — C5.f 67 
" " 3.75 = 11.044 

Difference, 14.S23, or area of rina. 
144 -± 12 feet. 

•Ji4S.'Jl2x.41015 = SS1.376.poM?ute. 



BRASS. 

To Compute the Weight of ordinary Bra?s Castings. 

Rule.— Ascertain the number of cubic inches in the piece, multiply them by 
.3112,* and the product will give the weight in pounds. 



* The weights of a cubic inch as here jriven are for the ordinary metals ; when, however, the spe- 
cific gravity of the metal under consideration is accurately known, the weight of a cubic inch of it 
should be substituted for the units here given. 



WEIGHTS OF IRON, STEEL, COPPER, ETC. 



113 



Weights of Wrought Iron, Steel, Copper, and. 
Brass Plates. 



SOFT KOLLED. 



He of 
Gauge 



Thickness determined by 


American Gauge 




Thickness of 




Plates — per Square Foot. 




<?ach Number. 


Wrought Iron. 


Steel. 


Copper. 


Brass. 


Inch. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


.46 


18.4575 


18.7036 


20.838 


19.688 


.40964 


16.4368 


16 


.6559 


18.5567 


17.5326 


.3648 


14.6376 


14 


.8328 


16.5254 


15.6134 


.32486 


13.0351 


13 


.2088 


14.7162 


13.904 


.2893 


11.6082 


11 


.7629 


13.1053 


12.382 


.25763 


10.3374 


10 


4752 


11.6706 


11.0266 


.22942 


9 


.2055 


9 


.3283 


10.3927 


9.8192 


.20431 


8 


.1979 


8 


3073 


9.2552 


8.7445 


.18194 


7.3004 


7 


.3977 


8.2419 


7.787 


.16202 


6.5011 


6 


.5878 


7.3395 


6.9345 


. 14428 


5 


.7892 


5 


.8664 


6.5359 


6.1752 


.12849 


5 


.1557 


5 


.2244 


5.8206 


5.4994 


.11443 


4 


.5915 


4 


.6527 


5.1837 


4.8976 


.10189 


4 


.0884 


4.1428 


4.6156 


4.3609 


.090742 


3 


.641 


3.6896 


4.1106 


3.8838 


.080808 


3 


.2424 


3.2856 


3.6606 


3.4586 


.071961 


2 


8874 


2.9259 


3.2598 


3.0799 


.064084 


2 


.5714 


2.6057 


2.903 


2.7428 


.057068 


2 


.2899 


2.3204 


2.5852 


24425 


.05082 


2 


.0392 


2.0664 


2.3021 


2.1751 


.045257 


1 


.8159 


1.8402 


2.0501 


1.937 


.040303 


1 


.6172 


1 


.6387 


1.8257 


1.725 


.03589 


1 


.44 


1 


.4593 


1.6258 


1.5361 


.031961 


1 


.2824 


1 


2995 


1,4478 


1.3679 


.028462 


1 


.142 


1 


.1573 


1.2893 


1.2182 


.025347 


1 


.017 


1.0306 


1.1482 


1.0849 


.022571 




.9057 




.9177 


1.0225 


.96604 


.0201 




.8065 




.8173 


.91053 


.86028 


.0179 




.7182 




7278 


.81087 


.76612 


.01594 




.6396 




6481 


.72208 


.68223 


.014195 




.5696 




5772 


.64303 


.60755 


.012641 


.5072 




.514 


.57264 


.54103 


.011257 


.4517 




4577 


.50994 


.4818 


.010025 


.4023 




4076 


.45413 


.42907 


.008928 


.3582 




.363 


.40444 


.38212 


.00795 


.319 




3232 


.36014 


.34026 


.00708 


.2841 




2879 


.32072 


.30302 


.006304 


.2529 




.2563 


.28557 


.26981 


.005614 




.2253 


.2283 


.25431 


.24028 


.005 




.2006 


.2033 


.2265 


.214 


.004453 


.1787 


.181 


.20172 


.19059 


.003965 


.1591 


.1612 


.17961 


.1697 


.003531 




.1417 


.1436 


.15995 


15113 


.003144 




.1261 




.1278 


.14242 


13456 



Specific Gravities 7 . 704 

Weights of a Cubic Foot. 481.25 
Inch. .2787 



7.806 

487.75 
.2823 



8.698 
543.6 
.3146 



8.218 
513.6 
.2972 



K* 



114 



WEIGHTS OF IRON, STEEL, COPPER, ETC. 



Weights of Wroviglit Iroix, Steel, Copper, and. 
I3x*ass Wire. 





Diameters and Thickness determined by American G 


%uge. 


No. of 


Diani of each 
Number. 




Wire— per 


Lineal Foot. 




Gauge. 


Wrought Iron 


Steel. 


Copper 


Brass. 




Inch. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


0000 


.46 


.56074 


.56603 


.640513 


.605176 


000 


.40964 


.444683 


.448879 


.507946 


.479908 


00 


.3648 


.352659 


.355986 


.40283 


.380666 





.32486 


.279665 


.282303 


.319451 


.301816 


1 


.2893 


.221789 


.223891 


.253342 


.239353 


2 


.25763 


. 175888 


.177548 


.200911 


.189818 


3 


.22942 


.13948 


.140796 


.159323 


.150522 


4 


.20431 


.110616 


.11166 


.126353 


.119376 


5 


.18194 


.08772 


.088548 


.1002 


.094666 


6 


.16202 


.069565 


.070221 


.079462 


.075075 


7 


.144-28 


.055165 


.055685 


.063013 


.059545 


8 


.12849 


.043751 


.044164 


.049976 


.047219 


9 


.11443 


.034699 


.035026 


.039636 


.037437 


10 


.10189 


.027512 


.027772 


.031426 


.029687 


11 


.090742 


.02182 


.022026 


.024924 


.023549 


12 


.080808 


.017304 


.017468 


.019766 


.0i8676 


13 


.071961 


.013722 


.013851 


.015674 


.014809 


14 


.064084 


.010886 


.010989 


.012435 


.011746 


15 


.057068 


.008631 


.008712 


.009859 


.009315 


16 


.05082 


.006845 


.006909 


.007819 


.007587 


17 


.045257 


.005427 


.005478 


.006199 


.005857 


18 


.040303 


.004304 


.004344 


.004916 


.004645 


19 


.03589 


.003413 


.003445 


.003899 


.003684 


20 


.031961 


.002708 


.002734 


.003094 


.00292 


21 


.028462 


.002147 


.002167 


.002452 


.002317 


22 


.025347 


.001703 


.001719 


.001945 


.001838 


23 


.022571 


.00135 


.001363 


.001542 


.001457 


24 


.0201 


.001071 


.001081 


.001223 


.001155 


25 


.0179 


.0008491 


.0008571 


.0009699 


.0009163 


26 


.01594 


.0006734 


.0006797 


.0007692 


.0007267 


27 


.014195 


.000534 


.0005391 


.0006099 


.0005763 


28 


.012641 


.0004235 


.0004275 


.0004837 


.000457 


29 


.011257 


.0003358 


.0003389 


.0003835 


.0003624 


30 


.010025 


.0002663 


.0002688 


.0003042 


.0002874 


31 


.008928 


.0002113 


.0002132 


.0002413 


.000228 


32 


.00795 


.0001675 


.0001691 


.0001913 


.0001808 


33 


.00708 


.0001328 


.0001341 


.0001517 


.0001434 


34 


.006304 


.0001053 


.0001063 


.0001204 


.0001137 


35 


.005614 


.00008366 


.00008445 


.0000956 


.00009015 


36 


.005 


.00006625 


.00006687 


.0000757 


.0000715 


37 


.004453 


.00005255 


.00005304 


.00006003 


.00005671 


38 


.003965 


.00004166 


.00004205 


.00004758 


.00004496 


39 


.003531 


.00003305 


.00003336 


.00003775 


.00003566 


40 


.003144 


.0000262 


.00002644 


.00002992 


.00002827 



Specific Gravities 7 . 774 

Weights of a Cub. Foot. 485.87 
Inch. .2812 



7.847 
490.45 
.2838 



8.88 
554.988 
.3212 



The Specific Gravities to determine these weights and the calculations were mad© 
by the author for Messrs. J. K. Urowne & Sharpe, Trovidence, R. I. 



WEIGHTS OF IRON, jSTEEL, COPPER, ETC. 



115 



Weights of AVronglit Iron, Steel, Copper, and 
13rass Plates. 





Thickness determined by Binning} 


am Gauge. 




No. of Th 


ckness of 
l Number. 




Plates — per Square Foot. 




Gauge. eac 


Iron. 


Steel. 


Copper 


Brass. 




Ins. 


Lbs. 


Lbs. 


' Lbs. 


Lba. 


0000 


.454 


18.2167 


18.4596 


20.5662 


19.4312 


000 


.425 


17.0531 


17.2805 


19.2525 


18.19 


00 


.38 


15.2475 


15.4508 


17.214 


16.264 





.34 


13.6425 


13.8244 


15.402 


14.552 


1 


.3 


12.0375 


12.198 


13.59 


12.84 


2 


.284 


11.3955 


11.5474 


12.8652 


12.1552 


3 


.259 


10.3924 


10.5309 


11.7327 


11.0852 


4 


.23$ 


9.5497 


9.6771 


10.7814 


10.1864 


5 


.22 


8.8275 


8.9452 


9.966 


9.416 


6 


.203 


8.1454 


8.254 


9.1959 


8.6884 


7 


.18 


7.2225 


7.3188 


8.154 


7.704 


8 


.165 


6.6206 


6.7089 


7.4745 


7.062 


9 


.148 


5. 9385 


6.0177 


6.7044 


6.3344 


10 


.134 


5.3767 


5.4484 


6.0702 


5.7352 


11 


.12 


4.815 


4.8792 


5.436 


5.136 


12 


.109 


4.3736 


4.4319 


4.9377 


4.6652 


13 


.095 


3.8119 


3.8627 


4.3035 


4.066 


14 


.083 


3.3304 


3.3748 


3.7599 


3.5524 


15 


.072 


2.889 


2.9275 


3.2616 


3.0816 


16 


.005 


2.6081 


2.6429 


2.9445 


2.782 


17 


.058 


2.3272 


2.3583 


2.6274 


2.4824 


18 


.049 


1 


.9661 


1.9923 


2.2197 


2.0972 


19 


.042 


1 


.6852 


1.7077 


1.9026 


1.7976 


20 


.035 


1 


.4044 


1.4231 


1.5855 


1.498 


21 


.032 


1 


.284 


1.3011 


1 .4496 


1.3696 


22 


.028 


1 


.1235 


1.1385 


1.2684 


1.1984 


23 


.025 


1 


.0031 


1.0165 


1.1325 


1.07 


24 


.022 




.8827 


.8945 


.9966 


.941<5 


25 


.02 




.8025 


.8132 


.906 


.856 


26 


.018 




.7222 


.7319 


.8154 


.7704 


27 


.016 


.642 


.6506 


.7248 


.6848 


28 


.014 


.5617 


.5692 


.6342 


.5992 


29 


.013 


.5216 


. 5286 


.5889 


.5564 


30 


.012 


.4815 


.4879 


.5436 


.5136 


31 


.01 


.4012 


.4006 


.453 


.428 


32 


.009 
.008 


.3611 
.321 


.3659 


.4077 


.3852 


33 


.3253 


.3624 


.3424 


34 


.007 


.2809 


.2846 


.3171 


.2996 


35 


005 


.2006 


.2033 


.2265 


.214 


36 


004 J 




.1605 


.1626 


.1812 


.1712 



Wire — per Lineal Foot. 

Diameter determined by Birmingham Gauge. 



.454 


.546207 


.55136 


.623913 


.58928G 


.425 


.478656 


.483172 


.546752 


.516407 


.38 


.38266 


.38627 


.437099 


.41284 


.34 


.30634 


.30923 


.349921 


.3305 


.3 


.2385 


.24075 


.27243 


.25731 


.284 


.213738 


.215755 


.244110 


.230590 


.259 


.177765 


.179442 


.203054 


.191785 



116 THICKNESS OF SHEET BRASS, SILVER, GOLD, ETC. 



Table— {Continued). 



No of 


Diameter of 
each Number. 




Wire — per Lineal Foot. 




Gauge. 


Wrought Iron. 


Steel. 


Copper. 


Brass. 




Ins. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


4 


.238 


.150107 
.12826 


.151523 
.12947 


.171461 
.146507 


.161945 


5 


.22 


.138376 


6 


.203 


.109204 


.110234 


.12474 


.117817 


7 


.18 


.08586 


.086667 


.098075 


.092632 


8 


.165 


.072146 


.072827 


.08241 


.077836 


9 


.148 


.058046 


.058593 


.066303 


.062624 


10 


.134 


.047583 


.048032 


.054353 


.051336 


11 


.12 


.03816 


.03852 


.043589 


.04117 


12 


.109 


.031485 


.031782 


.035964 


.033968 


13 


.095 


.023916 


.024142 


.027319 


.025802 


14 


.083 


.018256 


.018428 


.020853 


.019696 


15 


.072 


.013738 


.013867 


.015692 


.014821 


16 


.065 


.011196 


.011302 


.012789 


.012079 


17 


.058 


.008915 


.008999 


.010183 


.009618 


18 


.049 


.006363 


.006423 


.007268 


.006864 


19 


.042 


.004675 


.004719 


.00534 


.005013 


20 


.035 


.003246 


.003277 


.003708 


.003502 


21 


.032 


.002714 


.002739 


.0031 


.002928 


22 


.028 


.002078 


.002097 


.002373 


.002241 


23 


.025 


.001656 


.001672 


.001892 


.001787 


24 


.022 


.001283 


.001295 


.001465 


.001384 


25 


.02 


.00106 


.001070 


.001211 


.001144 


26 


.018 


.0008586 


.0008667 


.0009807 


.0009263 


27 


.016 


.0006784 


.0006848 


.0007749 


.0007319 


28 


.014 


.0005194 


.0005243 


.0005933 


.0005604 


29 


.013 


.0004479 


.0004521 


.0005116 


.0004832 


30 


.012 


.0003816 


.0003852 


.0004359 


.0004117 


31 


.01 


.000265 


.0002675 


.0003027 


.0002859 


32 


.009 


.0002147 


.0002167 


.0002452 


.0002316 


33 


.008 


.0001696 


.0001712 


.0001937 


.000183 


34 


.007 


.0001299 


.0001311 


.0001483 


.0001401 


35 


.005 


.00006625 


.00006688 


.00007568 


.00007148 


36 


.004 


.0000424 


.0000428 


.00004843 


.00004574 



Thickness of Slieet Brass, Silver, Gold, etc. 

By Birmingham Gauge for these Metals. 



No. 


Thickn.; 


No. 


Thickn. 


No. 


Thickn. 


No. 


Thickn. 


No. 


Thickn 


No. 


Tbiekn. 




Inch. 




Inch. 




Inch. 




Inch. 




Inch. 




Inch. 


1 


.004 


7 


.015 


13 


.036 


19 


.064 


25 


.095 


31 


.133 


2 


.005 


8 


.016 


14 


.041 


20 


.067 


26 


.103 


32 


.143 


3 


.008 


9 


.019 


15 


.047 


21 


.072 


27 


.113 


33 


.145 


4 


.010 


10 


.024 


16 


.051 


22 


.074 


28 


.120 


34 


.148 





.013 


11 


.029 


17 


.057 


23 


.077 


29 


.124 


35 


.158 


6 


.013 


12 


.034 


18 


.061 


24 


.082 


30 


.126 


36 


.167 



Braziers' and. Sheathing Copper. 

Braziers' Siiefts, 2x4 feet from 5 to 25 lbs., 2)£X5 fret from 9 to 150 lbs., and 
8X5 feet and 4x6 feet, from 16 to 300 lbs. per sbeet 

Sheathing Copper, 14x43 inches, and from i4 to 34 oz. per square foot. 
Yellow Metal, 14x48 inches, and from 10 to 34 oz. per square foot 



WEIGHTS OF ANGLE IRON AND COPPER BOLTS. 117 



Comparative Thicknesses of AVire Gauges. 





American. 






Birmingh 


am. 




No. 


Inch. . 


No. 


Inch. 


No. 


Inch. 


No. 


Inch. 


0000 

00 



2 


& + 


5 
8 

14 
20 


£ - 

%, + 

%. + 


0000 
00 

1 

3 


% + 

% + 
%, - 
% + 


7 

11 
16 
21 


V — 



Weight of Wrought Angle Iron, 
From l.}£ to 4.X Inches. 







ONE FOOT 


IN LENGTH. 






Thickness 


measured in 


he Middle of each Side. 




L Equal Sides. 




L Unequal Sides. 




Sides. 


Thickness. 


Weight. 


Sides. | Thickness. 


Weight. 


Ins. 


Ins. 


Lbs. 


Ins. 


Ins. 


Ins. 


1.25x1.25 


V 


1.5 


4. x3. 


% 


11. 


1.5 xl.5 


2. 


4. x3.5 


% 


11.5 


1.75x1.75 


-it 


3. 


4. x3.5 


u 


11.75 


2. x2. 


.* 


3.5 


4.5 x3. 


Vi 


11.75 


2.25x2.25 


-iZ 


4.5 


5. X 3. 


y* 


12.65 


2.5 x2.5 


•x 


5. 


5. x3. 


% 


13.7 


3. X3. 


• M 


7. 


5.5 X3.5 


% 


14.5 


3.5 x3.5 


•/16 


9. 


5.5 X3.5 


K 


15.6 


4. X4. 


* X A 


12.5 


6. X3.5 


% 


18. 


4.5 x4.5 


-K 


14. 


6. x4.5 


-% 


20. 


4.5 x4.5 


.%/ 


16. 


T2. X2.375* 


-% 


5.5 






2.5 X2.875 


i 

*/lG 


6.5 


L Unequal Sidi 


s. 


3.5 X3.5 


10.5 


3. X2.5 


M 


6.25 


4. X J£-i 

X3.5 x .% j 




13. 


3.5 x3. 


'1 

•/i6 


7.75 




3.5 x3. 


9.6 


4. x3.5 


■% 


13.5 


* This column gives the 


depth of the 


web added to the thickn 


ess of t 


he base or 


flange. 













Weight of Copper Rods or Bolts, 

From % to 4 Inches in Diameter. 



ONE FOOT IN LENGTH. 



Diflin 


Weight. 


Dinm. 


Weight. D 


iam. 


Weight. 


| Diam. 


Weight. 


Inch. 


Lbs. 


Inch. 


Lbs. I 


neh 


Lbs. 


Inch. 


Lbs. 


»K 


.0473 


•% 


1.9982 1 


X 


6.8109 


2.% 


22.8913 


9i 


.1064 


-K 


2.3176 


X 


7.3898 


>% 


25.0188 


m 


.1892 


.% 


2.6605 


% 


7.9931 


3. 


27.2435 


iX* 


.2956 


l. 


3.027 


% 


9 .2702 


-H 


29.5594 


.% 


.4256 


l-fts 


3.417 


% 


10.642 


•X 


31.9722 


.% 


.5794 


•X 


3.8312 2 




12.1082 


>% 


34.4815 


iK 


.7567 


J* 


4.2688 


% 


13.6677 


•S 


37.0808 


•X 


.9578 


•X 


4.7228 


K 


15.3251 


•# 


39.7774 


.% 


1.1824 


& 


5.214 


% 


17.075 


..& 


42.568 


*% 


1.4307 


•% 


5.7228 


M 


18.9161 


•K 


4;"). 45 5 


K 


1.7027 


'Ax, 


6.2547 


% 


20.8562 


4. 


48.433 



118 WEIGHTS OF IROX, COPPER, ETC., AND OF PIPES. 



Weignt of a Square Foot of* Cast and. Wrought 
Iron, Copper, Lead, Brass, and .Zinc. 







From Xe t° 1 I' iC h W 


TJtickness. 






Thickn. 


Cast Iron. 


Wrought Iron. 


Copper. 


Lead. | 


Brass. 


Zinc. 


Inch. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


15 


2.346 


2.517 


2.89 


3.691 


2.675 


2.34 


•Vs 


4.693 


5.035 


5.781 


7.382 


5.35 


4.68 


-% 


7.039 


7.552 


8.672 


11.074 


8.025 


7.02 


-K 


9.386 


10.07 


11.562 


14.765 


10.7 


9.36 


.2 


11.733 


12.588 


14.453 


18.456 


13.375 


11.7 


•% 


14.079 


15.106 


17.344 


22.148 


16.05 


14.04 


.% 


16.426 


17.623 


20.234 


25.839 


18.725 


16.34 


•M 


18.773 


20.141 


23.125 


29.53 


21.4 


18.72 


.% 


21.119 « 


22.659 


26.016 


33.222 


24.075 




.% 


23.466 


25.176 


28.906 


36.913 


26.75 




.% 


25.812 


27.694 


31.797 


40.604 


29.425 




;£ 


28.159 


30.211 


34.688 


44.296 


32.1 




30.505 


32.729 


37.578 


47.987 






•% 


32.852 


35.247 


40.469 


51.678 






•% 


35.199 


37.764 


43.359 


55.37 






1. 


37.545 


40.282 


46.25 


I 59.061 







NOTE.- 

per that 



-The Wrought Iron is that of hard rolled Pennsylvania plate?, and the Cop- 
of hard rolled plates from the works of Messrs. Phelps, Dodge & Co., Conn. 






"Weight of* Riveted Iron and Copper Pipes, 

From 5 to 30 Inches in Diameter, from }i to % 6 in Thickness 



ONE FOOT TN LENGTH. 



Diam. 


Thickn. | 


Iron. 


Copper. 1 


Diam. 


Thickn. 


Iron j 


Copper. 


Inch. 


Inch. 


Lbs. 


Lbs. 


Inch. 


Inch. 


Lbs. 


Lbs. 


5. 


Vs 


7.12 


8.14 


9. 


y 


25.01 


28.58 




X 


10.68 


12.21 




y 


26.33 


30.09 




y 


14.25 


16.28 


10. 


y 


27.75 


31.71 


5.# 


y% 


7.78 


8.89 


io. y 2 


y 


29.19 


33.22 




% 


11.66 


13.33 


ii. 


y 


30.49 


34.85 




y 


15.56 


17.78 


12. 


y 


33.13 


37.86 


6. 


% 


8.44 


9.64 


13. 


y 


35.88 


41. 




X 


12.65 


14.46 


14. 


y 


38.52 


44.02 




y 


16.88 


19.29 


15. 


y 


41.26 


47.15 


6-K 


X 


9.1 


10.4 




x 


51.57 


58.94 




X 


13.65 


15.6 


16. 


y 


43.9 


50.17 




y 


18.2 


20.8 




X, 


54.87 


62.71 


7. 


Vs 


9.78 


11.18 


17. 


y 


46.53 


53.18 




X 


14.68 


16.78 




X, 


58.17 


66.48 




M 


19.57 


22.37 


18. 


y 


49.17 


56.2 


7-K 


y 


10.49 


11.99 




X, 


61.47 


70.25 




X 


15.73 


17.98 


20. 


X, 


68.07 


77.79 




y 


20.89 


23.87 


24. 


X, 


81.33 


92.95 


8. 


y 


16.7 


19.08 


25. 


%> 


84.57 


96.65 




y 


22.26 


25.44 


28. 


X, 


94.56 


107.95 


•♦# 


y 


23.59 


| 26.96 


30. 


X, 


101.14 


115.59 



The abore weights include the laps of the sheets for riveting and calking. 

The weights of the rivets are not added, as the number per lineal foot of pipe de- 
pends upon the distance they are placed apart, and their diameter and length upon 
the thickness of the metal of the pipe. 



WEIGHT OF BOILER TUBES AND SHEATHING NAILS. 119 



Table of Standard. Dimensions of Wrought Iron 
Welded Tubes. 



i 


a 




| 


a 

s 


a* 

3 


6.3.5 




eS 


"o 


Eh 


"i"® 


3 


Q 


I 


A 





O 


«g £ 3 


2 


< 


a, 


1 


"3 
a 

1 


a 


H 


s 

£ 

a 


e 


"3 
g 
.2 

X 


tStn a 
^^2 




a 
"a 


[3d 

i 


ill 


Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


Ins 


Feet. 


Feet. 


Ins. 


Lbs. 




% 


.40 


.068 


.27 


.85 


1.27 


14.15 


9.44 


.057 


.24 


27 


% 


.54 


.088 


.36 


1.14 


1.7 


10.5 


7.075 


.104 


.42 


18 


% 


.67 


.091 


.49 


1.55 


2.12 


7.67 


5.657 


.192 


.56 


18 


% 


.84 


.109 


.62 


1.96 


2.65 


6.13 


4.502 


.305 


.84 


14 


% 


1.05 


.113 


.82 


2.59 


3.3 


4.64 


3.637 


.533 


1.13 


14 


1 


1.31 


.134 


1.05 


3.29 


4.13 


3.66 


2.903 


.863 


1.67 


UK 


1^ 


1.66 


.14 


1.38 


4.33 


5.21 


2.77 


2.301 


1.496 


2.26 


H^ 


Vti 


1.9 


.145 


1.61 


5.06 


5.97 


2.37 


2.01 


2.038 


2.69 


H^ 


2 


2.37 


.154 


2.07 


6.49 


7.46 


1.85 


1.611 


3.355 


3.67 


11^ 


2K 


2.87 


.204 


2.47 


7.75 


9.03 


1.55 


1.328 


4.783 


5.77 


8 


3 


3.5 


.217 


3.07 


9.64 


11. 


1.24 


1.091 


7.388 


7.55 


8 


3X 


4. 


.226 


3.55 


11.15 


12.57 


1.08 


0.955 


9.887 


9.05 


8 


4 


4.5 


.237 


4.07 


12.69 


14.14 


.95 


0.849 


12.73 


10.73 


8 


4^ 


5. 


.247 


4.51 


14.15 


15.71 


.85 


0.765 


15.939 


12.49 


8 


5 


5.56 


.259 


5.04 


15.85 


17.47 


.78 


0.629 


19.99 


14.56 


8 


6 


6.62 


.28 


6.06 


19.05 


20.81 


.63 


0.577 


28.889 


18.77 


8 


7 


7.62 


.301 


7.02 


22.06 


23.95 


.54 


0.505 


38.737 


23.41 


8 


8 


8.62 


.322 7.98 


25.08 | 27.1 


.48 


0.444 


50.039 


28.35 


8 


9 


9.69 


.344! 9. 


28.28 {30.43 


.42 


0.394 


63.633 


34.08 


8 


LO 


10.75 


.366! 


10.02 


31.47 


33.77 


.38 


0.355 


78.838 


40.64 


8 



Diameter and Weight of Lap-welded Iron Boiler 

Tubes — [Prosser's Patent.} 



External 


Thickness. 


Average 


Price 


External 


Thickness. 


Average 


Price 


Diameter 


Wire Gauge. 


Weight. 


pr. Foot 


Diameter. 


Wire Gauge. 


Weight. 


pr. Foot. 


Ins. 


No. 


Lbs. pr. Ft. 


Cents. 


Ins. 


No. 


Lbs.* per Ft. 


Cents. 


m 


16 


1 




3 


11 


3.5 




V4 


15 


1.16 




3^ 


11 


4 




1% 


14 


1.63 




4 


8 


6.4 




2 


13 


2 




5 


7 


9.1 




23€ 


12 


2.16 




6 


6 


12.3 




?H 


12 


2.56 




7 


6 


15.2 




2% 


11 


2.2 




8 


7 


16 





Weight of Composition Sheathing Nails. 



Xo. 


Length. 


Number in 
a Pound. 


No. 


Length. 


Number in 
a Pound. 


No. 


Length 




Ins. 






Ins. 






Ins. 


1 


% 


290 


6 


1 


190 


10 


iM 


2 


% 


260 


7 


m 


184 


11 


m 


3 


1 


212 


8 


w 


168 


12 


2 


4 


1% 


201 


9 


\% 


110 


13 


2^ 


5 


ix 


199 













Number in 
a Pound 



101 
74 
64 
59 



120 WEIGHT OF BOILER TUBES AND SHEET IRON. 

To Ascertain tlie Weight ofWrought Iron, Copper, 
or I3rass Tubes and. Pipes per Lineal Foot. 





From 


}/i an Inch in Internal Diameter to 6 Inches. 




Diam. 


Area of Plate. 


Diam. 


Area of Plate. 


Diam. 


Area of Plate. 


Diam. 


Area of Plato- 


Ins. 


Sq.Feet 


Ins. 


Sq. Feet. 


Ins. 


Sq. Feet. 


Ins. 


Sq. Feet. 


8 


.1309 


1% 


.3436 


2% 


.7199 


Vi 


1.1781 


/g 


.1473 


IK 


.36 


2% 


.7526 


±% 


1.2108 


1 


.1636 


J% 


.3764 


3 


.7854 


4M 


1.2435 


.18 


«* 


.3927 


3K 


.8181 


4% 


1.2763 


% 


.1964 


1M 


.4254 


BX 


.8508 


5 


1.309 


% 


.2127 


iM 


.4581 


o% 


.8836 


G& 


1.3417 


% 


.2291 


l« 


.4909 


3V 2 


.9163 


«X 


1.3744 


% 


.2454 


2 


.5236 


m 


.949 


*% 


1.4072 


l 


.2618 


2J6 


.5543 


3M 


.9818 


W% 


1.4399 


m 


.2782 


2^ 


.587 


4 


1.0472 


W* 


1.4726 


\H 


.2945 


2M 


.6198 


±Vs 


1.0799 


m 


1.5053 


IK 


.3105 


2K 


.6545 


4X 


1.1126 


m 


1.5381 


m 


.3272 


2% 


.6872 


4% 


1.1454 


6 


1.5708 



.Application of* tlie '-Talkie. 
When the Thickness of the Metal is given in the Divisions of an Inch. 
To the internal diameter of the tube or pipe add the thickness of 
the metal ; take the area of a plate in square feet, from the table for 
a diameter equal to the sum of the diameter and thickness of the tube 
or pipe, and multiply it by the weight of a square foot of the metal 
for the given thickness (see tables, page 118), and again by its length 
in feet. 

Illustration. — Required the weight of 10 feet of copper tube 1 inch in diameter 
and .125 of an inch in thickness. 

1-f % =1X== -2945 square feet for 1 foot of length. 

Weight of 1 squara foot of copper %t\\ of an inch in thickness, per table, page 118, 
= 5.7S1 lbs. ; then .2945x.5781 = 1.7025 lbs. 

When the Thickness of the Metal is given in Numbers of a Wire Gauge. 

To the internal diameter of the tube or pipe add the thickness of 
the number from table, pages 113-115; multiply the sum by 3.1416, 
divide the product by 12, and the quotient will give the area of the 
plate in square feet. Then proceed as before given. 

Illustration Required the weight of 10 feet of copper pipe 2 inches in diame- 
ter, and No. 2 American wire gauge in thickness. 

2 -f .25763 x 3.1416 -h 12 = 2.25T63 X 3.1410-^-12 = .591 square feet; then .591 
X 11.0706 (weight from table) — 6.897 lbs. 



Table showing the Thickness and. Weight of Gal- 
vanized Sheet Iron. 

Dimensions of Sheet, 2 Feet in Width by from 6 to 9 Feet in Length. 



Wire 


Weight per 


Wire 


Weight per" 


Wire 


Weight per 


Wire 


Weight p«r 


Gaupe. 


Sq. Foot. 


Gauge. 


Sq Foot. 


Gausre. 


Sq. Foot. 


Gau ere. 


Sq. Foot. 


No. 


Oz. 


No. 


Oz. 


No. 


Oz 


No. 


Or. 


30 


30 


26 


15 


22 


21 


18 


37 


29 


n 


25 


16 


21 


24 


17 


43 


28 


12 


24 


17 


20 


28 


16 


48 


27 


14 


23 


19 


19 


33 


14 


60 



WEIGHTS, ETC., OF BRASS TUBES AND TIN-PLATES. 121 

Talkie of Dimensions and. /Weights of Seamless 
Brass and. Copper Tubes. — [American Tube Works,} 



External Extreme 


Weight per 


Wire Gauge. 


External 


Extreme 


Weight per 


Wire Gauge. 


Piam. 


Length. 


Foot. 


Eng. 


Diam. 


Length. 


Foot. 


Eng. 


Ins. 


Feet. 


Lbs. 


No. 


Ins. 


Feet. 


Lbs. 


No. 


% 


8 


.375 


18 


2 


15 


2.05 


12 & 14 


ft 


11 


.5 


17 


2« 


12 


2.5 


12&14 


% 


7 


.625 


17 


22 


13 


2,375 


12 & 14 


1 


8 


.75 


16 


m 


12 


•2.5 


12 & 14 


VH 


13 


1.25 


■12.&14 


m 


12 


2.66 . 


12 & 14 


i& 


13 


1.5 


12 & 14 


z% 


12 


3. 


112 & 14 


i 


12 


1.625 


12&14 


3 


12 


3.33 


112 & 14 


13 


1.7.5 


12&14 


$H 


10 


3.875 


J12&14 


12 


1.813 


12 & 14 


?H 


9 


4.25 


12', . 


1% 


12 


1.875 


12&14 


4 


: % 


5. 


12 



Marks and Weic^t of English Tin-plates. 

Brand. Plates per Box. Length and Breadth 



Net Weight per 
Box. 



1 C or 1 Com 

2C 

3C 

HC 

HX 

IX 

2X 

3X 

1XX 

1XXX 

1 XXXX 

1XXXXX 

1XXXXXX. 

DC 

DX 

DXX 

DXXX 

DXXXX 

SDC 

SDX 

SDXX • 

SDXXX 

SDXXXX 

SDXXXXX 

SDXXXXXX 

Leaded IC ....... . 

" IX 

ICW 

IXW -.. 

CSDW 

CIIW 

XIIW 

TT 

XTT 



No. 

225 
225 
225 
225 
225 
225 
225 
225 
225 
225 
225 
225 
225 
100 
100 
100 
100 
100 
200 
200 
200 
200 
200 
200 
200 
112 
112 
225 
225 
200 
100 
100 
450 
450 



Ins. 

13% by 10 



13% 
12% 
13% 
13% 
13% 
13%. 
12% 
13% 
13%. 
13% 
13% 
13% 
16% 
16%: 
16% 
16%. 

s* 

15 

15 

15 

15 

15 

15 

20 

20 

13% 

13% 

15 

1«% 
16% 
13% 
13% 



9% 

l8* 

10 
10 
9%. 

^A 

.10 

10 .. 

10 

10 

10 

12K 

12K 

12* 

12* 

Hi 

ii 
ii 
n 
ii 
ii 
ii 
ii 

14 
14 
10 
10 
11 

PX 

10 
10 



Lbs, 
112 

105. 

98 
119 
157 
140 
133, 
126 
161 
182 
203 
224 
245 

98 
126 
147 
168 
189 
168 
188 
209 
230 
251 
272 
293 
112 
140 
112 
140 
168 
105 
126 
112 
126 



When the plates are 14 by 20 inches, there are 112 in a box. 
L 



122 WEIGHT OF LEAD AND TIN PIPE. CISTERNS. 



Weight of? Lead. and. Tin. Pipe per Foot. 

From % to 5 Inches in Diameter. 



WATER-PIPE. 



Internal 
Diam. 


Thickness. 


Weight. 


Internal 
Diam. 


Thickness. 


Weight. 


Internal 
Diam. 


Thickness. 


Wsight. 


Ins. 


Ins. 


Lbs. 


Ins. 


Ins. 


Lbs. 


Ins. 


Ins. 


Lbs. 


■ % 


.06 


1 .0424 


1. 


.10 


1.5 


2. 


.22 


7. 




%■ 


.08 


.625 


1. 


.11 


2. 


2. 


.27 


9. 




■% 


.12 


1. 


1. 


.14 


2.5 


2-K 


-%. 


8. 




% 


.16 


1.25 


1. 


.17 


3.25 


-S 


•M 


11. 




% 


.19 


1.5 


1. 


.21 


4. 


,s 


.%, 


14. 




%■ 


.07 


.0545 


1. 


.24 


4.75 


•X 


•% 


17. 




■n 


.09 


.75 


i.k 


.10 


2. 


3. 


•£ 


9. 




•k 


.11 


1. 


■2 


.12 


2.5 


3. 


•% 


12. 




M 


.13 


1.25 


.2 


.14 


3. 


3. 


•%, 


16. 




M 


.16 


1.75 


.& 


.16 


3.75 


3. 


•% 


20. 




M 


.19 


2. 


.% 


.19 


4.75 


3-M 


.% 


9.5 




% 


.08 


.0727 


•u 


.25 


6. 


•K 


•X 


15. 




% 


.09 


1. 


i-K 


.14 


3.5 


•X 


•% 


18.5 




% 


.13 


1.5 


,& 


.17 


4.25 


•X 


.% 


22. 




% 


.16 


2. 


.jf 


.19 


5. 


4. 


-X 


12.5 




% 


.20 


2.5 


•k 


.23 


6.5 


4. 


:£ 


16. 




% 


.22 


2.75 


\H 


.27 


8. 


4. 


•X 


21. 




Va. 


.08 


.0969 


i-K 


.13 


4. 


4. 


•K 


25. 




% 


.10 


1.25 


M 


.17 


5. 


4-K 


.% 


14. 




% 


.12 


1.75 


.% 


.21 


6.5 


•^ 


-X 


18. 




% 


.16 


2.25 


-K 


.27 


8.5 


5. 


•3€ 


20. 




% 


.20 


3. 


2. 


.15 


4.75 


5. 


•I 


31. 




% 


.23 


3.5 


2. 


.18 


6. 













WASTE-PIPE. 






Internal Diam 


Weight. 


Internal Diam. j Weight. 


Internal Diam. 


Weight. 


Ins. 


Lbs. 


Ins. 


Lbs. 


Ins. 


Lbs. 


M* 


2. 


4. 


5 


*-X 


8 


2. 


3. 


4. 


6 


0. 


8 


3. 


3.5 


4. 


8 


5. 


10 


3. 


5. 


4-K 


6 


5. 


12 







BLOCK-TIN PIPE. 






% 


.3594 


•% 


.5 


I.V 


1.25 


% 


.375 


•% 


.625 


-X 


1.5 


% 


.5 


M 


.625 


i.^ 


2. 


K 


.375 


.% 


.75 


•S 


2.5 


M 


.5 


1. 


.9375 


2. 


3. 


U 


.625 


1. 


1.125 


2-K 


3.75 



Capacity ojf Cistern in Gallons. 

For each 10 Inches in Depth. 



Diam. 


Gallons. 


Diam. 


Gallons. 


Diam. 


Gallons. 


Diam. 


Gallons. 


Diam. 


Gallons. 


Feet. 




Feet 




Feet. 




Feet. 




Feet. 




2- 


19.5 


4.5 


99.14 


7. 


239.88 


9.5 


441.4 


14. 


959.6 


2.5 


30.6 


5. 


122.4 


7.5 


275.4 


10. 


489.6 


15. 


1101.6 


3. 


44.07 


5.5 


H8.1 


8. 


313.33 


11. 


592.4 


20. 


1958.4 


3.5 


59.97 


6. 


176.25 


8.5 


353.72 


12. 


705. 


25. 


3059.9 


4. 


78.33 


6.5 


206.85 


9. 


396.56 


13. 


827.4 


SO. 


4406.4 



DIMENSIONS AND WEIGHTS OF BOLTS AND NUTS. 123 



Dimensions and. AVeights of Bolts and Nuts. 



SQUARE AND HEXAGONAL. 



Diam 


Depth of 
Nut. 


Width of 

Square 

Nut. 


* Diam. 

ofHexa'l 

Nut. 


f Width 

of 

Head. 




Volume. 




of Bolt. 


Square 
Nut. 


Hexagonal 
Nut. 


Hexagonal 
Head. 


1 Bolt per In. 
of Length. 


Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


Cub. Ins. 


Cub. Ins. 


Cub. Ins. 


Cub. Ins. 


•% 


.15 


.2 


-U 


.2 


.00416 


.00425 


.0045 


.01227 


.% 


.2 


.3 


.% 


.3 


.01248 


.01276 


.0152 


.62761 


'X 


.25 


.45 


-M 


.4 


.03835 


.02836 


.036 


.04908 


S 


.35 


.55 


.% 


•K 


.07903 


.06235 


.07 


.07669 


*% 


.4 


.6 


•X 


.6 


.09984 


.10209 


.1215 


.1104 


>%> 


.5 


.75 


•% 


.7 


.2061 


.17368 


.1929 


.1503 


•% 


.55 


.85 


l. 


-% 


.28941 


.25584 


.2531 


.1963 


V 


.6 


.95 


i.K 


.85 


.3924 


.34449 


.3658 


.2485 


•% 


.7 


1.1 


hX 


.95 


.6323 


.49625 


.5076 


.3067 


•% 


.75 


1.2 


i-M 


1.05 


.8016 


.64328 


.6822 


.3712 


M 


.8 


1.3 


l-M 


hH 


.9986 


.81664 


.8543 


.4417 


.% 


.9 


1.4 


$.f? 


l-H 


1.2977 


1.0782 


1.143 


.5184 


.% 


.95 


1.5 


IH 


1.35 


1.5663 


1.3199 


1.435 


.6013 


i. 


1.1 


1.75 


2. 


1-X 


2.5048 


1.996 


2.025 


.7854 


i.K 


1.25 


1.95 


*-H 


1.7 


3.5106 


2.8701 


2.926 


.994 


Wt 


1.37 


2.15 


*•& 


1» 


4.6518 


2.8846 


3.955 


1.227 


i-M 


1.5 


2.4 


*-X 


2.1 


6.414 


5.1474 


5.457 


1.484 


i.jl 


1.65 


2.6 


3. 


2.3^ 


8.2384 


6.737 


6.834 


1.767 


i.% 


1.8 


2.8 


S.% 


2.45 


10.381 


8.6267 


8.778 


2.073 


i.j! 


1.9 


3. 


B-K 


2.% 


12.53 


10.559 


10.853 


2.405 


1.% 


2.05 


3.25 


3.% 


2.8 


15.993 


13.058 


13.23 


2.761 


2 


2.2 


3.45 


4. 


3. 


19.275 


15.97 


16.2 


3.141 


2-K 


2.35 


3.7 


*-X 


3.& 


23.838 


19.257 


19.43 


3.546 


2 -^ 


2.5 


3.9 


4-K 


3-M 


28.085 


22.966 


23.066 


3.976 


2 -M 


2.6 


4.1 


4.% 


3.% 


32.188 


26.613 


27.128 


4.43 


2.£ 


2.75 


4.3 


5. 


3-5 


37.351 


31.19 


31.641 


4.908 


2-M 


2.9 


4.55 


5.3^ 


3.% 


44.345 


36.263 


36.628 


5.411 


2-M 


3. 


4.75 


5.& 


4-K 


49.871 


41.17 


42.114 


5.939 


2-% 


3.15 


4.95 


5.% 


4.% 


56.736 


47.249 


48.123 


6.491 


3. 


3.3 


5.2 


6. 


f.'S? 


65.908 


54.105 


54.675 


7.068 


3.^ 


3.6 


5.65 


6-K 


4.% 


85.059 


69.003 


69.514 


8.295 


3-K 


3.85 


6.1 


7. 


5.3^ 


106.218 


85.582 


86.822 


9.621 


S.Ji 


4.1 


6.5 


7.JI 


B.» 


127.945 


104.626 


106.787 


11.044 


4. 


4.4 


6.95 


8. 


6. 


157.241 


127.75 


129.6 


12.566 


4-3€ 


4.65 


7.35 


8-3^ 


*.% 


185.24 


152.411 


155.45 


14.186 


ft« 


4.95 


7.8 


9. 


6.% 


222.43 


181.893 


184.528 


15.904 


4.% 


5.2 


8.25 


9-K 


?•% 


261.781 


212.901 


217.023 


17.72 


5. 


5.5 


8.65 


10. 


7-X 


303.531 


249.507 


253.125 


19.635 


5.^ 


5.75 


9.1 


10. x 


7-K 


351.687 


287.589 


293.024 


21.647 


5-K 


6.05 


9.5 


11. 


8.% 


402.277 


332.097 


336.909 


23.758 


5-Ji 


6.3 


9.95 


ii. X 


*•% 


441.224 


377.872 


384.971 


25.967 


6. 


6.6 


10.4 


12. 


9. 


527.248 


431.152 


437.4 


28.274 



* Extreme diameter of nut. 

t Square or hexagonal, and the depth of it should be .8 of the diameter of the bolt. 

When the weight of a bolt and nut is required, Ascertain the volume 
for the bolt from the inside of the head to its point; add to this the 
volume obtained from the table for the diameter of bolt and descrip- 
tion of nut given; multiply the sum by the units in page 163 for the 
weight of a cubic inch of the metal of which the bolt and nut is made, 
and the quotient is the weight in pounds. 



124 DIMENSIONS, WEIGHTS, ETC., OF BOLTS AND NUTS. 

Illustration.— A wrought iron bolt and nut (hexagonal nut) is 1 inch in diameter 
and 10 inches in length from inside of head to end. 

Note. — The length of a bolt and nut is taken from the inside of the head to the 
inside of the nut, or its greatest capacity when in position. 

In a computation of the weight, it is necessary to measure the extreme length of 
the bolt, viz., from the inside of the head to the point. 

Volume for head 1.5x S = depth of head = 1.8 cub. ins. 

u " 1 inch of bolt 7354, which XlO — T.S54 " 

"" ofnut i : 1/J96 " 

11.050 " 
3-hich X.281G (page 161), the weight of a cubic inch of wrought iron bolt= 3.2S lbs. 

Table showing the Nmnher of Threads to an Inch, 
in V-thread Screws. 



Diara. 


Thre'ds.' 


Diam. 


Thr'ds. 


Diam. 


Thr'ds.! 


Diam. 


Thr'ds. 


Diam. 


Thr'ds. 


Diam. 


Thr'ds. 


Ins. 


No. 


Ins. 


No. 


In a. 


No. 


Ins. 


No. 


Ins. 


No. 


Ins. 


No. 


U 


20 


% 


10 


IK 


6 


*K 


4 


m 


3 


5 


2M 


X, 


18 


% 


9 


y% 





*% 


oK 


4 


o 


5« 


2M 


1 


16 


l 


8 


m 


5 


3 


3K 


w 


2« 


*X 


r% 


14 


\% 


4 


i% 


<*K 


Wa, 


BK 


V4 


2% 


o% 


2X 


H 


12 


}k 


7 


2 


±A 


*a 


m 


Mi 


Wa 


6 


% 


11 


\% 


6 


*H 


4 













Note. — The diameter of a screw, to work in the teeth of a wheel, should be such 
that the angle of the threads does not exceed 10°. 



Screw Threads, Bolt Heads and Nuts, 

As determined and recommended by Committee of Franklin Institute of 
Philadelphia, 1864. 







NTMBER OF THREADS PEE INCH 


ANGLl 


i 60°. 






Diam. of 
Bolt. 


Thre'ds. 


Diam. of 

Bolt. 


n.L M i,i. '! Diam. of 
Thre d3 ' Bolt. 


Thre'ds. 


Dinm. of 
1 Bolt. 


Thre'ds. 


Diam. of 
Bolt. 


Thre'ds. 


Ins. 


No. 


Ins. 


No. 


Ius. 


No. 


Ins. 


No.. 


Ins. 


No. 


K 


20 


% 


10 


m 


$4 


3 


3K 


m 


w* 


« 


18 


% 


9 


rU 


5 


W 


3K 


5 


*A 


H 


16 


l 


8 


W 


5 


*K 


3^ 


5« 


*A 


K 


14 


IVh 


* 


2 


4K 


3% 


3 


*>A 


*% 


% 


13 


1% 


t 


%% 


4K 


4 


3 


b% 


1% 


%/ 


12 


1% 


6 


2H 


4 


^ 


m 


6 


tu 


% 


11 


IX 


6 , 


m 


4 


! ^A 


2% 







Dimensions of Heads and Nuts. 

Rough Bolt. — The width between the parallel sides of both head and 
nut \)4 times the diameter of the bolt, to which is to be added j^th of 
an inch. 

The depth of the head .5 its width. The depth of the nut equal the 
diameter- of the bolt. 

Finished Boh. — The width between the parallel sides of both head 
and nut %-th of an inch less than for a rough bolt. 

The depth of the nut # a th of an inch less than the diameter of the 
bolt. 



SHIP AND RAILROAD SPIKES AND GAS PIPES. 125 

Dimensions and. Weigh. ts of* Bolts and. Nuts, 
Square and Hexagonal, by the preceding Rules. 



Diam. 
of Bolt. 


Width of 

Sq. Nut 

and Head. 


Diam. of 

Hexago'l 

Nut and 

Head. 




Volume. 




Depth 

of 
Head. 


Volume of 
Bolt per 
Inch of 
Length. 


Square 
Nut. 


Hexagonal 

Nut. 


Hexagonal i Square 
Head Head. 


Ids. 

1 

2 
3 
4 
5 
6 


Ins. 

-'% 


Ins. 

1.878 
3.613 
5.346 

7.08 

8.814 

10.548 


Cub. Ins. 

1.855 
13.248 

42.968 

99.8 
192.53 
329.95 


Cub. Ins. 

1.508 
10.681 
34.536 
80.079 

154.33 

264.3 


Cub. Ins. 

1.864 
13.253 

42.966 
99.794 
192.53 
329.98 


Cub. Ins. 

2.145 
15.529 
49.466 
114.89 
221.66 
379.9 


Ins. 

% 

4% 


Cub. Ins. 

.7854 

3.142 

7.068 

12.566 

19.635 

28.274 



Comparison of Weights between Bolts and Nuts, 
of the Proportion given in the preceding Table, 
and between those determined by the above 
Rules. 

VOLUiME OF IIEXAGONAL IIEAD AND NUT. 

Dimensions. 1 Inch. I 2 Inch. 3 Inch. 4 Inch. 5 Inch. 6 Inch. 



Cub. Ins. 

4.02 
3.37 



Cub. Ins. 

32.17 
23.93 



Cub. Ins. 

108.78 
77. .5 



Cub. Ins. 

257.35 
179.87 



Cub. Ins. 

502.6 
346.86 



Cub. Ins. 

868.55 
594.28 



Ordinary , 

Proposed 

The difference varying from 1.2 to 1.46 per centum in favor of the 
proposed dimensions for equal diameters of bolts. 



Ship and Railroad Spikes. 

Dimensions and Number tee Pound. — [P. C. Page, Mass.] 

Ship Spilces. 



H In- Sq. 


Yeln.Sq. 


&In.Sq. 


M In - Sq. 


&In.Sq. 


KIn.Sq. 


%I.Sq. 


A 


C-3 


A 


.S'-d 


A 


.St3 


a 


.StS 


J 


£ri 


* 


C-H 


A 


e-n 






bo 




bo 




bo 




bo 






C 




a 


§ 
►J 


it 


c 

9 


° 2 


a 
S 




a 




c 

IV 




a 
♦J 




S 
« 

Ins. 


© 2 


Ins. 




Ins. 




Ins. 




Ins. 




Ins. 




Ins. 






3 


19. 


3 


10. 


4 


5.4 


5 


3.4 


6 


2.2 


8 


1.4 


10 


.8 


3K 


15.8 


3K 


9.6 


4K 


5. 


*u 


3.1 


6X 


2. 


9 


1.2 


15 


.6 


4 


13.2 


4 


8.0 


5 


4.6 


6 


3. 


4 


1.9 


10 


1.1 


_ 


_ 


m 


12.2 


m 


6.0 


hi 


4.2 


6K 


2.8 


IK 


1.8 


11 


1. 


_ 


~ 


5 


10.2 


5 


5.8 


6 


4. 


7 


2.6 


8 


1.7 


_ 


- 


_ 


_ 


- 


- 


6 


5.2 


*V, 


3.2 


7* 


2.4 


«Wf 


1.6 


- 


- 


- 


- 














8 


2.2 


9 
10 


1.5 
1.4 


~ 


~ 


- 


- 



Railroad Spikes % '»• square x 5^ ins. 2 . per lb. 

" X i4 '! X5K •{ 1.6 " 



Thickness of Gas Pipes. 



Diam. 


Thickn. 


Diam. 


Thickn. 


Diam. 


Thickn. 


1% to 3 
4 " 6 




8 to 10 
12 " 13 


H 

% 


14 to 15 
I 16 « 48 


% 
% 



126 



SLATING. EARTH DIGGING. — HAY. 



Slating. 

A Square of slate or slating is 100 superficial feet. 

Tlie Lap of slates varies from 2 to 4 inches. The standard is as- 
sumed to be 3 inches. 

The Pitch of a slate roof should not be less than 1 in height to 4 of 
length. 

To Compute tlie Surface of a Slate Allien laid, and. tlie 
Number of Squares of Slating. 

Subtract the lap from the length of the slate, and half the remainder 
will give the length of the surface exposed, which, when multiplied by 
the width of the slate, will give the surface required, and for which the 
party requiring the slating only pays. 

Divide 14400 (the area of a square in inches) by the surface thus 
obtained, and the quotient will give the number of slates required for 
a square. 

Illustration.^- A slate is 24x12 inche?, and the lap is S inchrs. 
24 -3 = 21, and 21-^2 = 10.5, which X 12 = 126 inches; and 14400 -^ 120 = 114.29 
slates. 



Dimensions of Slates— [American.] 






Ids. 


Ins. 


14x7 


14x10 


14x8 


16x 8 


14x9 


1 16x- 9 



Ins. 

16x10 
18 x 9 
18x10 



Ins. 

18x11 
18x12 

: 20xio* 

WELSH. 



Ine. 

20x11 
20x12 
22x11 



Ins. 

22x12 
22x13 
24x12 



Ins. 

24x13 
24x14 
24x16 



Doubles 
Small " 

Plantations. 



Ins. 



13x 7 

11 X 7 
12x10 
13X10 
14x12 



Ladies. 



12x 8 
14 x 8 
16 x 8 
16x10 



Viscountess . . 
Countess .... 
Marchioness . 
Duchess 



18x10 
20x10 
22x12 
24x12 



The thickness of slates ranges from %. to %* of an inch, and their 
weight varies from 2.6 to 4.53 lbs. per square foot. 



Earth Digging. 

Number of Cubic Feet of various Earths in a Ton. 

Loose Earth 24 I Clay..... 18.6 I Clay with Gravel.14.4 

Coarse Sand 18.6 | Earth with Gravel. 17.8 | Common Soil. ...15.6 

The volume of Earth and Sand in bank exceeds that in embank- 
ment in the following proportions : 

Sand i| Clay i J | Gravel ^ 

and the volume of Rock in embankment quarried in large fragments 
exceeds that in bank fully one half. 



Hay. 

550 cubic feet of new meadow hay, and 440 and 500 from large or 
old stacks, will weigh a ton. 

577 to 604 cubic feet of dry clover weigh a ton. 



SPIKES, HORSESHOES. — NAILS. — HILLS. — CATTLE, ETC. 12? 



Patent Spikes and Horseshoes. 











[H. 


BUKDEN, ' 


Troy, N. r.] 


Boat 


Spikes. 


Ship Spikes. 


Hook Head. 


Horseshoes. 


Length. 


No. in Pound. 


Length. 


No. in Pd. 


Length. 


No. in Pd. 


Le'gth. 


No. in Pd 


Ins. 




Ins. 




Ins. 




Ins. 




3 


17.5 


4 


8 


4 X% 


5.55 


1 


.84 


*x 


14.68 


±A 


6.5 


±AxXe 


4.14 


2 


.75 


4 


12.57 


5 


4.37 


5 xA 


2.52 


3 


.65 


*A 


9.2 


*A 


4.3 


!>AxA 


2.41 


4 


.56 


5 


7.2 


6 


4.2 


SAx'A 

6 X^ 


1.87 


5 


.39 


*H 


6.3 


*A 


3.77 


1.72 


— 


— 


6 


4.97 


7 


2.75 


6 X% 


1.38 


— 


— 


6K 


4.78 


?A 


2.5 


7 X 9 Ae 


1.4 


— 


— 


7 


3.62 


8 


1.74 


8 X% 


1.1 


— 


— 


m 


3.37 


8A 


1.63 




— 


— 


— 


8 


2.95 


9 


1.55 


— 


— 


— 


— 1 


*A 


2.9 


10 


1.15 


— 


— 


— 


— 



Length of Horseshoe Nails. 



No. 5 1% inch 

" 6 1% " 



No. 7 l%inch 

44 8 2 



No. 9 23^ inch 

44 10 2^ " 



Leng 


th.s of Iron Nails, 


and. Number 


in a Pound. 


Size. |Lgth. 


No. 


Size. |Lgth.| No. 


Size. 


Lgth.| No. j| Size. 


Lgth.| No. || Size. |Lgth. 


No. 




Ins. 






Ins. 






Ins. 






Ins. 






Ins. 




3d. 


*** 


800 


4c?. 


v-4 


544 


6d. 


2 


250 


Sd. 


2K 


165 


12d. 


33< 


65 




1¥ 


400 


5 


^4 


200 




2 


272 


10 


3 


68 


20 


4 


34 




1U 


560 




W A 


272 


7 


m 


120 




3 


80 


30 


4K 


24 


4 


1% 


400 




1% 


480 




*% 


176 




3 


100 


40 


5 


18 




1% 


288 


6 


2 


152 


8 


m 


92 




3 


110 


50 


Hi 


14 




1A 


440 




2 


184 




%A 


140 


12 


m 


48 


60 


6 


10 





Hills in 


an Area of an Acre. 




Ft. apart. 


No. 


Feet apart 


No. 


Feet apart. 


No. 


"eet apart. 


No. 


1 


43560 


5 


1742 


9 


538 


16 


171 


IK 


19360 


$A 


1440 


$A 


482 


17 


151 


2 


10890 


6 


1210 


10 


435 


18 


135 


2X 


6969 


6X 


1031 


io$f 


361 


20 


108 


3 


4840 


7 


889 


12 


302 


25 


69 


%A 


3556 


7A 


775 


13 


258 


30 


48 


4 


2722 • 


8 


680 


14 


223 


35 


35 


4K 


2151 


SA 


692 


15 


193 


40 


27 



Transportation of Horses and. Cattle. 

The Space required on board of a Transport is : 

For Horses 30 inches by 9 feet. 

44 Beeves 32 u 9 44 

The Provender required per diem is : 

For Horses. . .Hay, 15 lbs. ; Oats, 6 quarts ; Water, 4 gallons. 
44 Beeves... 44 18 — " 6 44 



128 



SEEDS. SNOW LINE. BRIDGES. RIVERS. 



Nnmher of* several Seeds in a Bnshel, and Xum 
ber per Square Foot upon an Area of an Acre. 



Timothy .... 
Clover* 



No. 


ISq. Foot. ] 




1 No 


Sq.Foot. 


41823360 
16400960 


960 1 
1 376 1 


Rve 

Wheat .... 


888390 
.. 1 556290 


20.4 
12.8 



Snow Line, or Line of Perpetual Congelation, 



40° 
45° 



hat. 


Feet. 


Lat. 


Feet. 


5'- 
15 c 


15207 
14760 


25° 
35 c 


12557 
10287 



Feet. | 


Lat. 


Feet. 


Lat. | Feet. 


9000 
7670 J 


55° 

65° 


5030 

2230 


75° 1016 
85° I 117 



Limits of Vegetation in the Temperate Zone. 

The Vine ceases to prow at about 2300 feet above the level of the 
sea, Indian Corn at 2800, Oak at 3350, Walnut at 3600, Ash at 4800, 
Yellow Pine at 6200, and Fir at G700. 



Lengths of I3ridges. 



Feet. 



Avignon. . . 
Badajoz . . . 

Belfast 

Blackfriars 
Boston 



1710 
1S74 
25! »0 
905 
C4S3 



Bridges. 



Feet. | 



London 

Lyons 

Menai 

N.Y. and Brooklyn 
Pont St Esprit... 



950 
1560 

1'5' 

5063 
3060 



Bridges. 



Feet 



Potomac 

Riga 

Strasbourg . . , 
Vauxhall 

Westminster , 



5300 
2600 
3390 
S60 
1223 



St. Lawrence River, 9144 feet. 
Lengths of Spans of Bridges. 



Brid . 


Feet. 


Bridges. 


[ Feet. 


Bridges. 


Feet. 


Britannia 


460 
400 

53<> 


Mag'a at the Falls 
14 at Queens- 
town 


1268 

1040 


Schuylkill 

Southwark 

Wheeling 


340 
240 


Menai 


1010 



New York and Brooklyn, 1595 feet. 
Lengths of Rivers. 



Rivers. 


Miles. 


Rivers. 


Miles. 


Rivers. 


Miles. 


EUROPE. 

Danube 

Gaudiana 

Po 


1S00 
500 
42i 
S40 
51<> 
45. 
150 
ISO 
25' » 
190 
700 

2500 
140 

25O0 
3040 


Jordan 

Kiang 

Tigris 

Yeneisy & Selenga 

AFRICA. 

Gambia 

Niger 

Nile 

NORTH AMERICA. 

Arkansas 

Colorado 

Columbia 

Connecticut 

Delaware 

Hudson and Mo- 


176 

3290 
1160 
3530 

1000 
24(10 
3240 

2070 
1050 
1100 

420 

1400 

350 

2S00 


Mississippi 

Missouri and Mis- 

sissi ppi 

Ohio <fe Alleghany 

Potomac 

Red.- 

Rio Bravo 

St. l^iwrence .... 

Susquehanna 

Tennessee 

SOUTH AMERICA. 

Amazon and lieni. 

Efsequibo 

Mag lalena 

Orinoco 


1350 

4300 

14^1 


Rhine 


420 
1520 


Seine 


2300 




1450 


Tay 


620 




790 


Tiber 




Vistula 


4000 


Volga, Russia 

Wve 


520 
900 




1600 




Platte 


2700 


Euphrates 

Ganges 

Hoang Ho 


Kansas 


Rio Madeira 

Rio Negro 

Uruguay 


2300 


La Platte 

Mackenzie's 


1650 
1100 



SEA DEPTHS. — AGES OF ANIMALS. — RAIN. 



129 



Sea Depths. 



Feet. 



Feet. 




Feet. 




Off Cape Carnaveral 


2400 


16000 


" Charleston 


4200 


27000 


u Cape Hatteras.. 


3120 


4200 


" Cape Henry 


4200 


3720 


u Sandy Hook. .. . 


2400 


1950 







Baltic Sea 

Adriatic... 

English Channel. . . 
Straits of Gibraltar. 
Eastward of " 
Coast of Spain 



123 

130 
3')0 
100 

3000 
6000 



W. of Cape of Good 
Hope. ...... 

W. of St. Helena. . . 
Tortugas to Cuba. . 

Gulf of Florida 

Off Cape Florida. . . 



Estimated depth of the Atlantic 26 000 feet. 

" u Pacific 29 000 " 

250 miles off Cape Cod, no bottom at 7800 feet. 

Course of the A tlantic Telegraph from Ireland to Newfoundland. 
Longitude, 15°, 2 443 feet ; 20°, 9 258 ; 30°, 12 000 ; 40°, 9 000; 47°, IS 000 ; 50°, 6 600. 

-A.ges of Animals, etc. 
Baar, 20 years ; Cat, 15; Cow, 20; Camel, 100; Deer, 20 ; Eag^e, 100; Elephant, 
400; Fox, 15; Hare, Rabbit, and Squirrel, 7; Horse, 30; Lion, 70; Porpoise, 30; 
Raven, 100; Rhinoceros, 20; Sheep, 10; Swine, 20; Tortoise, 100 to — ; Swan. 
300 ; Whale, estimated 1000 ; and Wolf, 20. 



Alabama 

Albany 

Alleghany 

Antigua 

Auburn 

Baltimore 

Barbadoes 

Bath, Me 

Belfast 

Bombay 

Boston 

Buffalo 

Burlington, Vt. . . 

Calcutta 

Cape St. Francois. 
Charleston, S. C. . 

Demarara 

" 1849... 
Dover (Engl.).. .. 

Dublin 

Dumfries 

East Hampton . . . 

Edinburgh < 



30.17 

41.35 

46.66 

45. 

30.17 

30 9 

55. S7 

G4.58 

39.46 

110. 
39.23 
27.27 
32. 
81. 

150. 
54. 
91.2 

132.21 
37.52 
30.87 
30.92 
38.52 
24.5 
29. 

Globe, mean depth 
Cape of Good Hope 
At Khassaya, in 6 r 



Rain. 

Annual Fall at different Places. 

Ins. Location. Ins. Location. I Ins. 



England < 

Fairfield 

Ft. Crawford, Wis. 
Ft. Gibson, Ark. . . 
Ft. Snelling, Iowa. 
Fortr. Monroe, Va. 
Gordon Castle, Sc'd 

Glasgow...... 

Granada 



Great Britain . , 

Greenock 

Hudson 

Key West 

Khassaya, Calc'tta 

Lewiston 

Liverpool 

London •[ 

Louisiana 

Michigan, mean . . 



31. 

35. 

32.93 

29.54 

30.64 

30.32 

52.53 

29.3 

21.3 

31. 

105. 

126. 
31.88 
36. 
61.8 
39.32 
31.39 

610. 
23,15 
34.12 
20.68 
24. 
51.85 
33.5 



Madeira . 



in 1846 

ainy months. 



. 550 ins. 



Manchester < 

Mississippi 

Mol.ile, 1842 

Newburg 

New York 

Ohio, mean 

Petersburg (Eng.). 

Philadelphia 

Foughkeepsie 

Plymouth (Engl.). 

Providence. 

Rochester 

Home 

Savannah 

Schenectady 

Sierra Leone 

State of N. V .,mean 

Utica 

Vera Cruz 

West I oint 

Washington 

36. 

. in 3 days, 6.2 ins. 
; in 1 day, 25.5 " 



36.14 

43. 

45. 

54.94 

40.5 

36. 

36. 

16. 

49. 

32.06 

44. 

36.74 

29. 

39. 

55. 

47.77 

S4. 

33.79 

39.3 

62. 

48.7 

34 64 



Volume oF Rain Fall. 

Rain fall in inches, X 2323200 = cubic feet per square mile. 
u " X 17.3787 = millions of gallons per mile. 

u u X 3630 = cubic feet per acre. 

The average fall of rain for the southern and eastern counties of Great Britain is 
about 34 inches; but in the western and hilly counties it is from 4S to 50 inches. 
The mean quantity of water in a cubic foot of air in that climate is 3.789 grains, 



130 



MOUNTAINS. COLUMNS. SPIKES, ETC. 



Heights ofobtained Elevations, and. various Places 
and. Points above the Sea. 



Locations. I Feet. . 


Locatioos. 


Feet. 


Locations. 


Feet. 


Balloon(GayLuss:,,c) 22900 
Brazil, Quito and / 6000 to 
Mexico plains.. \ S000 
Dent's Bridge, Alps . 11000 
Gibraltar 1400 


Isthmus of Darien. 
Laguna,Teneriffe. . 
Lake Erie 


645 

2)00 

558 

538 

234 

647 

64 

2200 


Mexico, city of . . . . 

Paris, city 

Quito 

St. Bernard' 8 Mon'y 
Volcano, Cotopaxi . 
Volcano, Mt. Etna . 

" " Hecla. 

u Vesuvius 


7525 

115 

13500 


" Huron 

u Ontario 

u Superior 

London, city ...... 

Madrid 


S040 
1SS6S 


Geneva Lake 10:6 

Humboldt's highest 
elevation 19400 


11(00 
5000 
3600 



Heights of Mountains above the Level of the Sea. 



Mountains 



EUKOPE. 

Barthelemy, France 

Ben Nevis 

Etna 

Guadarama, Spain . 

Hecla 

Ida 

Mount Cenis 

" Blanc 

Nephin, Ireland 

Olympus 

Parnassus 

Plynlimmon, Wales . 

St. Bernard 

u Gothard 
Sea Fell, England 
The Cylinder, Pyr. 
Vesuvius 



7365 
433) 

10 26 
S520 
5000 ! 
496) 
67S:) ! 

15572 
2634 
6510 
6000 
2463 
8172 

11000 
3266 

10 30 
3731 



Mountains. 



| Feet. 



Crows' Nest, 



ASIA. 

Ararat 

Caucasus 

Dhawalagheri 

Geta, Java 

Himalaya 

Mount Libanus. . . . 

Olympus 

Petcha 

AFRICA. 

Atlas 

Compass, Caps of 

Good Hope 

Dianai Peak, St. He- 
lena 

Geesh 

Ruivo, Madeira 

Teneriffe Peak 

Highlands, N.Y.... 



12700 
16433 
28077 
8500 
25„53 
9523 
8000 
15000 

13000 

10000 

2700 
15000 

5160 
12300 



Mountains. 


Feet. 


AMERICA. 

Alleghanies 

Blue Mount, Jam' a. 
Catskill 


3500 
8000 
3S04 


Chimborazo 

Cotopaxi 


21441 
1S900 


Great Peak, New 
Mexico 


197SS 


Mount Elias 

" Washington 
Nevado de Sorata. . 


1S0S7 

6225 

2524S 

17371 


Passages of the ( 
Cordilleras ... \ 

Popocatapetl 

: Fotosi 


15225 
13525 
17716 
18000 


; Sierra Nevada 

i Tahiti 


15700 
118-5 






1370 feet. 



Heights of* Columns, Towers, Domes, Spires, etc. 

Locations Feet. Locations. Feet. 



OOLTT-MNS. 

Alexander St.Petersb. 

Bunker Hill Mass 

Chimn ;y, St. Rollox. Glasgow . . 
u Musprat's. Liverpool . 

City London . . . 

J uly Paris 

Napoleon Paris 

Nelson's Dublin . . . 

Nelson's London . . . 

Place Vendome Paris 

Pompey's Pillar Egypt .... 

Trajan Koine 

Washington Wash'gton 

York London . . . 

TOWERS AND DOMES. 

Babel 

Balbec 

Capitol Wash'gton 

u Diam. Dome, l * 

Cathedral Antwerp . . 

u Cologne... 

" Cremona.. 

u Escurial . . 

u Florence.. 



175 

221 

455)^ 

4% 

202 

157 

132 

134 

171 

136 

114 

145 

138 

6>0 
500 

124X 

476 

5>1 

392 

200 

3^4 



Cathedral Milan 438 

" Petersb'rg. 363 

Leaning Pisa 188 

Porcelain China 200 

St. Paul's London ... 366 

Strasbourg 486 

St. Mark's Venice ... 328 

Utrecht 464 

SPIRES. 

Cathedral, new New York. 325 

Grace Church " 216 

Salisbury 450 

St. John's New York . 210 

St. Paul's " 200 

St. Mary's Liibeck ... 404 

St, Peter's Rome 391 

St. Stephen's Vienna . . . 465 

Trinity Church* . . . New York. 2S6 
Balustrade of Notre 

Dame Paris 216 

Hotel des Invalides. M 344 

Pyramid of Cheops. Egypt 520 

Pyram. of Sakkara . Egypt .... 356 

St. Peter's Rome 518 

* From high-waUr ltfv«l t 336 feet. 



DOMES. — TUNNELS. — BELLS. — OCEANS, ETC. 



131 



Location. ' 



Cascades and. Waterfalls. 

Feet. Location. Feet. | Location. 



Arve, Savoy 

Cascade, Alps 

Cataracts of the 

Nile 

Mohawk 



1600 
2400 



Missouri . 



Montmorency. , 
Niagara 



(50 
^80 
(94 
250 
164 



Passaic 

Potomac . . 

Ribbon, Yosemite 

Valley 

Yosemite Valley.. . 



74 
74 

3300 
2600 



Diameters of Domes. 



Domes. 


1 Feet. II Domes. 


1 Feet. 


1 Domes. 


j Feet. 


Capitol, Washington 


|l24X|| St. Paul's 


. . 1 112 | 


St. Peter's 


.. 1 139 



Lengths of Tunnels. 



Tunnels. 


Feet. 


Tunnels. 


Feet. | 


Tunnels. 


Feet. 


Blaizy 

Blue Ridge 


13455 

4280 


Nerthe 

Nochistongo .... 


15153 
21659 | 


Riquivel 

Thames & Medw. 


18623 
11SS0 



Mont Cenis, 7.5 miles 242 yards. 



Bells. 



Pekin 

Fire Alarm, 33d St. 
Linden, Germ'y.. . 

Lewiston, Me 

Montreal, C. E. . . . 

Moscow, Russia . . . 



"Weights of Bells. 

Pounds. Bells. Pounds, 



130000 
21612 
10854 
10233 
28560 

432000 



Oxford, " Great 

Tom," Eng 

Olmutz, Bohemia . 

Rouen, France 

St. Paul's, Eng 

St. Ivan's, Moscow. 



18000 
40000 
40000 
11470 

127830 



Bells. 



St. Peter's, Rome. 

Vienna 

Westminster, u Big 
Ben," England. . 
Worcester u 
York u . . 



18600 
40200 

30350 
6600 
6384 



Areas of Oceans. 



Oceans. 


Sq Miles. 


Oceans. 


Sq. Miles. 


Oceans. 


Sq. Miles. 


Antarctic 

Arctic 

Atlantic 


30,000,000 

8,400 
25,000,000 


Baltic 

Black Sea 

Caspian 


175,000 
150,000 
120,000 


Indian 

Mediterranean 
Pacific 


17,000,000 

1,006,000 

50,000,000 



Northern Lakes of the United States. 



Lakes. 


Length. 


Breadth. 


Mean Depth. 


Height above 
the Sea. 


Area. 


Erie 

Huron 

Michigan 

Ontario 


Miles. 
250 
200 
360 
180 
355 


Miles. 

80 
160 
109 

65 
160 


Feet. 
200 
120 
900 
500 
988 


Feet. 
555 
574 
587 
282 
627 


Sq. Miles. 

6000 

20000 

20000 

6000 


Superior 


32000 



Sheet Lead. 

Sheet Lead is designated by the weight of a square foot of it, and it 
usually ranges from 2)4 to 10 lbs. per square foot. 



132 BRICKS.' — LIME AND LATHS. 

Bricks. 

The variations in the dimensions of bricks by the various manufac- 
turers, and the different degrees of intensity of their burning, render a 
table of the exact dimensions of the different classes of bricks altogether 
impracticable. 

As an exponent, however, of the ranges of their dimensions, the 
following averages are given : 



Description. 



Baltimore front. . 
Philadelphia li .. 
Wilmington " .. 
Croton li 
Colabaugh 



8^X4 X2M 
S%X3 5 AX2% 



Description. 



Maine 

Milwaukee . . 
North River . 

Ordinary . . . 



leches. 



7^X3%X2% 

8^X4^x2^ 

S X3%X2% 

\1%X?>%X2}£ 

»8 X±%X2% 



Stourbridge fire-brick $%X±%X2% inches. 

American (N. Y.) • S%x4^X2% » 

In consequence of the variations in the dimensions of bricks, and 
the thickness of the layer of mortar or cement in which they may be 
laid, it is impracticable to give any rule of general application for the 
volume of laid brick-work. It becomes necessary, therefore, when it 
is required to ascertain the volume of bricks in masonry, to proceed as 
follows : 

Xo Compute tlie Volume of Bricks and. the Nuiri* 
ber in a Cubic Foot of Masonry. 

Rule. — To the face dimensions of the particular bricks used, add 
one half the thickness of the mortar or the cement in which they are 
laid, and compute the area ; divide the width of the wall by the num- 
ber of bricks of which it is composed ; multiply this area by the quo- 
tient thus obtained, and the product will give the volume of the mass 
of a brick and its mortar in inches. 

Divide 1728 by this volume, and the quotient will give the number 
of bricks in a cubic foot. 

Example. —The width of a wall is to be 12% inches, and the front of it laid with 
Philadelphia hricks in courses 3^ of an inch in depth; how many bricks will there 
be in face and backing in a cubic foot ? 

Philadelphia front brick, 8^X2% ins. face. 

8.25 -{- . 25x2 -=-2 — 8.25 +.25 = 8.5 = length of brick and joint; 
2,375 -f .25X2-H2 = 2.375 -f . 25 = 2. C25 = width of brick and joint. 
Then 8.5x2.625 = 22.3125 inches = area of face; 12. 75 -h 3 (number of bricks in 
width of wall) = 4.25 inches. 
Hence 22.3125x4.25 — 9483 cubic inches; and 172S-=-94.S3 = lS.22 bricks. 

Lime and. Laths. 

A Cask of Lime = 240 lbs., will make from 7.8 to 8.15 cubic feet 
of stiff paste. 

A Cask of Cement = 300* lbs., will make from 3.7 to 3.75 cubic 
feet of stiff paste. 

See Limes, Cements, and Mortars, pages 499 to 508. 

Laths are 13^ to 1}£ inches by four feet in length, are usually set }i 
of an inch apart, and a bundle contains 100. 

* 300 lbs. net is the standard ; it usually overruns S lbs. 






ANCHOKS. — CHAINS, ETC. 



133 



ANCHORS AND KEDGES. 

To Compute the "Weight of a Bower Anchor for a "Vessel 
of a, given Character aixdL R.ate- 

Rule.— Multiply the square of her extreme breadth by the unit of the character 
and rate in the following table, and the product will give the weight in pounds, ex* 
elusive of the stock. 

Example. — The extreme breadth of a side-wheel and bark-rigged steamer is 4$ 
feet. 

402 x 3 = 1600 X 3 = 4S00 lbs. 

The weight of Anchor and K edge is given exclusive of that of its stock 
Bower and Sheet Anchors should be alike in weight. 
Stream Anchors should be % the weight of the best bower. 
Kedges. — When 1 is used, J the weight of the Bower 



h A» 

i/i i. 

6' 8' 10' 
i l JL JL 
7' 8' 10' 14' 
1 1 1 JL JL 
7' 8' 9' 10' 14' 



Table showing tlie Units to determine tlie Weights 
of Anchors, also the Number required, to each 
Class and. Rate. 

NAVAL ANn MERCHANT SERVICE. 

Number allowed. 



Class of Vessel. 



Unit. 



Bowers. Sheet. Stream. I Kedges 



SAILING VESSELS. 

Ship of the Line ... 

Frigate 

Razeed Frigate.... 

Ship 

Sloop of War 

Bark 

Ve§sel, full rig, 550 to 300 Tons ... 

" »• 300 to 200 " ... 

4 •'• 200 to 100 " ... 

" 4i less than 100 " ... 

PROPELLER STEAMERS. 

Frigate, Ship, or Sloop of War, 
ship or bark rigged 

Sloop, lighter-rigged. . . 

Vessel, light rig, 1200 to 900 Tons 

M 



3.5 

3. 
(3. 
J 2.8 

2.8 
j 2.8 
( 2.G 
j 2.6 
(2.3 
j 2.4 
1 2.1 
j 2.2 
1l.9 

if:. 



3. 
1 2.8 
2.5 
2.4 
2.3 
2.2 



4 
4 
3 

2 

3 
3 
2 
3 
2 
2 
1 
2 
1 
1 
1 



134 



CHAINS. — CABLES. ANCHORS, ETC. 



Taole — (Continued). 



Class of Vessel. 



Unit. 



Number allowed. 



Bowers. Shoot. Stream. Kedges 



Vessel, light rig, 900 to 600 Tons 

" " 600 to 500 " 

" " less than 500 " 
" without any rig 

SIDE-WHEEL STEAMERS. 

Ship or Bark 

Brig or Brigantine 

Vessel 700 to 500 Tons 



« 500 to 350 " 
" 350 to 200 " 
'« less than 200" 



Steam-boat without any rig, hull 
much above water 



IRON-CLADS. 

Hull much above water.. 
" alike to a Monitor.. 



BOATS. 

Any description , 



j 2.3 
( 2.2 

is 

2. 
1.8 

p. > 

(2.8f 

ft. 

ft. 

fts 



2.2 
1. 

1.2 



Note. — The Tonnage as above given is computed by the dd U. S. Measurement. 

2. For a Comparison between the tonnages of all classes of vessels under the old 
and new law, see page 105. 

To Compute trie Diameter of* a Chain Cable corre- 
sponding to a Given Weight of* Anchor. 

Rule. — Cut off the two right-hand figures of the anchor's weight in 
pounds ; multiply the square root of the remainder by 4 ; and the prod- 
uct, subtracting 3 when the weight of the anchor exceeds 8000, 2 when 
it is between 8000 and 7000, and 1 when it is between 7000 and 4000, 
will give the diameter of the chain in sixteenths of inches. 

Examft.Ei — The weight of an anchor Is 2500 lhs. 
-/25.00X4 = 5x4 = 20, and 20 — (weight less than 4000) = 20 sixteenths — 1^ 
inches. 

Note.— The diameter of a chain Messenger should be % that of the chain Cable to 
which it is to be applied. 

2. The tensile proof of chains in the English Merchant service is for a diameter 
of chain of 1 inch and under, about 44800 lbs. (20 tons) per square inch of ar?a of a 
half link, and for diameters exceeding this it is reduced gradually to 42560 (19 tons' 
per square inch of area. 



ANCHORS. ROPES, HAAVSERS, AND CABLES. 135 

3. When proved chains are used, they may be one sixteenth of an inch less in di- 
ameter, from 1 to \% inches in diameter, and one eighth of an inch less in those 
above 1% inches. 

4. The proof in the U, S. naval service is about G7500 lbs. per square inch of area 
of link for diameters of \% inch and less, and i34500 lbs. for the larger diameters. 

5. The British Admiralty proof is 630 lbs. per square of diameter of link in eighths 
of an inch. 

ANCHORS. 

From Experiments of a Joint Committee of Representatives of Ship- 
owners and the Admiralty of Great Britain. 

An anchor of the ordinary or Admiralty pattern, the Trotman or 
Porter's improved (pivot fluke), the Honiball, Porter's, Aylin's, 
Rodgers's, Mitcheson's, and Lennox's, each weighing, inclusive of 
stock, 27000 lbs., withstood without injury a proof strain of 45000 lbs. 

Comparative Resistance to Dragging. 

Dry Ground. 

Rodgers's dragged the Admiralty anchor at both long and short 
stay, and Aylin's at long stay ; at short stay Rodgers's and Aylin's 
gave equal resistance. 

Mitcheson's dragged Aylin's at both long and short stay, and Ay- 
lin's dragged the Admiralty's at short stay, they giving equal resist- 
ance at long stay. 

Ground under Water. 

Trotman's dragged Aylin's, Honiball's Mitcheson's and Lennox's ; 
Aylin's dragged Rodgers's ; Mitcheson's dragged Rodgers's ; and 
Rodgers's and Lennox's dragged the Admiralty's. 

The breaking weights between a Porter and Admiralty anchor, as 
tested at the Woolwich Dock-yard, were as 43 to 14. 



ROPES, HAWSERS, AND CABLES. 

Hopes of hemp fibres are laid with three or four strands of twisted 
fibres, and run up to a circumference of 12 inches. 

Hawsers are laid with three strands of rope, or with four rope 
strands. 

Cables are laid with three strands of rope only. 

Tarred ropes, hawsers, etc., have 25 per centum less strength than 
white ropes; this is in consequence of the injury the fibres receive 
from the high temperature of the tar =290°. 

Tarred hemp and Manila ropes are of about equal strength. Ma- 
nila ropes have from 25 to SO per centum less strength than white ropes. 

Hawsers and Cables, from having a less proportionate number of 
fibres, and from the increased irregularity of the resistance of the 
fibres, have less strength than ropes, the difference varying from 35 to 
45 per centum, being greatest with the least circumference. 

Ropes of four strands up to 8 inches are fully 16 pet centum stronger 
than those having but three strands. 



136 



ROPES, HAWSERS, AND CABLES. 



Hawsers and cables of three strands up to 12 inches are fully 10 pet 
centum stronger than those having four strands. 

The absorption of tar in weight by the several ropes is as follows : 

Bolt rope 18 per centum I Cables 21 per centum 

Shrouding... 15 to 18 " | Spun yarn... 25 to 30 M 

White ropes are more durable than tarred. 

The greater the degree of twisting given to the fibres of a rope, etc. 

the less its strength, as the exterior alone resists the greater portio 

of the strain. 






To Compnte tlie Strain that cart be'borne with 
safety by new Ropes, Hawsers, and Cables. 

Deduced from the experiments of the Russian Government upon the 
relative strength of different Circumferences of Ropes, Hawsers, etc. 

TheU. S. Navy test is 4200 lbs. for a White rope of three strands of 
best Riga hemp, of\% inches in circumference ( = 17000 lbs. per square 
inch), but in the following table 14000 lbs. is taken as the unit of strain 
that can be borne with safety. 

_ Rule. — Square the circumference of the rope, hawser, etc., and mul- 
tiply it by the following units for ordinary ropes, etc. 

Table showing the Units for computing the safe 
strain that may be borne by Ropes, Hawsers, 
and. Cables. 







Ropes. 




Haw 
White. 


sers. 


Cables. 


Description. 


White. 


Tarred. 


Tarred 


White 


Tarred 




3 strands 


4 strands 3 str'ds 4 str'ds 


3 str'ds 


'str'ds. 3 str'ds 


3 str'da 


Inches Circumference. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


White rope, 2.5 to 6 ins. 


1140 


1330 


_ 


_ 


600 








" 6 " 8 " 


1090 


1260 


_ 


_ 


570 


_ 


510 




" 8 "12 « 


1045 


880 


_ 


_ 


530 


_ 


530 




" " 12 "18 " 


- 


- 


_ 


_ 


550 


_ 


550 


_ 


ii w 18 u 26 u 


- 


- 


_ 


_ 


_ 




560 




Tarred " 2.5 " 5 " 


- 


- 


855 


1005 


_j 


460 




J 


14 U 5 U g »« 


- 


- 


825 


940 


_ 


480 


_ 


_. 


» 8 " 12 u 


- 


- 


780 


820 


_ 


505 


_ 


505 


14 12 14 ^ 44 


- 


- 


- 


_ 


_ 




_ 


525 


" 18 u 26 " 
















550 


Manila 2.5 " 6 u 


810 


950 


_ 


_ 


440 






" M Q 41 -J9 " 


760 


835 


- 


_ 


465 


_ 


510 


_ 


II 14 12 44 Jg U 


- 


- 


_ 


_ 


_ 


_ 


535 




44 18 Ugg „ 


- 


- 


- 


- 


- 


- 


560 


- 



When it is required to ascertain the weight or strain that can be borne 
1*1 ropes, etc., in general use, The above "Units should be reduced one 
third, m order to meet the reduction of their strength by chafing and 
exposure to the weather. 

KxAMn ( K.— What is the weight that can he home with safety by a Manila rope 
ot 6 straudd having a circumference of C inches' 



rcumference of C inches 

6 2 XT60=27 360 J&s. 



WIRE ROPE, ETC. 137 

Ex. 2. What is the weight that can be borne by a tarred hawser 10 inches iu cir- 
cumference, in general use t 

102X (505 - _i>j =s 1C0X33C.C7 — 33G67 lbs. 

To Compute the Cii'cumfererice of a Rope, Haw* 
ser, or Catole fox* a Given Strain. 

Rule. — Divide the strain in pounds by the appropriate units in the 
above table, and the square root of the product will give the circum- 
ference of the rope, etc., in inches. 

Examine. — The stress to be borne in safety is 1G5550 lbs. ; what should be the 
circumference of a tarred cable to withstand it ? 

1G5552 -r- 550 = 301, and V301 = 17.35, say 17% ins. 

Ex. 2. — What should be the circumference of a Manila cable to withstand a strain, 
in general use, of 14933G lbs. ? 
Assuming the circumference to exceed 18 ins , the unit = 500. 

14C33G I*- (580 — b ^\ = 14933G £- 373.34 & 400, and ^400 = 20 ins, 

WIRE ROPE. 

Wire rope of the same strength as new Hemp rope will run on the 
game-sized sheaves ; but the greater the diameter of the sheaves, the 
longer it will wear. Short bends should be avoided, and the wear 
increases with the speed. It is better to increase load rather than 
speed. The adhesion is the same as that of hemp rope. 

Wire rope should not be coiled or uncoiled like hemp rope, but 
should be wound upon a reel. 

When substituting wire rope for hemp rope, it is well to allow for 
the former the same weight per foot which experience has approved 
of for the latter. As a general rule, one wire rope will outlast three 
hemp ropes. To guard against rust, stationary rope should be oiled 
once a year with linseed-oil, or kept well painted or tarred. Running 
rope, while in use, requires no protection. 

Where great pliability is required, the centre or core of wire rope is 
made of hemp, and small-sized rope is generally made with hemp 
centres. 

Running rope is made of fine w T ire, and standing rope of coarse 
wire. 

Wire rope made from charcoal-made iron is fully one fourth stron- 
ger than the ordinary rope. 

The standing rigging of a vessel when composed of wire rope is one 
fourth less in weight than when of hemp. 

Results of an Experiment with Galvanized Wire. 

A strand of 2-inch wire rope broke with a strain of 13564 lbs., and a 
piece of a like rope, when galvanized, withstood a strain of 14796 lbs. 
before breaking. 

M* 



138 



CHAINS, CABLES, AND ANCHORS. 



Taole showing the Diameter, Length., and Weight 
of Chains, and. the Circumference and. Weight 
of Cables corresponding to a Given Weight of 
Bower Anchor. 







Length of Chain. 


Cables. 


Weight 


per Fath. 


Weight of 


Diam. of 
















Anchor. 


Chain. 


Bower 


Sheet 


Stream 


Hemp or 
Manilla. 


Stream of 
Manilla. 


Close- 
linked- 


Stud. 


Lbs. 


Ins. 


Fath 


Fath. 


Fath. 


Circumf 


Circumf. 


Lbs. 


Lbs 


50 


& 


60 


45 


45 


2.5 


2.21 


3.5 


_ 


75 


x 


60 


45 


45 


3. 


2.75 


5.5 


_ 


100 


% 


75 


60 


60 


3. 


2.75 


5.5 


_ 


130 


X, 


75 


60 


60 


4. 


3. 


6.25 


_ 


160 


% 


75 


60 


60 


4. 


3. 


9.5 


- 


200) 
250j 


% 


75 


60 


60 


4.5 


3.25 


9.5 


- 


300! 
350j 
400 


X, 


75 


60 


60 


5. 


3.75 


13.5 


- 


k 


75 


60 


60 


5.5 


4.5 


17. 


^ 


450) 
500 f 


% 


75 


60 


60 


6. 


5.25 


21. 


- 


550 1 
600 f 


% 


75 


60 


60 


6 75 


5.5 


26. 


1 


650 


% 


75 


60 


60 


6.75 


5.5 


26 : 


_ 


700 


% 


75 


60 


60 


7.5 


6. 


SO. 


_ 


800 


% 


90 


75 


60 


8. 


6. 


37. 


34 


900 


% 
% 


90 


75 


60 


8.5 


6.5 


42. 


40 


1000 


90 


75 


60 


9. 


8. 


48. 


44 


1200 


%. 


105 


90 


75 


10. 


8.75 


55. 


51 


1400 




105 


90 


75 


10.5 


9.5 


63. 


59 


1600 


i& 


120 


105 


75 


11. 


9.5 


70. 


66 


1800 


m 


120 


105 


75 


12. 


10.5 


79. 


75 


2000 


i% 


135 


105 


75 


12. 


10.5 


79. 


75 


2250 


!« 


135 


105 


75 


12.5 


10.5 


88. 


82 


2500 


ix 


135 


120 


90 


13. 


11. 


98. 


91 


2750 


\u 


135 


120 


90 


13.5 


11.5 


98. 


91 


3000 
3250 




150 


120 


90 


15. 


12. 


118. 


113 


3500 


m 


150 


135 


90 


15.5 


12.5 


_ 


120 


3750 


m 


150 


135 


90 


16. 


13. 


_ 


132 


4000 


1% 


150 


135 


90 


16.5 


13. 


_ 


145 


4300 


1% 


150 


135 


90 


16.5 


13.5 


_ 


145 


4600 
4900 




150 


135 


90 


17. 


13.5 


- 


156 


5200 
5500 




165 


150 


105 


17.5 


14. 


- 


(162 
1l75 


6000 


f 


165 


150 


105 


19. 


15. 


_ 


189 


6500 


J 65 


150 


105 


19.5 


15. 


__ 


205 


7000 




165 


150 


105 


20.5 


16. 


,. 


240 


7500 
8000 


k 


180 


165 


120 


21.5 


17. 


r- 


- 


8500 


*% 


180 


165 


120 


22. 


17.5 


_ 


_ 


9000 


"iy 


180 


165 


120 


22.5 


18. 


_ 


_ 


9500 


23? 


180 


165 


120 


23. 


18.5 


_ 


_ 


10000 


2^ 


180 


165 


120 


24. 


19 


- 


- 



CHAINS, CABLES, AND ANCHOKS. 



139 



Table of ilie Maximum. Breaking Strain, of* 
Wrought-Iron Cliain Rigging. 









CLOSE-LINKED. 








Plant 

of Iron 


Strain. 


Diam. 
of Iron. 


Strain. 


Diam. 

of Iron. 


Strain. "?"*■ 

of Iron, 


Strain. 


Ins. 


Lbs. 


Ins. 


Lbs. 


Ins. 


Lbs. 


Ins. 


Lbs. 


X 


2464 


Vi 


15680 


% 


44800 ! 


m 


100800 


X 


3920 


% 


22400 


% 


51520 


m 


120960 


X, 


6720 


% 


26880 


% 


58240 


ik 


143360 ' 


% 


8960 


% 


31360 


l 


62720 


*% 


168000 


S 


13440 


% 


38080 


*H 


82880 


m 


201580 



Note.— The minimum breaking strain is about 9 per cent, less than this. 

Close-linked chain is heavier than stud-linked. 
The strength of iron chain rigging for general use, compared with 
tarred rope, also for general use, is as 3.57* to 1 for each part of a link. 

To Compute the Circumference of a Link of Chain 
of equal Strength, of a Tarred. Rope. 

Rule. — Divide the strain in pounds by 4000, t and the square root 
of the product will give the circumference of one part of the link of 
chain in inches. 

Example. — The stress to be borne is 50000 lbs. ; what is the circumference of one 
part of a link of chain rigging of equivalent resistance ? 

50000 -h 4000 = 1 2.5, and V12.5 = 3.535 ins. 
The diameter of 3.535 ins. circumference p \% ins. 



CHAINS, CABLES, AND ANCHORS. 

In the Table, page 138, the weight of the best Bower anchor is made 
the exponent of the requirement of dimensions of Cables, etc., and not the 
Tonnages of the vessel, as hitherto. 

The adoption of a new and essentially different admeasurement of ton- 
nages sets aside the propriety of a reference to the tonnage of a vessel un- 
der the old measurement ; added to which, the beam of a vessel, in connec- 
tion with her rig and extent of hull above water, is made the sole basis of 
tlte computation of the weight of an anchor. 

The number and weight of anchors, and the length of chains here 
given, exceed the usual practice of our Merchant service, but the pro- 
priety, if not the necessity of the weights and lengths given, is no less 
apparent. 

* Chain links \% ins. in diameter will hear an average maximum strain of 43500 
lbs., or a minimum of 37500 lbs. per square inch of section, from which is to be de- 
ducted % for general use == 25000 lbs. 

White rope of three strands will bear 14000 lbs. per square inch, from which is 
te be deducted %, to reduce it to the resistance of tarred rope = 10500 lbs., and also 

25000 
% for general use = 7000 lbs. ; hence ' —3.57. 

t The constant of 4000 represents for both parts of a link of iron chain the varying 
units in the table, page 137, for ropes, etc., the occasion of the variance in the hitter 
case arising from the difference of the strength of a rope, etc., whether of three or 
four strands, or whether rope or hawser ltiid, or in its circumference, its proportion- 
ate strength being inversely as its circumference. 



140 



WIRE ROPE. 



Table of t lie Relative Dimensions of Wire Rope 
(Coarse and Fine laid), and of Ropes, Hawsers, 
and Cables, with their Breaking Strain. 

R-J A. Roebling. N r NEWA ll & Co. AG-Admiraltv, and Garnock, Bibby & Co. 

(COARSE LAID.J 

Circumference of equnl Resist- 
ance for General Use. 



No. 

27 
26 

25 

24 
23 
22 



21 
20 



19 
18 



16 
15 

14 
13 
12 
11 



R 

R 

. R 

N 
R 
N 
R 
N 
R 
N 
N 

AG 
R 
R 
N 
N 

AG 
N 
R 
R 

AG 
N 

AG 
R 

AG 
N 
N 

AG 
R 
N 
R 

AG 

AG 
N 
R 

AG 
N 
R 

AG 
N 
R 

AG 
N 
R 



Ins 

.25 

.26 

.3 

.32 

.35 

.36 

.39 

.4 

.41 

.44 

.48 

.49 
.52 
.52 
.56 

.64 
.6 

.68 



.95 



1.03 

M< 

l.U 

1.19 

1.% 

1.27 
1.4 



.94 
1. 
1.11 

UH 

1.23 

*M 

1.31 

1.x 
1.x 

1.53 

!•% 

J:S 

V.« 

1.9 

»:% 

2.4 

2-X 

2-M 

2-% 

2.% 

2.68 

3. 

•2.98 

3. 

8.* 

3-.¥ 

8-X 

3.i4 

3-K 

3-K 

3.% 

8.J£ 

4. 
4. 
4. 
4.45 



.16 

.19 

.21 

.23 
.25 

.34 



.34 
.42 
.42 

.58 



.58 
.75 
.75 

.92 

.92 

1.08 

1.08 

1.25 

1.25 
1.42 
1.42 

1.67 
1.66 

2. 

2. 

2.33 
2.33 



«a 


l _ i 


to 








*l 


*>a 




0- 


■* a 


^^3 


e ° 












a 


p« 


Lbs. 


lbs. 


1120 


_ 


1620 


_ 


2060 


_. 


4480 


_ 


2760 


_ 


5018 


- 


3300 


— 



5600 

4260 

6182 

6720 

6720 

5660 

8180 

8960 

11200 

11200 

15680 

11600 

15200 

15680 

20160 

17600 
19400 
24640 
29120 

24600 
33600 
32000 



38080 
40000 
40880 
44800 
50000 

53760 
60000 

62720 
72000 



5040 



7280 



9632 



11870 



14124 



16464 



19152 
23744 



26208 



30240 



34272 



TABBED 

Haw': 



Ins. 
IX 

IX 

2Xc 

2X 
3 



0% 
% 

4^ 5 
5X 

5% 

CX 

6X 
0% 

6X 

7X 
7W 



OX 
9X 



2X 
2K 
2X 

2X 

2K 
2K 



3X 

3X 

4^ 



4% 
5X 



GX 



8X 

8X 

9X 

ox 



3^ 
3X 
3% 

*%« 



4% 
*% 

6 

*X 

6 

OX 

7^ 

8X 



*X 
Wz 

10 
10 

10% 

11% 

11% 



WIRE ROPE. 



141 



Table— (Continued). 

(COARSE-LAID ) 











1 


to . 


• 

1 


B 

a 




a 
£ 


fa 

6 


'3 a 


a 
• 

OS 


a 
a 
N 


1 

3 

a 


«£ 
g 

a 


Pi 

.2° 


C»3 

£ 


h 


s 


Q 


O 


£ 


« 


No. 




Ins. 


Ins. 


Lbs. 


Lbs. 




AG 


- 


i.yi 


2.67 


- 


_ 


N 


1.35 


i-% 


2.5 


67200 


_ 


N 


1.43 


*-i 


2.66 


71680 


_. 


AG 


_ 


4-K 


3. 


- 


_ 


N 


1.51 


4-M 


3. 


80640 


_ 


AG 


._ 


i.% 


3.33 


- 


_ 


AG 


_ 


5. 


3.66 


- 


_ 


N 


1.59 


5. 


3.66 


98560 


_ 


AG 


_ 


5.3^ 


4. 


108400 


_ 


N 


1.75 


5 s 3i 


4.41 


118720 


_ 


AG 


_ 


5^>| 


4.33 


- 


_ 


_ 


_ 


5.% 


_ 


130530 


_ 


N 


1.91 


6. 


5.25 


141120 


_ 


AG 


_ 


6. 


5. 


- 


_ 


- 


_ 


6-K 


_ 


165555 


_ 


_ 


_ 


7. 


_ 


192080 


— 


_ 


_ 


7.# 


- 


215048 


- 


- 


- 


8. 


- 


256880 





Circumference of equal 


Resist' 


o 


ance for General Uf.e 


,2 




TARRED 


•~ 5 
^'3 


Ropes. 


Hnw'rs Cables 




£ a 
ft £ 






hi 




co 


CO 


co 


CO 


Lbs. 


Ins. 


Ins. 


Ins. 


Ins. 


38752 


9% 


9^ 


mc 


12^ 


- 


9% 


9% 


V2}£ 


12^ 


- 


io>£ 


9K 




123^ 


42232 


jo^ 


w^ 


- 


12M 


- 


iom 


10* 


- 


13>g 


48944 


n 


10% 


- 


13^ 


54656 


- 


11% 


H% 




- 


- 


H% 


11^ 


- 


63392 


- 


12% 


12^ 


- 


72240 


- 


- 


- 


- 


80640 


- 


- 


- 


- 


- 


- 


- 


- 


- 



FINE-LAID. 



R 


.5 


1.37 


- 


7500 


- 




3K 


2% 


R 


•% 


1.68 


- 


9680 


- 




3% 


3K 


U 


-% 


2.12 


- 


11600 


- 


- 


%% 


m 


R 


.% 


2.45 


- 


17280 


- 


- 


4% 


±% 


R 


•% 


2.56 


- 


22800 


- 


- 


^A 


5 


R 


1. 


2.98 


- 


32800 


- 


- 


m 


6K 


R 


*.» 


3.36 


- 


40400 


— 


- 


8% 


7 


R 


i.U 


3.91 


- 


54400 


- 


- 


8% 


R 


1.x 


4.5 


- 


70000 


- 


_ 


10 


9% 


R 


I'M 


4.9 


- 


87200 


- 


- 


11% 


10% 


R 


*.M 


5.44 


- 


108000 


- 


- 


12K 


12^ 


R 


i-X 


6.2 


- 


130000 


- 


- 


- 




R 


2.^ 


6.62 


- 


148000 


- 


- 


- 


- 



\% 
ii 



In the above table the determination of the circumference of the 
rope, etc., is based upon the Breaking Weight or Tensile resistance of 
the wire being reduced by one fourth, and the units or the ultimate re- 
sistance of the rope, etc., are reduced one third. 

In the U. S. Navy the relative dimensions of Hemp Cable and of Wire 
Rope are an follows : 

Circumference in Inches. 

Hemp... 3, 4, 5, f% 6, 6%, 7}£ 8, 9, 10 ; 10^,11, 12, 

Wire.... 1%, 2y 8l 2%, 3, 3J£ 3%, 4, 4%, 4%, 5K, 5%, 6, 6^'. 

Note. — The difference between tho dimensions of the wire rope here given and 
In the preceding table, of one fourth in area, is in consequence of the high estimato 
of strength given to the hemp rope made in the U. S. Service. 



142 HEMP AND MANILA ROPES, SHROUDS, ETC. 

The circumferences given are for Tarred ropes, etc., alone ; if, there- 
fore, the circumferences for White and Manila ropes are required 
proceed as follows : 

Xo Compute the Circumference of a ^White or Ma- 
nila Rope, Hawser, or Cable compared, with, one 
of Tarred. Hemp. 

Rule.— Multiply the square of the circumference of the given rope 

by the unit for the circumference, from the table, page 136 ; divide the 

product by the unit for the circumference of the rope, etc., required, 

and the square root of the product will give the circumference required. 

Note— If the circumference is required for a rope in general use, reduce the units 
in the table one third. 

Illustration.— Required the circumferences for a white rope and a Manila haw- 
ser, for general use— equivalent to a tarred rope of 3 strands, and 9% inches in cir- 
cumference. 

Units of tarred rope of 9% ins. = 780 — % — 520. 

" white rope of about 9% " = 1045 — % = 697. 

" Manila hawser of about 9% " = 760 — % — 507. 

Then 9.52x520 = 43030, which -^ 697 = GT. 33, and a/67.33 
the white rope. 



Again, 9.52x520 = 4G330, which -4-507 : 
for the Manila hawser. 



S.2, say S% ins. for 
92.56, and V92.56 = 9.62, say 9% ins 



Proof and. Breaking Strain of* Chain CaToles. 


Diameter 
of Chain. 


Proved. 


Breaking Strain. 


Diameter 
of Chain. 


Proved. 


Breaking Strain. 


Ins. 


Lbs. 


Lb8. 


Ins. 


Lbs. 


Lbs. 


% 


16750 


33500 


1% 


56675 


113350 


l 


21700 


43400 


ifl 


65750 


1315(10 


iy s 


27500 


55000 


i% 


75650 


151300 


\X 


33300 


66600 


2 


86100 


172200 


i% 


40450 


80900 


2>£ 


97375 


194750 


ix 


48150 


96300 


23^ 


109090 


218180 



The proof of British Navy Chain is % the breaking strain. 

To Compute the Circumference of* tlie Shrouds of "Ves- 
sels, and. to Ascertain their UN" umber of Shrouds. 







Number of Pairs 






Number of Pairs 




Unit. 


Shrouds. 


Vessels. 


Unit. 


Shrouds. 


Vessels. 






c 






• 






o 
fa 


c 
'5 


i 






CD 

i 

fa 


"3 

s 


<0 


SAILING \H3SELS. 










Over S00 Tons . . 


.82 


6 


7 


5 


Ship 

Bark 


.64 


r> 


7 


5 


Under 800 " .. 


.70 


6 


7 


- 


.63 


5 


5 


4 


Under 400 ■" .. 


.0 


4 


3 


- 


Brig 


.55 


4 


5 




SIDE- WHEEL STEAM- 
ERS. 










Schooner 


.32 


3 


4 


_ 










Sloop 


.5 to .3 


4 to 2 


- 


- 


First Rate 


.96 


9 


10 


r> 


SCBEW STEAMERS. 










Over 1400 Tons . . 
Under 1400 " . . 


.81 
.70 





7 

7 


5 
5 


First Rate 


.96 


10 


10 


6 


Over 800 " .. 


.7 


5 


6 


4 


Over 1500 Tons . . 


.95 


9 


10 


6 


Under 800 " .. 


.05 


5 


5 




Under 1500 " . . 


.81 


8 


9 


6 


Under 400 » .. 


.55 


4 


3 


- 



Note.— The extreme Unit and Number of pair of Shrouds are given in each case. 



I 



WEIGHT OF ROPES, HAWSERS, CABLES, ETC. 143 

Rule. — Multiply the mean extreme length of the fore and main 
mast (measuring from the keelson) in feet by the unit in the preced- 
ing table, and the square root of the product will give the circumfer- 
ence of the fore and main shrouds in inches. 

Example. — What should be the circumference of the fore and main shrouds of a 
screw steamer of 1600 tons, the mean length of her fore and main masts being 110 
feet? 

110 X .95 = 104.5, and VW4.5 = 10.22, say 10J£ ins. 

Note. — When a mast does not step upon the keelson, assume its length to extend 
to it. 

2. These units are somewhat too large for the circumference of the mizzen shrouds. 

Thus, by the above rule, the circumference of the mizzen shrouds of a first-class 
Frigate would be 9 inches, whereas 8 inches is the proper circumference. 

To Compute the TV^eight of Ropes, Hawsers, and. 

Cables. 

Rule. — Square the circumference? and multiply it by the appropri- 
ate unit in the following table, and the product will give the weight 
per foot in pounds : 

ROPES. HAWSERS. CABLES. 

3-strand Hemp 032 .031 .031 

3-strand tarred Hemp 042 .041 .041 

3-strand Manila ....032 .031 .031 

4-strand Hemp . 033 — — 

4-stiand tarred Hemp 048 — — 

4-strand Manila 035 .034 .034 

The units for Thread Ropes is the same as that for Ropes of like 
material. 

Example — What is the weight of a coil of 10-inch Manila rope of four strands of 
i20 fathoms ? 

102 X .035=i3.5, and 120x6x3.5=: 2520 lbs. 



"Weight of Mien and. Women. 

The average weight of 20,000 men and women, weighed at Boston, 
1864, was— men, Hl% lbs. : women, 124% lbs. 



"Weight of Horses.— (XT. S.) 

The weight of horses ranges from 800 to 1200 lbs. 

WEIGHT OF CATTLE. 
To Compute the Dressed Weight of Cattle* 
Rule. — Measure as follows • 

1. The girt close behind the shoulders. 

2. The length from the fore-part, of the shoulder-blade along the 
back to the bone at the tail, in a vertical line with the buttocks. 

Then multiply the square of the girt in feet by 5 times the length 
in feet, and divide the product by 1.5; the quotient will give the 
dressed weight of the quarters. 



144 WEIGHT AND DIMENSIONS OF SHOT AND SHELLS, ETC. 

Example — The girt of a beeve is 6.5 feet, and the length, measured as above, is 
5. 25 feet. 

X5 = 42.25X2G.25 = 115?4^ = T39.3T5^ 



T5 -— a^.,^ — 



Note.— With very fat cattle divide by 1.425, and with very lean by 1.5T5. 

2. The quarters of a beeve exceed by a little half the weight of the living animaL 

3. The hide weighs about the eighteenth part, and the tallow the twelfth part. 



WEIGHT AND DIMENSIONS OF SHOT AND SHELLS. 

The weights of these may be ascertained by rules for the Mensura* 
tion of Solids ; also, by inspection in the tables, pages 543 and 544. 

To Compute the "Weight of a Cast-iron Shot from 
its Diameter. 

A cast-iron shot of 4 inches in diameter weighs 8.736 lbs. 

Therefore, -^- of the cube of the diameter is the weight of a shot of any diameter, 
for the weights of spheres are as the cubes of their diameters. 

Rule.— Multiply the cube of the diameter in inches by .1365,* and 
the product is the weight. 

Example.— What is the weight of a cast-iron shot 10 inches in diameter? 
10 3 X.1365 = 136.5Z6s. 

To Compute tlie TJiameter from, the "Weight. 

Rule.— Divide the weight in pounds by .1365, and the cube root 
of the quotient is the diameter. 
Example.— What is the diameter of a cast-iron sbot, its weight being 99.5 lbs. ? 
99.5-^.1365 = 729, and -^729 = 9 ins. 

To Compute tlie "Weight or Diameter of a Leaden 

Snot. 

A lead shot 4 inches in diameter weighs 13.744 lbs. 

13 744 

Therefore, -^- of the cube of the diameter is the weight of a shot of any diameter. 

Rule.— Multiply the cube of the diameter in inches by .2147, and 
the product is the weight. Or, divide the weight in pounds by .2147, 
and the cube root of the quotient is the diameter. 

Example.— What is the weight of a lead shot 10 inches in diameter? 
103 X . 2147 = 214.7 Hw. 

To Compute the Weight of a Cast-iron Shell. 

Rule.— Multiply the difference of the cubes of the exterior and in- 
tenor diameter in inches by .1365. 

Example.— What is the weight of a cast-iron shell having diameters of 10 and 
S.5 inches. 

103 _8.53 = 1000- 614.125 = 3S5.S75, w h ichx . 1335^52. 672 lbs. 



* .1305 represents a cubic inch of cast iron = .2607 lbs., and .1474 a cubic inch of 
wrought iron = .2816 lbs. 



SHOT AND SHELLS. FRAUDULENT BALANCES. 145 



PILING OF SHOT AND SHELLS. 

To Compete the IN"um ber of Shot in a Triangular 

File. 

Rule. — Multiply continually together the number of shot in one 
side of the bottom course, and that number increased by 1 ; and again 
by 2, and one sixth of the product will give the number. 

Example.— What is the number of shot in a triangular pile, each side of the br.se 

containing 30 shot ? 

30x30 + 1x30 + 2 29760 

— = — ; — =; 4900 6/iot. 

o o 

To Compute the Number of Shot in a Square File. 

Rule. — Multiply continually together the number in one side of the 
bottom course, that number increased by 1, double the same number 
increased by 1, and one sixth of the product will give the number. 

Example. — How many balls are there in a square pile of 30 courses? 

30x30+1x30x2+1 56730 o _'; 

■ — 5 ! — == — — = 9455 shot 

o o 

To Compute the Number of Shot in an Oblong 

3?ile. 

Rule. — From 3 times the number in the length of the base course 
subtract one less than the number in the breadth of it; multiply the 
remainder by the number in the breadth, and again by the breadth, 
increased by 1, and one sixth of the product will give the number. 

Example. — Required the number of balls in an oblong pile, the numbsrs in the 
base course being 16 and 7 ? 

16x 3-f^TxTxf+T = gS52 = 392 shot 
6 

To Compute the Number of Shot in an Incom- 
plete Pile. 

Rule. — From the number in the pile, considered as complete, sub- 
tract the number conceived to be in that portion of the pile which is 
wanting, and the remainder will give the number. 



FRAUDULENT BALANCES. 

To Detect them. 

After an equilibrium has been established between the weight and 
the article weighed, transpose them, and the weight will preponderate 
if the article weighed is lighter than the weight, and contrariwise if it 
is heavier. 

To .Ascertain the True "Weight. 

Rule. — Ascertain the weight which will produce equilibrium after 
the article to be weighed and the weight have been transposed ; reduce 
these weights to the same denomination, multiply them together, and 
the square root of their product will give the true weight, 

N 



146 BOARD AND TIMBER MEASURE. 

Example.— If the first weight is 32 lbs., and the second, or weight of equilibrium 
after transposition, is 24 lbs. 8 oz., what is the true weight? 



24 lbs. S oz. = 24.5 lbs, 
Then 32 X 24.5 = 784, and V 7 S4 .= 2S lbs. 

Or, when a represents longest arm, f A greatest weight, and 

6 lt shortest arm, B least weight. 

Then Wa = A&, and W6 = Ba ; multiplying these two equations, W a a& = ABai 
or W* = AB, and W == V AB. 

Illustration. — A =82; B = 24.5; W = 2S. Assume the length of the longe: 
arm = 10. 

Then 32 : 28 : : 10 : 8.75. 

Hence a = 10, 5 = 8.75, or 2S2 = 32x24.5, and 28=V32x24.5. 






To Ascertain tlie W eiglit of a Bar, Beam, etc., "by 
tlie Aid of a known "W^eiglit, as the Body of a 
Man, etc. 

Operation. — Balance the bar, etc., over a suitable fulcrum, and 
note the distance between it and the end of its longest arm. Suspend 
the known weight from the longest arm, and move the bar, etc., upon 
the fulcrum, so that the bar with the attached weight will be in equi- 
librio ; subtract the distance between the two positions of the fulcrum 
from the longest arm first obtained ; multiply this remainder by the 
weight suspended, divide the product by the distance between the ful- 
crums, and the quotient will give the weight required. 

Example. — A piece of tapered timber 24 feet in length is balanced over a fulcrum 
when 13 feet from the less end ; but when the body of a man weighing 210 lbs. is 
suspended from the extreme of the longest arm, the piece and the weight are bal- 
anced when the fulcrum is 12 feet from this end. What is the weight of the timber? 
13-12 = 1, and 13 — 1 = 12 /eg*. 

Then, 12x210 -M = 2520 lbs. 



BOARD AND TIMBER MEASURE. 

BOAED MEASURE. 
In Board Measure, all boards are assumed to be 1 inch in thickness. 

To Compute tlie Measure or Surface in Square 

Feet. 

When all the Dimensions are in Feet. 

Rule.— Multiply the length by the breadth, and the product will 
give the surface required. 

When either of the Dimensions are given in Inches. 
Rule. — Multiply as above, and divide the product by 12. 

When all the Dimensions are in Inches. 
Rule. — Multiply as before, and divide the product by 144. 
Example.— What are the number of square feet in a board 16 feet in length and 
16 inches in width? 

15x16 = 240, and 240 -4- 12 = 20 feet. 



BOARD AND TIMBER MEASURE. 147 

TIMBER MEASURE. 
To Compute tlie Volume of Round. Timber. 

When all the Dimensions are in Feet. 
Rule.— Add together squares of diameter of greater and lesser ends, 
and product of the two diameters ; multiply sum by .7854, and product by 
one third of length. 

Or, a-f-a'+a"Xo=V. « ««^ a' representing areas of ends, and a" area 
of mean proportional. 
Note.— Mean proportional is the square root of product of area of both ends. 

Or, c 2 +c' 2 +cXc'X.0795SXo-=V. c and c' representing circumference of ends. 

Example — Diameter of a log of timber is 2 and 1.5 feet and length 15; what is 
its volume ? 



15 



22+1.52+2x1.5=6.25+3=9.25, which X.7854 and j=ZQ.32 feet. 

When the Length isgivenin Feet, and the Diameter, Area, or Circumference 
in Inches. 

Rule.— Multiply as above, and divide by 144. 

When all the Dimensions are in Inches* 

Rule.— Multiply as before, and divide by 1728. 

Sawed or Hewed Timber is measured by the cubic foot. 

To Compute tlie "Volume of Square Timber. 
When all the Dimensions are in Feet. 

Rule. — Multiply the product of the breadth by the depth, by the 
length, and the product will give the volume in cubic feet. 

When either of the Dimensions are given in Inches. 
Rule. — Multiply as above, and divide the product by 12. 

When any two of the Dimensions are given in Inches. 
Rule. — Multiply as before, and divide by 144. 

Example — A piece of timber is 15 inches square, and 20 feet in length; required 
its volume in cubic feet. 

15x 15x 29 = 4500, and 4500 + 144 = 3 1.25 cubic feet. 

SPARS AND POLES. 

Pine and Spruce Spars, from 10 to V/i inches in diameter inclusive, 
are to be measured by taking their diameter, clear of bark, at one 
third of their length from the abut or large end. 

Spars are usually purchased by the inch diameter ; all under 4 inch- 
es are termed Poles. 

Spars of 7 inches and less should have 5 feet in length for every 
inch of diameter, and those above 7 inches should have 4 feet in length 
for every inch of diameter. 



148 HYDROMETERS. 

HYDROMETERS. 

The U. S. Hydrometer (Tralle's) ranges from (water) to 100 (pare 
spirit); it has not any subdivision or standard termed "Proof," but 
50, upon the stem of the instrument, at a temperature of 60°, is the 
basis upon which the computations of duties are made. 

In connection with this instrument, a Table of Corrections, for differences in the 
temperature of spirits, becomes necessary ; and one is furnished by the Treasury De- 
partment, from which all computations of the value of a spirit are made. 

Illustration. — A cask contains 100 gallons of whisky at 70°, and the hydrome- 
ter sinks in the spirit to 25 upon its stem. 

Then, by table, under 70°, and opposite to 25, is 22.99, showing that there are 
22.99 gallons of pure spirit in the 100. 

The Commercial Hydrometer (Gendar's) has a "Proof" at 60°, which 
is equal to 50 upon the U. S. Instrument and its gradations, run up 
to 100 with it, and down to 10 below proof, at upon the U. S. In- 
strument; or the of the Commercial Instrument is at 50 upon the 
U. S. Instrument, from which it progresses numerically each way, each 
of its divisions being equal to two of the latter. 

In testing spirits, the Commercial standard of value is fixed at proof; 
hence any difference, whether higher or lower, is added or subtracted, 
as the case may be, to or from the value assigned to proof. 

A scale of Corrections for temperature being necessary, one is fur- 
nished with the Thermometer. 

Application of the TJiermometer. — The elevation of the mercury indicates the cor- 
rection to be added or subtracted, to or from the indication upon the stem of the hy- 
drometer. 

When the elevation is above 60°, subtract the correction ; and when below, add it. 

Illustration. — A hydrometer in a spirit indicates upon its stem 50 below proof, 
and the thermometer indicates 4 above 00° in the appropriate column. 

Then 50 — 4 = 46 = strength below proof. 

To Compute tine Strength, ofa Spirit, or the Vohime of its 
pnre Spirit, "by a Commercial Hydrometer, and. convert 
it to the Indication ofa XJ. S. Hydrometer. 

When the Spirit is above Proof. Rule. — Add 100 to the indication, and divide 
the sum by 2. 

When the Spirit is below Proof. Rule.— Subtract the indication from 100, and 
divide the remainder "by 2. 

Example.— A spirit is 11 above proof by a Commercial Hydrometer ; what propor- 
tion of pure spirit does it dontain ? 

11 + 100 -4- 2 = C5.5 per cent. 

To Compute the Strength, etc., "by a XJ. S. Hydrometer. 

Whpn the Spirit is above Proof. Rule. — Multiply the indication by 2, and sub- 
tract 100. 

When the Spirit is below Proof. Rule. — Multiply the indication by 2, and sub- 
tract it from 100. 

Exavple.— A spirit is 55.5 by a U. S. Hydrometer; what is its per centage above 
proof? 

55.5x2 — 100 = 11 per cent. 

The Commercial practice of reducing indications ofa hydrometer is as follows : 

Multiply the number of gallons of spirit by the per centage or number of degrees 
above or below proof, divide by 100, and the quotient will give the number of gallons 
to be added or subtracted, as the c se may be. 

Illustration. — 50 gallons of whisky are 11 per cent, above proof. 

Then 50x11 -r-100 = 5.5, which, added to £0 — 55.5 gallons. 



U. S. ENSIGNS, PENNANTS, AND FLAGS. 149 

U. S. ENSIGN, PENNANTS, AND FLAGS. 

Ensign (Head, Lepth, or Hoist).— Ten nineteenths of its length. 
• Field. — Thirteen horizontal stripes of equal breadth, alternately red 
and white, beginning with red. 

Union. — A blue field in the urjper quarter, next the head, A of the 
length of the field, and 7 stripes in depth, with white stars ranged in 
equidistant, horizontal, and vertical lines, equal in number to the num- 
ber of states of the Union. 

Pennant (Narrow).— Head — 6.24 inches to a length of 70 feet ; 5.76 inches to a 
length of 55 feet ; 5.24 inches to a length of 40 feet ; 4.8 inches to a length of 30 feet ; 
and 42 inches to a length of 25 feet. 

Night — 3.6 inches to a length of 20 feet. Boat— 3 inches to a length of 9 feet, and 
2.42 inches to a length of 6 feet. Union — A blue field at the head, one fourth the 
length, with 13 white stars in a horizontal line. Field.— A red and white stripe ta- 
pered to a point, red uppermost, each of the same breadth at any part of the length. 

Night and Boat Pennants. — Union to have but 7 stars. 

Jack. — Alike to the Union of an Ensign. 

Vice-Admiral's Flag — A Rectangle. Blue. Three five-pointed 
stars set as an equilateral triangle 18 inches from centres, the upper 
star 18 inches from the head and 27 inches from the tabling. 

Rear-Admiral's Flag — A Rectangle. Field Blue, Red, or White. 

Stars. — Two set vertically 18 inches from centres, the upper star set 
18 inches from the head and 18 inches from the tabling. White when 
the flag is Blue or Red, and Blue when it is White. 

Boat and Night Flags. — The distances between the stars to be proportionably less 
than above. 

Commodore's Pennant (Broad). — Blue, Red, or White, with stars 
ranged in equidistant vertical and horizontal lines, equal in number to 
the States of the Union, to be white in the blue and red pennants, and 
blue in the white. 

Swallow-tailed, the angle at the tail to be bisected by a line drawn at a right an- 
gle from the centre of the depth or hoist, and at a distance from the head of three 
fifths of the length of the pennant ; the lower side is to be rectangular with the 
head or hoist ; the upper side is to be tapered, running the width of the pennant at 
the tails .1 the hoist. The stars to be ranged in the field alone in equidistant verti- 
cal lines, and horizontally to taper with the pennant. Head. — .6 their length. 

Signals (Numbers). — Head. — .8 their length. Repeaters. — Head. 
— Eleven twentieths of their length. Quarantine Flag — A Rectan- 
gle. Field yellow. Head. — Nine elevenths of their length. 

Divisional Mark. — A Triangle. Field of three stripes or divi- 
sions, but of two colors only ; the centre one being of a color different 
from the others, to be in the form of a wedge, the bar being one third 
of the head, and the point extending to the extremity of the fly. 

1st Division — blue, white, and blue; I 3d Division — white, blue, and white; 
2d " red, white, and red ; I Service office — white, yellow, and white. 

When the Lengths are 6.4, 5.6, 4.8, and 4 feet, the Heads are 8, 7, 6, and 5 feet. 

Secretary of the Navy's Flag — Blue. Head. — 10.25 feet; fly 
14.4 feet, with a white foul anchor 3 feet in length, set vertically in 
the centre. 

Storm Flag, same, with a head of but 5.4 feet, and a fly of 7.6 feet. 

N* 



150 MEASURES, WEIGHTS, PRESSURES, ETC. 

MEASURES, WEIGHTS, PRESSURES, ETC. 

Equivalents of Old. and. "New XT. S. and of French 
Measures, "Weiglits, Pressures, etc. 

MEASURES. 

By Act of Congress, July, 1S68. By French Computation. 

1 Litre per foot, yard, acre, etc. = Gallons per foot, yard, acre, etc. 

. • 61.022 \' 77 ■ Iv'i 61.025387 -, „ 

1:1:: — — — : .2642 gallon, etc. 1:1:: — — : .2642 gallon, etc 

231 231 

1 Gallon per foot, acre, etc. = Litres per foot, acre, etc.. 
901 9m 

1 : 1:: „ ^^ : 3.7855 liters, etc. 1 : 1:: M t ^ w ^ : 3.7853 litres, etc. 
61.022 ' I 61.025387 ' 

1 Litre per square meter = Gallons per square foot. 

144 I 144 

231 : 6202 : : --^- : .0245 gallon. 231 : 61.025387 : : „ : .0245 gall 

looO 1550.031 

1 Gallon per square foot = Litres per square metre. 

61.022: 231::^?: 40.7466 liters. 161.025387 231:; 15 ° a08t : 40.7 154 litres, 
144 144 

1 Litre per cubic metre = Gallons per cubic foot. 

231 : 61.022 : : A~^ : 9.3292 gallons. I 231 : 61025.387 :: : 9.392 nails. 

61.022 u | 61025.387 J 

1 Square foot per acre = Square metres per hectare. 

1550 : 144 :: — — -: .2296 sq. meter. 1550.031: 144 : : —1— : .2296 so. metre. 
2 .471 | 2-4711 ? 

WEIGHTS AND PRESSURES. 

Note 30 inches of mercury at 62° = 14 7 lbs. per sq. inch; hence 1 lb. = 2.0408 

inches, and a centimetre of mercury = 30 -H. 3937 for U. S. computation and 30 -r- 
39.370432 for French. 

1 Centimetre of mercury per square inch = Pounds per square inch. 
2.0408 : .3937::1 : .19291 pounds. \ 2.0408 : .39370432::1 : .19292 pounds. 
1 Atmosphere (14.7 pounds per square inch) == Kilogrammes. 
14.7 -4- 2.2046 = 6.6679 kilograms. | 14.7 -4- 2.20462 = 6.6678 kilograms. 
1 Inch of mercury per sq. inch = Centimetres of mercury per sq. inch. 

1 -f- .3937 = 2.54 centimeters. | 1 -4- .39370432 == 2.54 centimetres. 
1 Pound per square inch = Grammes per square inch. 
15.432 : 7000 :: 1 : 453.6029 grams. |15.43235 : 7000 :: 1 : 453. 5926 grammes. 
1 Cubic foot per ton = Cubic metres per tonneau. 

9204 6 I 9904 69 

61022 : 1728 :: ~~ : .0279 cubic 61022.387 : 1728:: ™*'°* : .0279 cubic 
2240 2240 

meter. I metre. 

POWER AND WORK. 

1 Horse-power = Cheval, or Cheval-vapeur = (4500 k x m). 



83000 ,=1.01388 



33000 



4500 x 2.20462 x (39.370432 -r- 12) " 
1.01387 chevaux. 



4500 x 2.2046 x (39.37-4-12) 
chevaux. 
1 Cheval, or Cheval-vapeur = Horse-power. 

4500 x 2.20462 x (39.370432-4-12) 



4500 X 2.2046 x (39.37-4-12) 

33000 = - 986 ° 

horse. 



33000 
.9863 horse. 



MEASURES, WEIGHTS, PRESSURES, ETC. 151 

By Act of Congress, July, 1866. By French Computation. 

1 Foot-pound = Kilogrammetre (k x m) = 7.233 foot-pounds. 

1 -h 2.2046 x 3.280833 == .13826 kilo-\l ~ 2.20462 x 3.280869 = .13825 jb'fo- 
gram. \ grammetre. 

1 Pound per horse-power — Kilogrammes per cheval. 

2.2046 : 1 : : ., 1QQO : .44738 kilogram. 2.20462:1 : : } ; Ail kilogramme. 
l.Olooo I l.Oloo/ 

1 Kilogram per cheval = Pounds per horsepower. 

1 : 2.2046 : : -J— : 2.2352 pounds. 1 : 2.20462 : : -~L~ : 2.2353 pounds. 
.9863 I .9863 

1 Square foot per horse-power = Square metres per cheval. 

...... '.. 1 

1550 : 144 : : r-^- : .09163 sq. meter. 
l.Olooo 



1550.031 : 144 :: * , ; .09163 53. 
l.Oloo/ 



1 Caloric, or French unit = Heat units. 

3.968 £ea£ units. \ 3.968 Aea* wmte. 
1 Heat unit = Calorics. 

1 -h 3.968 = . 252 caforzc. | 1 -r- 3.968 = .252 caloric. 
1 U. S. mechanical equivalent (772 foot-pounds) == Kilogrammefres. 
772-^-7.233 = 106.733 kilograms. |772^-7.233 = 106.733 kilogrammetres. 

1 Heat unit per pound = Calorics per kilogramme. 

2.20462 



1 : .252 :: -^— — : .5551 kilogram. 



1 : .252 :: -~r — : .5556 kilogramme* 



1 Caloric per kilogram = Heat units per pound. 

.252 : 1 :: — ; 1.8 heat units. .252 : 1 :: — J — ■ : 1.8 heat units. 

2.2046 I 2.20462 

1 French mechanical equivalent (423.55 k x m) = Foot-pounds. 
3.280833 x 2.2046 X 423.55 = 3063.5 13.280869x2.20462x423.55=3063.566 
foot-pounds I foot-pounds. 



1 Foot per second = Metres per second. 

1 -r- 3.280833 = .3047 meter. | 1 -^- 3.280869 = .3047 metre. 
1 Metre per second = Feet per second. 

3.280833 -r-il = 3.280833 feet. | 3.280869 + 1 = 3.280869 feet. 
1 Metre per hour = Metres per second. 

3.280833 : 5280 :: i : 26.8224 meters. I 3.280869 : 5280 :: i : 26.8222 metrh: 
60 ! 60 

"Weights, etc., of Principal Race and Trotting 
Courses of the TJnitecL States, etc. 

RACING.— Weights Feather.— By weight, 75 lbs. ; by custom, a Jockey 

who is not weighed. 

Welter — 40 lbs. added to weight for age, except by South Caroli7ia,hy State Agricu'l 
Soc'ij, Sacramento, and by Louisville Jockey Club, where it is 28 lbs. 

Distances. — Maryland, Louisiana, Louisville, Savannah, and Central 
Georgia Jockey Clubs, and Saratoga Ass'ri, and State AyricxCl Soc'y, Sacramento. 
— 1 mile, 40 yards ; 2 miles, 50 ; 3 miles, GO ; 4 miles, 70. 

Kentucky Ass^n and South Carolina Jockey Club. — 1 mile, 50 yards; 2 miles, 
60 ; 3 miles, 80 ; 4 miles, 100. 

Nashville Blood-horse Awfn. — 1 mile, 50 yards ; 2 miles, 65; 3 miles, 80 ; 4 miles, 
100 ; and 3 single mile heats in 5, 50. 

.A»ge.— Age of all horses dates from 1st of January. 



152 



EACING AND TROTTING COURSES. 



General Rules. — In racing, a horse is not allowed to start with 5 lbs. overweight. 
If a rider returns 2 lbs. short of his weight, he loses the heat ; and if 3 lbs., he is dis- 
tanced. Fillies and Geldings are allowed 3 lbs. in all cases. 

No distance: in a Dash, and none in a match unless specified. 

No two horses from one stable in a heat race, and no two riders except by per- 
mission of the Judge. Only one horse to start from one stable, except in a Dash. 

No article is allowed to make weight from which a liquid can be wrung. 

One pound is allowed for weight of curb or a double bridle. 

Bridle is not included in weight to be carried. 

No distance in a third heat. 

In England, a yearling's course is 2 furlongs ; 2 years, 6 furlongs ; 3 years, 1 mile; 
4 years, 2 miles; 5 years and upward, 4 miles. A rider carrying more than 2 lbs. 
over his assigned weight is distanced. A Feather is 66 lbs., and no horse can start 
with a less weight. 

TROTTING.— "VST eights.— National Association.— Wagon or Sulky, 150 lbs., 
exclusive of harness ; Saddle, 145 lbs., inclusive of saddle and whip. 
If 20 lbs. over weight, it is to be announced from the Judges' stand. 
TJistanoes.— 1 mile heats, 80 yards; 2, 150; 3, 220; 3 mile heats in 5, 100. 

Time TDet^^een. Heats. — 1 mile. 20 minutes ; 2 miles, 30; 3 miles, 35; 
4 miles, 40 ; 3 single mile heats in 5, 25. 

"Whips. —Saddle, 2 feet 10 ins. ; Sulky, 4 feet 8 ins. ; Wagon, 5 feet 10 ins. ; 
Double Team, 8 feet ; Tandem and Four-in-hand, unlimited. 



i 



Lengths of English. Race-courses. 



NEWMARKET. 

Beacon 

last 3 miles 

Cambridgeshire. . 

Cesarewitch 

Ditch Mile 

Round 

Rowley Mile 

Yearling 



Miles. 
4.206 
3.034 
1.136 
2.266 
1.136 
3.579 
1.009 
.277 



DONC ASTER. 

St. Leger 

Cup Course 

TYC 



EPSOM. 

Craven 

Derby and Oaks. . 
Metropolitan 



Miles. 

1.825 

2.634 

.996 

1.25 

1.5 

2.25 



GOODWOOD. 

Cup Course.... 



Liverpool, new.. 

YORK. 

Stakes Course . . . 



Miles. 
2.5 



1.5 



1.75 



Dimensions of Canal Locks.-(U. S.) 



Length I Breadth, 



Depth. L'gth Canal. 



Albemarle and Chesapeake 



Black River, Crooked Lake,} 
Chenango, Chemung, and > 
Genesee Valley ) 



Chesapeake and Delaware* . 

Cham plain 

Cayuga and Seneca 

Delaware and Raritan 

Dismal Swamp 

Erie 

Falls of the Ohio, Ky 

Oneida 

" Improvement 

Oswego 

Welland 



Feet 

220 



90 



220 
110 
110 
220 

90 
110 
350 

90 
120 
110 
270 



Feet 

40 



15 



24 

18 

18 

24 

17.5 

18 

80 

15 

30.5 

18 

45 



Feet. 



9 
4 
7 
7 

5.5 
7 

2 to 60 
4 

4.5 
4 
14 



Miles. 

14 

( 77 

8 
< 97 
| 33 
U13.75 

14 

66.75 

24.75 

43 

44 
352 



The length of vessel that can be 
the lengths of the locks. 



7 

19 

38 

28 

transported is somewhat less than 



* Height from under side of Summit Bridge to surface of water, 76 feet 10 inch**. 



VETERINARY. 153 

VETERINARY. 

Horses.- Cathartic Ball— Cape Aloes, 6 to 10 drs. ; Castile Soap, 1 dr. ; Spirit 
of Wine, 1 dr. ; Sirup to form a ball. If Calomel is required, add from 20 to 50 grains. 
During its operation, feed upon mashes and give plenty of water. 

Cattle.— Cathartic. — Cape Aloes, 4 drs. to 1 oz. ; Epsom Salts, 4 to 6 oz. ; pow 
dered Ginger, 3 drs. Mix, and give in a quart of gruel. For Calves, one third o" 
this will be sufficient. 

Dogs. Cathartic. — Cape Aloes, X a dr. to 1 oz. ; Calomel, 2 to 3 grs. ; Oil of 

Caraway, 6 drops ; Sirup to form a ball. Repeat every 5 hours till it operates. 

Horses and. Cattle.— Tome— Sulphate of Copper, 1 oz. to 12 drs. ; Sugar, 
X an oz. Mix, and divide into 8 powders, and give one or two daily in food. 

Cordial— Powdered Opium, 1 dr. ; powdered Ginger, 2 drs. ; Allspice, powdered, 

3 drs. ; Caraway Seeds, powdered, 4 drs. Make into a ball with sirup, or give as a 
drench in gruel. 

Cordial Astringent Drench for Diarrhcea, Purging, or Scouring.— Tincture oi 
Opium, X an oz - i Allspice, 2% drs. ; powdered Caraway, X an oz - » Catechu Pow- 
der, 2 drs. ; strong Ale or Gruel, 1 pint. Give every morning till the purging ceases. 
For Sheep this will make 4 doses. 

Alterative.— Kthiop's Mineral, >£ an oz - '■> Cream of Tartar, 1 oz. ; Nitre, 2 drs. Di- 
vide into from 16 to 24 dozes, one morning and evening in all cutaneous diseases. 

Diuretic Ball.— Hard Soap and Turpentine, each 4 drs. ; Oil of Juniper, 20 drops 
powdered Resin to form a ball. 

For Dropsy, Water Farcy, Broken Wind, or Febrile Diseases, add to the above All- 
spice and Ginger, each 2 drs. Divide into 4 balls, and give one morning and evening. 

Alterative or Condition Powder. — Resin and Nitre, each 2 oz. ; levigated Anti- 
mony, 1 oz. Mix for 8 or 10 doses, and give one morning and evening. When given 
to Cattle, add Glauber Salts, 1 lb. 

Fever Ball. — Cape Aloes, 2 oz. ; Nitre, 4 oz. ; Sirup to form a mass. Divide into 
12 balls, and give one morning and evening till the bowels are relaxed ; then give 
an Alterative Powder or Worm Ball. 

Tar or Hoof Ointment. — Tar and Tallow, each 1 lb. ; Turpentine, % a lb. Melt. 

Dogs — Emetic. — 2 to 4 grs. of Tartar Emetic in a meat ball, or a teaspoonful 
or two of common salt. Give twice a week if required. 

Distemper Powder. — Antimonial Powder, 2, 3, or 4 grs. ; Nitre, 5, 10, or 15 grs. ; 
powdered Ipecacuanha, 2, 3, or 4 grs. Make into a ball, and give two or three times 
a day. If there is much cough, add from % a gr. to 1 gr. of Digitalis, and every 3 
or 4 days give an Emetic. 

Mange Ointment. — Powdered Aloes, 2 drs.; White Hellebore, 4 drs.; Sulphur, 

4 oz. ; Lard, 6 oz. 

Red Mange, add 1 oz. of Mercurial Ointment, and apply a muzzle. 

Note Physic, except in urgent cases, should be given in the morning, and upon 

an empty stomach ; and, if required to be repeated, there should be an interval of 
several days between each doae. 

To Ascertain a Horse's Age. — A foal of six months has six grinders in each jaw, 
three in each side, and also six nippers or front teeth, with a cavity in each. 

At the age of one year, the cavities in the front teeth begin to decrease, and he 
has four grinders upon each side, one of the permanent and the remainder of the 
milk set. 

At the age of two years, he loses the first milk grinders above and below, and the 
front teeth have their cavities filled up alike to the teeth of horses of eight years of age. 

At the age of three years, or two and a half, he casts his two front uppers, and in 
a short time after, the two next. 

At four, the grinders are six upon each side ; and, about four and a half, his nip- 
pers are permanent by the replacing of the remaining two corner teeth ; the tushes 
then appear, and he is no longer a colt. 

At five, a horse has his tushes, and there is a black-colored cavity in the centre 
of all his lower nippers. 

At six, this black cavity is obliterated in the two front lower dippers. 

At seven, the cavities of the next two are filled up, and the tushes blunted ; and 
at eight, that of the tAvo corner teeth. The horse may now be said to be aged. Tho 
cavities in the nippers of th« upper jaw are not obliterated till the horse is about ton 
years old, after which time the tushes become round, and the nippers project and 
change their surface. 



154 GRAVITIES OF BODIES. SPECIFIC GRAVITIES. 



GRAVITIES OF BODIES. 

Gravity acts equally on all bodies at equal distances from the 
earth's centre; its force diminishes as the distance increases, and in- 
creases as the distance diminishes. 

The gravitating forces of bodies are to each other, 

1. Directly as their masses. 

2. Inversely as the squares of their distances. 

The gravity of a body, or its weight above the earth's surface, de- 
creases as the square of its distance from the earth's centre in semi- 
diameters of the earth. 

Illustration. — If a body weighs 900 lbs. at the surface of the earth, what will it 
weigh 2000 miles above the surface ? 

The earth's semi-diameter is 3993 miles (say 4000). 

Then 2000 -f 4000 = 6000, or 1 . 5 semi-diameters, 

and 900 -r- 152 — 1|^ = 400 lbs. 

Inversely, if a body weighs 400 lbs. at 2000 miles above the earth's surface, what 
will it weigh at the surface ? 

400X1.52 = 900 lbs. 

2. A body at the earth's surface weighs 360 lbs. : how high must it be elevated to 
weigh 40 lbs. ? 

-— =9 semi-diameters, if gravity acted directly; but as it is inversely, as the square 

of the distance. 
Then V 9 = 3 semi-diameters = 3 X 4000 = 12000 miles. 

3. At what height must a body be raised to lose half its weight ? 

As v^l : y/% : : 4000 : 5656 = as the square root of one semi-diameter is to the square 
root of two semi-diameters, so is one semi-diameter to the distance required. 
Hence 5656 — 4000 = 1656 = distance from the earth's surface. 
The diameters of two Globes being equal, and their densities different, 
the weight of a body on their surfaces will be as their densities. 

Their densities being equal and their diameters different, the weight of 
them will be as their diameters. 

The diameters and densities being different, the weight will be as their 
product. 

Illustration. — If a body weighs 10 lbs. at the surface of the earth, what will it 
weigh at the surface of the sun, their densities being 392 and 100, and their diame- 
ters 8000 and SS3000 miles? 

883000x100 -5- SjOu x 392 = 28. 15T = quotient of product of diameter of the sun and 
its density, and product of diameter of the earth and its density. 
Then 28. 157 X 10 = 2S1.57 lbs. 

Note. — The gravity of a body is .00346 less at the Equator than at the Poles. 






SPECIFIC GRAVITIES. 



The Specific Gravity of a body is the proportion it bears to the 
weight of another body of known density. 

If a body float on a fluid, the part immersed is to the whole body as 
the specific gravity of the body is to the specific gravity of the fluid. 

When a body is immersed in a fluid, it loses such a portion of its 
own weight as is equal to that of the fluid it displaces. 



SPECIFIC GRAVITIES. 155 

An immersed body, ascending or descending in a fluid, has a force 
equal to the difference between its own weight and the weight of its 
bulk of the fluid, less the resistance of the fluid to its passage. 

Water is well adapted for the standard of gravity ; and as a cubic 
foot of it weighs 1000 ounces avoirdupois, its weight is taken as the 
unit, viz., 1000. 

To Ascertain, the Specific Gravity of a Body 
heavier than Water. 

Rule. — Weigh it both in and out of water, and note the difference ; 

then, as the weight lost in water is to the whole weight, so is 1000 to 

., . * i t , ^ WxlOOO - 
the specific gravity of the body. Or, — — =G, w representing the 

weight in water, and G the specific gravity. 

Example What is the specific gravity of a stone which weighs in air 15 lbs., In 

water 10 lbs. ? 

15—10 = 5; then 5 : 15 : : 1000 : : 3000 spec. grav. 

Xo Ascertain, the Specific Oravity of a Body 
lighter than Water. 

Kule. — Annex to the lighter body another that is heavier than wa- 
ter, or the fluid used ; weigh the piece added and the compound mass 
separately, both in and out of water, or the fluid ; ascertain how much 
each loses in water, or the fluid, by subtracting its weight in water, 
or the fluid, from its weight in air, and subtract the less of these dif- 
ferences from the greater ; then, 

As the last remainder is to the weight of the light body in air, so is 
1000 to the specific gravity of the body. 

^ Example. — What is the specific gravity of a piece of wood that weighs 20 lbs. in 
air; annexed to it is a piece of metal that weighs 24 lbs. in air and 21 lbs. in water, 
and the two pieces in water weigh 8 lbs. ? 

20 -f 24 — 8 = 44 — 8 = 36 = loss of compound mass in water ; 
24 — 21 = 3 = loss of heavy body in water. 

33 : 20 : : 1000 : 606 = 24 spec. grav. 

To Ascertain the Specific Oravity of a Fluid. 

Rule. — Take a body of known specific gravity, weigh it in and out 
of the fluid ; then, as the weight of the body is to the loss of weight, 
so is the specific gravity of the body to that of the fluid. 

Example. — What is the specific gravity of a fluid in which a piece of copper (spec, 
grav. =9000) weighs 70 lbs. in, and 80 lbs. out of it? 

80 : 80 — 70 = 10 : : 9000 : 1125 spec. grav. 

To Compute the Proportions of two Ingredients 
in a Compound, or to discover Adulteration in 
Metals. 

Kule. — Take the differences of each specific gravity of the ingredi- 
ents and the specific gravity of the compound, then multiply the gravi- 
ty of the one by the difference of the other ; and, as the sum of the 
products is to the respective products, so is the specific gravity of the 
body to the proportions of the ingredients 



156 



SPECIFIC GRAVITIES. 



Example.— A compound of gold (spec, rjrav. = 1S.SSS) and silver (aper. grav. — 

10.535; has a specific gravity of 14; what is the proportion of each metal? 

1 8.838 — 14 = 4.SSSX 10.535 — 51.495 

14. — 10. 535 = 3. 465X18. S83 = 65. 447 

65.447 4-51.495 : 65.447 : : 14 : 1.S15 gold t 

65.447 4- 51.495 : 51.495 : : 14 : 6.165 silver. 



To Compute the Weights of tlie Ingredients, tliat 
of* the Compound being given. 

Rule. — As the specific gravity of the compound is to the weight of 
the compound, so are each of the proportions to the weight of its ma- 
terial. 

Example. — The weight, as above, being 23 lbs., what are the weights of the ingre- 
dients ? 



i±. 9Q • . / 7-835: 15.67 qolch 
14. 28. . | 6>165 . 1^33 



Iver. 



Proof of Spirituous Liquors. 

A cubic inch of proof spirits weighs 234 grains ; then, if an im- 
mersed cubic inch of any heavy body weighs 234 grains less in spirits 
than air, it shows that the spirit in which it was weighed is proof. 

If it lose less of its weight, the spirit is above proof; and if it lose 
more, it is below proof. 

Illustration — A cubic inch of glass weighing 700 grains weighs 500 grains when 
weighed in a certain spirit ; what is the proof of it ? 

700 — 500 = 2(J0 = grains = weight lost in the spirit. 

Then 200 : 234 : : 1. : 1.17= ratio of proof of spirits compared to proof spirits, 
or 1. =.17 above proof. 

Note For Hydrometers and Rules for ascertaining the Proof of Spirits, see page 

14S ; and for a very full treatise on Specific Gravities and on Floatation, see Jamie- 
son's Mechanics of Fluids. Lond., 1837. 

Solids. 

Rule. — Divide the specific gravity of the substance by 16, and the 
quotient will give the weight of a cubic foot of it in pounds. 

Specific | ™JgJf 



Specific 
Gravity. 


Weigbt 

of a Cub. 

Inch. 


2560 


.0926 


6712 


.2428 


5763 


.2084 


470 


.017 


9823 


.3553 


8S32 


.3194 


7S20 


.2S28 


83S0 


.3031 


8214 


.2972 


8700 


.3147 


2000 


.0723 


3000 


.1085 


S650 


.3129 


15S0 


.057 


5300 


.2134 


809S 


.2929 



METALS. 

Aluminum 

Antimony 

Arsenic 

Barium 

Bismuth 

Brass, copper 84 
44 tin 16 
44 copper 67 
44 zinc 33 

44 plate 

44 wire 

Bronze, gun metal 

Boron 

Bromine 

Cadmium 

Calcium 

Chromium 

Cinnabar 



METALS. 

Cobalt 

Columbium 

Gold, pure, cast 

44 hammered 

44 22 carats fine 

44 20 44 44 

Copper, cast 

44 plates 

44 wire 

Iridium 

44 hammered 

Iron, cast 

44 44 gun metal . . . 

44 hot blast 

14 cold 44 

44 u wrought bars 

14 ;4 wire 

44 rolled plates 



8600 


.3111 


6)00 


.217 


1925S 


.6965 


19361 


.7003 


17486 


.6225 


15709 


.56S2 


8788 


.3179 


869S 


.3146 


83S0 


.1212 


1S6S0 


.6756 


23000 


.8319 


7-207 


.2607 


730S 


.264 


7065 


. r 555 


7218 


.2611 


7788 


.2817 


7774 


.2SH 


7704 


.27S7 



SPECIFIC GRAVITIES. 



157 



Table— {Continued). 



Gravity. Inch _ 



«.•»&' 



METALS. 

Lead, cast 

U rolled 

Lithium 

Manganese 

Magnesium 

Mercury — 40° 

-f-32° 

" 60° 

" 212° 

Molybdenum 

Nickel 

" cast 

Osmium 

Palladium 

Platinum, hammered. . . . 

" native 

" rolled 

Potassium, 53° 

Red-lead 

Rhodium 

Ruthenium 

Selenium 

Silicium 

Silver, pure, cast 

u u hammered . 

Sodium 

Steel, plate3 

" soft 

" tempered and hard- 
ened 

u wire 

Strontium 

Tin, Cornish, hammered . 

u " pure 

Tellurium 

Thalium 

Titanium 

Tungsten 

Uranium 

Wolfram 

Zinc, cast 

" rolled 

woods (Dry). 

Alder 

Apple 

Ash | 

Bamboo 

Bay 

Beech 

u 

Birch '..[]....]['.'.]]'.'..'. 
Box, Brazilian 

" Dutch 

11 French 

Bullet-wood 

Butternut 

Campeachy 

Cedar 

" Indian 



11352 

113SS 

590 

8000 

1750 

15632 

13598 

13580 

133T0 

8600 

8800 

82T9 

10000 

11350 

2033T 

16000 

22069 

865 

8940 

10650 

8609 

4500 

104T4 

10511 

970 

7806 
7S33 

7S18 

7847 

2540 

7390 

7291 

6110 

11S50 

5300 

17000 

18330 

7119 

6S61 

7191 



800 
793 
845 
6 
400 
82 1 
852 
690 
567 

1031 
912 

1328 
923 
376 
913 
561 

1315 



.4106 

.4119 

.0213 

.2S94 

.0633 

.5661 

.4918 

.4912 

.4836 

.3111 

.3183 

.2994 

.3613 

.4105 

.7356 

.5787 

.7982 

.0313 

.324 

.3852 

.3111 

.1627 

.3788 
.3S02 
.0351 

.2823 
.2833 

.2828 

.2838 

.0918 

.2673 

.2637 

.221 

.4286 

.1917 

.6149 

.6629 

.2575 

.2482 

.26 

Cubic 
Foot. 
50 

1 49! 562 
52.812 
43.125 
25. 

51.375 
53.25 
43.125 
35.437 
64.437 
57. 
83. 
58. 
2R.5 
5^062 
35.062 
182.157 



o 



woods {Dry). 

Charcoal, pine 

u fresh burned. 

" oak 

u soft wood 

" triturated 

Cherry 

Chestnut, sweet 

Citron 

Cocoa 

Cork 

Cypress, Spanish 

Dog-wood 

Ebony, American 

" Indian 

Elder 

Elm J 

Filbert 

Fir (Norway Spruce) . . . 

Gum, blue 

u water 

Hackmatack 

Hazel 

Hawthorn 

Hemlock 

Hickory, pig-nut 

u shell-bark 

Holly 

Jasmine 

Juniper 

Lance- wood 

Larch < 

Lemon 

Lignum-vitse 

Lime 

Linden 

Locust 

Logwood 

Mahogany i 

" Honduras. 

" Spanish . . 

Maple 

' k bird's-eye.... 
Mastic 

Mulberry -j 

Oak, African 

M Canadian 

" Dantzic 

u English 

" green 

u heart, 60 years . 

4 ' live, green 

" u seasoned . . 

" white 

Orange 

Pear 

Persimmon 

Plum 



441 

380 
1573 
280 
1380 
715 
610 
726 
1040 
240 
644 
756 
1331 
1209 
695 
570 
671 
600 
512 
843 
1000 
592 
860 
910 
368 
792 
690 
760 
770 
566 
720 
544 
560 
703 
1333 
804 
604 
728 
913 
720 
1063 
560 
S52 
750 
576 
849 
561 
897 
823 
872 
759 
932 
1146 
1170 
1260 
1008 
860 
705 
661 
710 
7S5 



158 



SPECIFIC GRAVITIES. 



Ta."ble— (Continued). 

Gravlt y| Foot 



Gravit y Foot. 



woods (Dry). 



Pine, pitch . . . 

" red 

u white... 

u yellow . . 
Pomegranate . 

Poon 

Poplar 

" white . 

Quince 

Rose- wood 

Sassafras 

Satin-wood . . . 

Spruce 

Sycamore 

Tamarack. 



Teak (African oak) -j 



Walnut . 



" black . 



Willow 

Yew, Dutch 

U Spanish 

(Well Seasoned.*) 

Ash 

Beech 

Cherry 

Cypress 

Hickory, red 

Mahogany, St. Domingo 

Pine, white 

" yellow 

Poplar 

White Oak, upland 

M James River. 

STONES, EAETHS, ETC. 

Agate . 

Alabaster, white . 
yellow. . 
Alum . 

Amber 

Ambergris 

Asbestos, starry . 

Asphaltum 



Barytes, sulphate . 

Basalts 

Borax 

Brick 



fire 

work in cement . 



Carbon. 



,.{ 



660 
590 
554 
461 
1354 
530 
3S3 
529 
705 
T28 
4S2 
S85 
500 
623 



STONES, EARTHS, ETC. 



41.25 

G6.S75 
34.625 
2S.812 
84 625 
36.25 
23.937 
33.0J2 
44.062 
45.5 
30. 125 
55.312 
31.25 
38.937 



3S3 


20.937 


657 


41.062 ; 


745 


46.562 


671 


41.937 


500 


31.25 


48;» 


30.375 


5S5 


36.562 


78S 


49 25 


807 


50.437 


722 


45.125 


624 


39. 


606 


37.875 


441 


27.562 


838 


52.375 


720 


45. 


473 


29.562 


541 


33.812 


537 


36.6S7 


637 


42.937 


759 


42.437 


2590 




2730 


170.625 


2699 


10S.6S7 


1714 


107.125 


1073 


67.375 


866 





3073 


192.062 


905 


51562 


1650 


103.125 


4000 


250. 


4865 


304.002 


2740 


171.25 


2864 


179. 


1714 


107.125 


1900 


113 75 


1367 


85.437 


2201 


137.562 


1S0O 


112.50 


1600 


100. 


2000 


125. 


3500 


21S.75 



Cement, Portland. 
" Roman . . 

Chalk 

Chrysolite 

Clay 

" with gravel. . 

Coal, Anthracite.. 

" Borneo 

u Cannel 



Caking , 

Cherry 

Chili I 

Derbyshire. . 
Lancaster . . . 
Maryland . . . 

Newcastle 

Rive de Gier 

Scotch 



" Splint 

" Wales, mean 

Coke 

" Nat'l, Va 

Concrete, mean 

Copal 

Coral, red 

" white /#" 

Cornelian - btl0 . 1 

Diamond, Oriental 

" Brazilian 

Earth, t common eoil . . . 

u loose 

" moist sand 

u mould, fresh . . . 

u rammed 

" rough sand 

" with gravel 

Emery 

Flint, black 

u white 

Fluorine 

Glass, bottle 

u Crown. 



1300 
1560 
1520 
2784 
2782 
1930 
2480 
1436 
1640 
121)0 
1238 
1318 
1277 
1276 
1290 
1292 
1273 
1355 
1270 
1300 
1259 
1300 
1302 
1315 
1000 
746 
2000 
10-tK 



S1.25 
97.25 
95. 
174. 

120.62C 

155. 
89.75 

102.5 
80.625 
77.375 
82.375 
79.812 
79.75 
80.625 
80.75 
79.562 
84.6S7 
79.375 
S1.25 
7S.6S7 
81.25 
SI. 375 
82.187 
62.5 
46.64 

125. 
65.*" rt 



flint. 



" green 

u optical 

" white 

u window 

G:\rnet 

u black 

Granite, Egyptian red. . 

" Patapsco 

M Quincy 

" Scotch 

u Susquehanna. . 



its, see 


page 


*i6l3 T ~ 




3521 




8444 


— 


2194 


137.125 


1500 


93.75 


2050 


12S.125 


2050 


12S.125 


1600 


100. 


1920 


120. 


2020 


126.25 


4000 


250. 


2582 


161.375 


1594 


162.125 


1320 


82.5 


2732 


170.75 


2487 


155.437 


2933 


1S3.312 


3200 


196. 


2642 


165.125 


3450 


215.625 


2892 


1S0.75 


2642 


165.125 


41S3 


— 


3751 





2654 


105.875 


2640 


165. 


2652 


165.75 


2025 


164.062 


2704 


169. 



' 



* Or.lnance Manu.il, 1841. 

f Spec. gray, of the earth. is variously estimated nt from 5450 to 5600. 



SPECIFIC GRAVITIES. 



159 



Table— (Continued). 



STONES, EARTHS, ETC. 

Gravel, common 

Grindstone 

Gypsum, opaque 

Hone } white, razor 

Hornblende 

Iodine 

Jet 

Lime, hydraulic 

44 quick..'. 

Limestone, green 

44 white 

Magnesia, carbonate 

Marble, Adelaide . . 

44 African. . 

u Biscayan, black. 

44 Carara 

44 common 

u Egyptian 

44 French 

u Italian, white .. . 

44 Parian 

44 Vermont, white . 

Marl, mean 

Mica 

Mortar 1 

Millstone 

Mud...... 

44 " tempeic 

ened. . . •; 

Ov &•*»--«$<«»«« . . . 

Paving-stone 

Pearl, Oriental 

Peat | 

Phosphorus 

Plaster of Paris 

Plumbago 

Porphyry, red 

Porcelain, China 

Pumice-stone 

Quartz 

Rotten-stone 

Red-lead 

Resin 

Rock, crystal . . , 

Ruby 

Salt, common 

Saltpetre 

Sand, coarse 

44 common 

!* damp and loose. . . 

44 dried and loose . . . 

" dry 

44 mortar, Ft. Richm'd 

u u Brooklyn .. 

14 silicioua 

Sapphire 

Shale 

Slate I 



1749 
2143 
2168 
2876 
3540 
4940 
1300 
2745 

804 
3180 
3156 
2400 
2715 
2708 
2695 
2716 
26S6 
2668 
2649 
2708 
2838 
2650 
1750 
2800 
1384 
1750 
2484 
1630 
1900 
2114 
2092 
2416 
2650 

600 
1329 
1770 
1176 
2100 
2765 
2300 

915 
2660 
19S1 
8940 
1089 
2735 
42S3 
2130 
2090 
1800 
1670 
1392 
1560 
1420 
1659 
1716 
1701 
3994 
2600 
2900 
2672 



109.312 

133.937 

1355 

179.75 

221.25 



171.562 
50.25 
198.75 
197.25 
150. 
169. 6S7 
169.25 
168.437 
169.75 
167.875 
166.75 
165 562 
169.25 
177.375 
165.57 
109.375 
175. 
86.5 
109.375 
155.25 
101. S75 
118.75 

130.75 
151. 

37.5 

S3. 062 
IK '.625 

73.5 
131.25 

172.812 
143.75 

57.187 
166.25 
123.812 
55S.75 

68.062 
170.937 

133,125 
130.625 
112.5 
104.375 
87. 
97.5 
88.75 
103.66 
107.25 
106.33 

162.5 

181.25 

167. 



|e'« 



STONES, EARTHS, ETC. 



Slate, purple 

Smalt 

Stone, Bath Engl. 

u Blue Hill 

" Bluestone (basalt) 

" Breakneck.. N.Y. 

" Bristol Engl. 

u Caen, Normandy. 

u Common 

" Craigleth...Engl. 

u Kentish rag " 

41 Kip's Bay. . N.Y. 

44 Norfolk (Parlia- 
ment House). . 

" Portland ... Engl. 

44 Sandstone, mean. 

44 u Sydney 

" StatenIsVd,N.Y. 

" Sullivan Co. " 

Schorl 

Spar, calcareous 

44 Feld, blue 

" " green.. 

44 Fluor 

Stalactite 

Sulphur, native 

Talc, mean 

Tale, black 

Tile 

Topaz, Oriental 

Trap 

Turquoise 



MISCELLANEOUS. 

Asphaltum < 

Atmospheric Air 

Beeswax 

Butter 

Camphor 

Caoutchouc 

Egg 

Fat of Beef 

" Hogs 

44 Mutton 

Gamboge 

Gum Arabic 

Gunpowder, loose 

44 shaken 



solid , 



Gutta-percha . 

Horn 

Ice, at 32°.... 

Indigo 

Isinglass 

Ivory 

Lard 

Mastic 

Myrrh 

Opium 



2784 
2440 
1961 
2640 
2625 
2704 
2510 
2076 
2520 
2316 
2651 
2759 

2304 
236S 
2200 
2237 
2976 
2688 
3170 
2735 
2693 
2704 
3400 
2415 
2033 



174. 

152.5 

122.562 

165. 

164.C62 

169. 

156.875 

129.75 

157.5 

144.75 

165.687 

172. 

144. 

148. 

137.5 

139.812 

186. 

168. 

198.125 

170.937 

168.312 

169. 

215.5 

150.937 

127.062 



2500 156.25 



2900 
1815 
4011 
2720 
2750 



905 

1650 

001205 

965 

942 

988 

903 

1090 

923 

936 

923 

1222 

1452 

900 

1000 

1550 

1S00 

980 

16S9 

920 

1009 

1111 

1825 

947 

1*74 

1360 

1330 



1S1.25 
1 13.431 



170. 



."6.562 
103.125 

.07529 

60.312 

58.875 

61.75 

56.437 

57.6S7 

5S.5 

57.687 

90.75 

56.25 

62.5 

96.875 
112.5 

61.25 
105.562 

57.5 

63.062 

69.437 
114.062 

59. 1S7 

67.125 

85. 

S3.5 



160 



MISCELLANEOUS. 



Soap, Castile . 
Spermaceti . . . 

Starch 

Sugar 



.66 . 



Tallow . 
Wax. . . 



LIQUIDS. 



Acid, Acetic 

u Benzoic 

u Citric 

" Concentrated 

u Fluoric 

u Muriatic 

u Nitric 

" Phosphoric 

1-1 wt solid . . . 

tt Sulphuric 

Alcohol, pure, 60° 

" 95 per cent 

14 8.) " 

u 50 " 
" 40 " 
u 25 " 
" 10 " .... 
" 5 " 

" proof spirit,* 50 
per cent. ..60° 
" proof spirit, 50 
percent. ..80° 
Ammonia, 27.9 per cent. . 

Aquafortis, double 

r single....... 



SPECIFIC GRAVITIES. 



Ta"ble— (Continued). 



peoific 


Weight 
of a Cub. 




Foot. 


1071 


56.037 


943 


58.937 


950 


59-375 


16U6 


100.375 


1326 


82.875 


972 


60.25 


941 


5S.812 


964 


60.25 


970 


60.625 


1062 


66.375 


667 


41.6S7 


1034 


64.625 


1521 


1)5.062 


1500 


93.75 


1200 


75. 


1217 


76.062 


1558 


97.375 


2800 


175. 


1849 


115.5^2 


794 


49.622 


816 


51. 


803 


53.937 


934 


53.375 


951 


59.437 


970 


60.625 


986 


61.625 


992 


62. 


934 


5S.375 


S75 


54.6S7 


891 


55.687 


1300 


81.25 


1200 


75. 



Beer 

Bitumen, liquid 

Blood (human) 

Brandy, % or 5 of spirit. 

Cider 

Ether, acetic 

u muriatic 

'* sulphuric 

Honey 

Milk 

Oil, Anise-seed 

u Codfish 

u Cotton-seed 

u Linseed 

" Naphtha , 

" Olive 

" Palm 

" Petroleum 

" Rape 

u Sunflower 

" Turpentine 

u Whale 

Spirit, rectified 

Tar 

Vinegar 

Water, Dead Sea 

" 60° ... 

" 212° ... 

u distilled, 39°f . . 
41 Mediterranean.. 

u rain 

u sea 

Wine, Burgundy 

u Champagne 

u Madeira 

u Port 



Specific 7 ft j? h * 
Gravity. ofaCul >- 



1034 
848 

1054 
924 

1018 

866 

845 

715 

'1450 

1032 
986 
923 

940 

84S 

915 

969 

878 

914 

926 

870 

9-23 

824 

1015 

10S0 

1240 

919 

£57 

998 

1029 

1000 

1026 

992 

997 

1038 

997 




64.625 

53. 

65.S75 

57.75 

63.625 

54.125 

52.S12 

44.687 

H0.625 

64.5 

61.625 

57.687 

5S.75 

53. 

57.187 

60.562 

54875 

57.125 

57.875 

54.375 

57.6S7 

51.5 

63.43T 

67.5 

77.5 

62.449 

59.812 

62.379 

64.312 

62.5 

64.125 

62. 

64.375 

62.312 

62.312 



' 



Compression of the following fluids under a pressure of 15 lbs. per 
square inch : 

Alcohol 0000216 I Mercury 00000265 

Ether 00006158 Water 00004663 



Elastic Fluids. 

1J Cubic Foot of Atmospheric Air weighs 527.04 Troy Grains. 
Its assumed Gravity of 1 is the Unit for Elastic Fluids. 



Atmospheric air, 34°. 

Ammonia 

Azote 

Carbonic acid 

u oxyd 

Carbureted hydrogen. 

Chlorine 

Chloro-earbonic 

Cyanogen 



1. 


.539 


.976 


1.52 


.972 


.559 


2.47 


3.38') 


1.815 



Gas, coal 

Hydrogen 

Hydrochloric acid . 
Hydrocyanic u . 

Muriatic acid 

Nitrogen 

Nitric oxyd 

Nitrous acid 



.4 
.752 

.07 
1.278 

.942 
1.247 

.972 
1.094 
2.638 



* Specific gravity of proof spirit according to Ure's Table for Sykes's Hydrometer, 920 

f 1 cubic inch =".252.69 Troy firrairjB. t Equal to .075-29143 lbs. avoirdupois. 



WEIGHTS AND VOLUMES OF VARIOUS SUBSTANCES. 161 



Nitrous oxyd* 

Oxygen 

Phosphureted hydrogen 

Sulphureted u 

Sulphurous acid 

Steam,* 212° 

Smoke, of bituminous coal . 

u coke 

44 wood 

Vapor of alcohol 



Table— (Continued). 



bisulphuret of carbon. . 2.64 



1.T2T 

1.102 

1.77 

1.17 

2.21 
.4S83 
.102 
.105 
.09 

1.613 



Vapor of bromine 

'' chloric ether 

u ether 

" hydrochloric ether . . 

': iodine 

u nitric acid 

u spirits of turpentine. 

u sulphuric acid 

u ether 

sulphur 



water . 



5.1 
3.44 

2.5SG 
2.255 
8.675 
3.75 
4 763 
2.7 
2.586 
2 214 
.623 



Weights and. Vol mines of various Substances in 
Ordinary Use. 



Cubic Foot. 


Cub. Inch 


Lbs. 


Lbs. 


488.75 


.2S29 ! 


543.75 


.3147 \ 


513.6 


.297 


524.16 


.3 r !33 i 


547.25 


.3179 


543.625 


.3167 


450.437 


.2607 ! 


466.5 


.27 


4T9.5 


.2775 | 


481.5 


.2787 


4S6.75 


.2816 ! 


709.5 


.4K-6 


711.75 


.4119 I 


84S.7487 


.491174 


487.75 


.2823 


489.562 


.2833 


455. 6S7 


.2637 ! 


428.S12 


.24S2 1 


449.437 


.2601 




Cub. Feet 




in a Ton. 


52.812 


42.414 


51.375 


43.C01 


15. 


149.333 


35.062 


6X886 


88.125 


58.754 


49.5 


45.252 


43.125 


51.942 


83.312 


26.8S6 


57.062 


39.255 


35. 


64. 


66 437 


33.714 


54.5 


41.101 


5S.25 


38.455 


66.75 


33.558 


53.75 


41.674 


42.937 


52 169 


41.25 


54.303 


36.875 


60 745 


34.625 


64 693 


29.562 


75.773 



I Cubic 
Cubic Foot Feet in a 
Ton. 



METALS. 

" gun metal . 

41 sheets 

" wire 

Copper, cast 

44 plates 

Iron, cast 

44 gun metal 

44 heavy forging . . 

44 plates 

44 wrought bars. . . 

Lead, cast 

44 rolled 

Mercury, 60° 

Steel, plates 

44 soft 

Tin 

Zinc, cast 

44 rolled 

WOODS. 

Ash 

Bay 

Cork 

Cedar 

Chestnut 

Hickory, pig-nut 

44 shell-bark . . 

Lignum-vitse 

Logwood '. 

Mahogany, Hondur's < 

Oak, Canadian 

44 English 

44 live, seasoned. . . 

44 white, dry 

44 " upland... 
Pine, pitch 

44 red 

14 white 

44 well seasoned. . 



WOODS. 

Pine, yellow 

Spruce 

Walnut, black, dry. . 

Willow 

44 dry 

MISCELLANEOUS. 

Air 

Basalt, mean 

Brick, fire 

44 mean 

Coal, anthracite . . 1 

44 bitumin., mean 

44 Canuel 

44 Cumberland . . . 

44 Welsh, mean.. 

Coke 

Cotton, bale, mean . . 

44 4t pressed < 

Earth, clay 

44 common soil. . 

44 " gravel 

44 dry, sand 

44 loose 

44 moist, sand. . . 

44 mold 

44 mud 

44 with gravel . . 

Granite, Quincy 

44 Susqueh'na. 

Hay, bale 

44 pressed 

India rubber.. * 

44 vulcanized 

Limestone 

Marble, mean 

Mortar, dry, mean. . . 
Water, fresh 

44 salt 

Steam 



Lbs. 
33.812 
3125 
31.25 
36 562 
30.375 



.075291 
175. 
137.502 
102. 

89.75 
102.5 

SO. 

94.875 

84.687 

81.25 

62 5 

14.5 

20. 

25. 
120.025 
137.125 
109.312 
120. 

93.75 
128.125 
128.125 
101.S75 
126. f . 5 
165.75 
169. 
9.525 

25. 

56.437 

197.25 
167.S75 

97. 9S 

62.5 

64 126 
.036747 



66.24S 

71.68 

71.68 

61/265 

73.744 



12.8 

16.284 

•21.961 
24.958 
21.854 
28. 

23.609 
26.451 
27.569 
35.S4 

154.48 

114. 
89.6 
18.569 
16.: 35 
20.49 
18 667 
23.893 
17 482 
17.4S2 
21.9S7 
17.742 
13.514 
13.254 

23 517 

89 6 
39.69 

11.355 
13.343 
22.862 
35.84 
34,931 



* Weight of a cubic foot, 257.363 Troy grain* , 

o* 



162 



BALLOON.- -WEIGHT OF PATTERNS. 



Application, of tlie Tables. 

When the Weight of a Substance is required. Rule. — Ascertain tl 
volume of the substance in cubic feet ; multiply it by the unit in tl 
second column of tables, and divide the product by 10 ; the quotient 
will give the weight in pounds. 

When the Volume is given or ascertained in Inches. Rule. — Multiply 
it by the unit in the third column of the tables, and the product will 
be the weight in pounds. 
Example. — What is the weight of a cube of Italian marble, the sides being 3 feet ? 

33x270S = 7311C oz., which -f- 16 = 4569.75 lbs. 
Or of a sphere of cast iron 2 inches in diameter ? 

2 3 X.5236X.26 weight of a cubic inch =1.089 lbs. 

Comparative Weight of Timber in a Green and. 
Seasoned. State. 





Timber. 


Weight of £ 


Cub. Foot. 




Weight of 


a Cub. Ft. 




Green 


Seasoned 




Green. 


Seasoned 




;an Pine 


Lbs Oz 
44.12 
58. 3 
60. 


Lbs. Oz. 

30.11 

50. 
53. 6 


Cedar 


Lbs Oz. 
32. 
71.10 
48.12 


Lbs. Oz. 
28 4 


Ash . 


English Oak 

Kiga Fir 


43. S 


Beech 


35. S 



To Compute tlie Capacity of a ISalloon. 

Rule. — From the specific gravity of the air in grains per cubic foot 
subtract that of the gas with which it is inflated ; multiply the remain- 
der by the volume of the balloon in cubic feet ; divide the product by 
7000. and from the quotient subtract the weight of the balloon and its 
attachments. 

Example.— The diameter of a balloon is 26.6 feet, its weight is 100 lbs., and the 
specific gravity of the gas with which it is inflated is .06 (air being assumed at 1); 
what i- it> capacity? 

^-.04- S 1.62x2 a .«^.^0_ 100 = 495.42x9854.726_ 100 = 59t461 ^ 



7000 



T0U0 



To 



-j 



Compute the Diametei? of* a Balloon, tlie 
Weight to "be raised, being given. 

By inversion of the preceding rule, 

[Wx7000-f-*=? ; ■'.. . ;•";.;: 

- = a, s and s representing the weight of air and gas 



.5236 

in grains per cubic foot, and d the diameter of the balloon in feet, 
Example.— Given the elements in the preceding case. 



Then 



V 



507.4 6 -f 100 X 7000 -i- 527.04 — -51.62 
.5236 



= yiSS21.03 = 26.6/eef. 



To Compute the ^vVeight of Cast Metal by the 
T^eight of the Fattern. 

When the Pattern is of White Pine. 
Rule. — Multiply the weight of the pattern in pounds by the follow- 
ing multiplier, and the product will give the weight of the casting: 
Iron, 14; Brass, 15; Lead, 22 ; Tin, 14; Zinc, 13.5. 



GEOMETRY. 163 

When there are Circular Cores or Prints. — Multiply the square of the 
diameter of the core or print by its length in inches, the product by 
.0175, and the result is the weight of the pattern of the core or print 
to be deducted from the weight of the pattern. 

It is customary, in the making of patterns for castings, to allow for 
shrinkage per lineal foot of pattern : 

Iron and Lead |th of an inch, Brass and Zinc ^ths, and Tin xa tn * 



GEOMETRY. 

Definitions. — A Point has position, but not magnitude. 

A Line is length without breadth, and is either Right, Curved, or Mixed. 

A Right Line is the shortest distance between two points. 

A Carved Line is one that continually changes its direction. 

A Mixed Line is composed of a right and a curved line. 

A Superficies has length and breadth only, and is plane or curved. 

A Solid hsis length, breadth, and thickness. 

An Angle is the opening of two lines having different directions, and is 
either Right, Acute, or Obtuse. 

A Right Angle is made bj r a line perpendicular to another falling upon it. 

An Acute Angle is less than a right angle. 

An Obtuse Angle is greater than a right angle. 

A Triangle is' a figure of three sides. ^ 

An Equilateral Triangle has all its sides equal. 

An Isosceles Triangle has two of its sides equal. 

A Scalene Triangle has all its sides unequal. 

A Right-angled Triangle has one right angle. 

An Obtuse-angled Triangle has one obtuse angle. 

An Acute-angled Triangle has all its angles acute. 

A Quadrangle or Quadrilateral \s a figure of four sides, and has the fol- 
lowing particular designations, viz. : 

A Parallelogram, having its opposite sides parallel. 

A Square, having length and breadth equal. 

A Rectangle, a parallelogram having a right angle. 

A Rhombus or Lozenge, having equal sides, but its angles not right an- 
gles. 

A Rhomboid, a parallelogram, its angles not being right angles. 

A Trapezium, having unequal sides. 

A Trapezoid, having only one pair of opposite sides parallel. 

Note. — A Triawjle is sometimes called a Trigon, and a Square a Tetragon. 

Polygons are plane figures having more than four sides, and are either 
Regular or Irregular, according as their sides and angles are equal or un- 
equal, and they are named from the number of their sides or angles. 
Thus : 



A Pentagon has five sides. 
A Hexagon u six " 
A Heptagon " seven " 
An Octagon u eight " 



A Nonagon has nine sides. 
A Decagon " ten " 
An Undecagon " eleven " 
A Dodecagon M twelve " 



A Circle is a plane figure bounded by a curved line, called the Cir- 
cumference or Periphery. 

A Diameter is a right line passing through the centre of a circle or 
sphere, and terminated at each end by the periphery or surface. 

An Arc is an}' part of the circumference of a circle. 

A Chord is a right line joining the extremities of an arc. 

A Segment of a circle is any part bounded by an arc and its chord. 



164 



GEOMETRY. 



The Radius of a circle is a line drawn from the centre to the circumference. 

A Sector is an}' part of a circle bounded by an arc and its two radii. 

A Semicircle is half a circle. 

A Quadrant is a quarter of a circle. 

A Zone is a part of a circle included between* two parallel chords. 

A Lune is the space between the intersecting arcs of two eccentric circles. 

A Gnomon is the space included between the lines forming two similar 
parallelograms, of which the smaller is inscribed within the larger, so as 
to have one angle in each common to both. 

A Secant is the line running from the centre of the circle to the extrem- 
ity of the tangent of the arc. 

A Cosecant is the secant of the complement of an arc, or the line running 
from the centre of the circle to the extremity of the cotangent of the arc. 

A Sine of an arc is a line running from one extremity of an arc perpen- 
dicular to a diameter passing through the other extremity, and the sine 
of an angle is the sine of the arc that measures that angle. 

The Versed Sine of an arc or angle is the part of the diameter intercept- 
ed between the sine and the arc. 

The Cosine of an arc or angle is the part of the diameter intercepted be- 
tween the sine and the centre. 

The Coversed Sine of an arc or angle is the part of the secondary ra- 
dius intercepted between the cosine and the circumference 

A Tangent is a right line that touches a circle without cutting it. 

A Cotangent is the tangent of the complement of the arc. 

The Circumference of every circle is supposed to be divided into 360 
equal parts termed Degrees ; each degree into 60 Minutes, and each min- 
ute into 60 Seconds, and so on. 

The Complement of an angle is what remains after subtracting the angle 
from 90 degrees. 

The Supplement of an angle is what remains after subtracting the angle 
from 180 degrees. 

To exemplify these definitions, let A c b, in the following diagram, be an 
assumed arc of a circle described with the radius B A : 

A c b, an Arc of the circle A C E D. 

A h, the Chord of that arc. 

B A, an Initial radius. 

B C, a Secondary radius. 

e D d, a Segment of the circle. 

A B b, a Sector. 

A D E, a Semicircle. 

C B E, a Quadrant. 

A e d E, a Zone. 

n o ?i, a Lune. 

B r/, the Secant of the arc Ac b; written 
Sec. 

b k, the Sine of the arc A c b; written Sin. 

A fc, the Versed Sine of the arc Acb ; writ- 
ten Versin. 

B k or m b, the Cosine of the arc Acb. 

A p, the Tangent of « " 

C B b, the Complement, and b B E, tht 
Supplement of the arc A c b. 

C s, the Cotangent of the arc ; written Cot, 

B .*, the Cosecant of the arc ; written Cosec 
m C, the Coversed sine of the arc, or, by convention, of the angle A B b ; written 
Coversin. 

The Vertex of a figure is its top or upper point. In Conic Sections it is 
the point through which the generating line of the conical surface always 
passes. 

The Altitude, or height of a figure, is a perpendicular let fall from its 
vertex to the opposite side, called the base. 







GEOMETRY. 



165 



The Measure of an angle is an arc of a circle contained between the two 
lines that form the angle, and is estimated by the number of degrees in 
the arc. 

A Segment is a part cut off by a plane, parallel to the base. 

A Frustrum is the part remaining after the segment is cut off. 

The Perimeter of a figure is the sum of all its sides. 

A Problem is something proposed to be done. 

A Postulate is something required. 

A Theorem is something proposed to be demonstrated. 

A Lemma is something premised, to render what follows more easy. 

A Corollary is a truth consequent upon a preceding demonstration. 

A Scholium is a remark upon something going before it. 



Lengths of the following Elements, Radius = 1. 



Sine 

Cosine 

Versed Sine . . 
Coversed Sine 



Secant 1.414214 2 



Angle 45o, 



.707107 
.707107 
.292893 

.292893 



Angle 60°. 



.866025 
.5 
.5 
.133975 



Cosecant 

Tangent 

Cotangent.. . 

Chord 

Arc 



Angle 45°. 



1.414214 

1. 

1. 

.765366 



.785398 1.0472 



Angle C(K'. 



1.1547 
1.73205 
.577349 
1. 



SCALES. 

To Construct a Diagonal Scale -upon any Line, as A. 33- 

Fig. 1. 

(U __ _____________ 

"""" '& 

I 



A ab 




E G 










c 


) 


kl\\\\ 


1 1 
















\ 


1 11 111 


















\ 


111 1 


















\ 


111 










i 








\ 


DIXlL 


















\ 


Hi "11 1 


















\ 


mill 


















\ 


Ijtlll 


















\ 


nil 


















\ 


cl "M 


\\ 
















\ 



123 



D 



Divide the line into as many divisions as there are hundreds of feet, spaces of ten 
feet, feet, or inches required. 

Draw perpendiculars from every division to a parallel line, C D. Divide them 
and one of the divisions, A E, G F, into spaces of ten if for feet and hundredths, and 
twelve if for inches ; draw the lines A 1, a 2, 6 3, etc., and they will complete the 
scale. 

Thus : The line A B representing ten feet; A to E, E to G, etc., will measure one 
foot ; A to a, C to 1, 1 to 2, etc., will measure l-10th of a foot. The several lines A 1, 
a 2, etc., will measure upon the lines kk,ll, etc., 1 -100th of a foot; and op will 
measure upon k k,ll, etc., divisions of l-10th of a foot. 



POLYGONS. 
To Circumscribe a Pentagon about a Griven Circle — Fig. 2. 



Divide the circumference of a circle into five points, 
defining them s r v m n. 

From the centre o, draw o r, o v, etc. 
Through n th, etc., draw A B, B C, etc., perpendicu- 
lar to o w, o m, etc. , and complete the figure. 




Note.— Any other polygon may be made in a similar 
manner by drawing tangents to the points, first defining 
them in the circle. 



166 



GEOMETRY. 



To Inscribe a Pentagon in a Gfiven Circle—Fig. 3. 
(3.) A 



Draw the diameters, A p and m n, at right angles to 

each other: bisect o n in r, and with r A describe As; 

7i from A with A s describe s B. 

Join A B, and the distance is equal to one side of the 
/ pentagon required. 




To Describe a Pentagon upon a Griven Line— Fig. 4. 
(4.) C 



Draw B m perpendicular to A B, and equal to one half 
of it ; extend A m until m n is equal to B m. 




From A and B, with the radius B n, describe arcs cut- 
ting each other in o; then from o, with the radius* o B, 
describe the circle AGB, and the line A B is equal to 
one side of the pentagon upon the circle described. 



To Describe an Octagon upon a Griven Line A. 33— Fig. G. 



From the points A B erect indefinite perpendiculars 
A/,Be; produce A B to m and ?j, and bisect the angles 
m Ao and n B p with A H and B C. 

Make A H and B C equal to A B, and draw H6,CD, 
parallel to A/, and equal to A B. 

From G and D, as centres with a radius equal to A B, 
describe arcs cutting A/, B e, in / and e. Join G/, /«/, 
and e D, and the figure is complete. 




SQUARES. 
To Describe a Square about a Griven Circle 



-Fig. 6. 



K*-) 



/"" 3 \ x \ 



B Aj< 




Draw the line A B through 
the centre of the circle. 

Take any radius, as A«; 
describe the arcs A e e, B e e; 
join e f, continuing the line, 
...y b to C D. 

Describe B r and D r; draw 
and extend B r and Dr, and 
the sides A and C, drawn par- 
allel to them, will describe the 
square. 



To Inscribe a Square in a Griven Circle — Fig. 7. 

Draw the line A B through the centre of the circle; take any radii "; \e, and 
describe the arcs A e e, B e e; join e r, continuing the line to C and D ; *<*" froi».A J> 
etc., and the square is inscribed. 






GEOMETRY. 



167 



To Describe an Octagon, in a Gfrven Square — Eig. 8, 

Let A B C D be the given square. 

Describe A o, B o, etc. ; join the intersections r r r r, etc., and the figure formed 
is the octagon required. 

CIRCLES. 

To Describe a Circle tliat shall pass through any three 
Given IPoints, as J± IB C— Fig. O. 





Upon the points A and B, with any opening of a dividers, describe two arcs to 
intersect each other, as at e e. 

On the points A C describe two more to intersect each other in the points c c. 

Draw the lines e e and c c, and where these two lines intersect o, place one leg 
of the dividers, and extend the other to A, B, or C, and it will describe a circle through 
the three given points. 

To Inscribe an Equilateral Triangle in a Given Circle- 
Fig. 10. 

From any point A, with A o equal to the radius of the circle, describe e o e; from 
e and e describe er, e r; join A ?\ r r, and r A, and the triangle is formed. 

Note — All figures of 10 or 20 sides are readily determined from the side of a pen- 
tagon, being" halved or quartered ; and in like manner, all figures of 6, 12, or 24 sides 
are readily determined from the radius of a circle, being equal to the side of a hex* 
agon. 



ELLIPSES. 

To Describe an Ellipse to any- Length and. Breadth 
given — Eig. 11. 



(11.) 




Let the longest diameter given be equal to a 5, and the shortest to c o. 
Bisect f i qually at right angles in s. 

Ma*V — tal to co; divide rft into three equal parts, set off two of thope parts 

s ""-- d from s to u; then, with the distance ui\ make the two equilateral tit- 



168 



GEOMETRY. 



angles ucr and nor; these angles are the centres, and the sides being continuec 
are the lines of direction for the several arcs of the ellipse acbd. 

Note — When it is required to work an Architrave, etc., of this form : by the use 
of the four centres w, c, r, o, and the lines of direction h c, of, o i, and g c, describe 
another line around the former, and at any distance required, aahifg. 

To Describe an Ellipse to any Length and. Breadth re- 
quired, another Way — Fig. IS. 

Let the longest diameter be C D, and the shortest E F. 

Take the distance C o or o D, and with it, from the points E and F, describe the 
arcs h and / upon the diameter C D. 

Insert pins at h and at /, and put a striDg around th-- m of such a length that it 
will just reach to E or F. 

Introduce a pencil, and bearing upon the string, cany it around the centre o, and 
it will deseribe the ellipse required.* 

To Describe an Ellipse of* any G-iven Length, -without re- 
gard to Breadth — Eig. 13. 

(13.) 



Let A B be the given le'ngth. 

Divide it into three equal parts, as A s i B. 

B With the radius A «, describe the circle A F o i 
n C ; and from ?', B D n s o E. 

With n F and o C draw F E and C D, and the 
ellipse is described. 



To Ascertain the Centre and two Diameters of an Ellipse 
—Eig. 14. 

Let A B c u be the ellipse. 





Draw at pleasure two lines, gg^Tn o, par- 
allel to each other ; bisect them in the points 
h w, and draw the line P E ; bisect it in k, 
and upon Ar, as a centre, describe a circle a 
pleasure, as / I r, cutting the figure in th 
points / 1 . 

Draw the right line/ I; bisect it in i, and 
through the points i k draw the greatest di- 
ameter A B, and through the centre, k, draw 
the least diameter c it, parallel to the line 



To Draw a Spiral Line about a G-iven Eoint — Eig. IS. 



ts 

= 



(15.) 




Let c be the centre. 

Draw A h, and divide it into twice the number r . 
parts that there are to be revolutions of the line. U^l 
c describe k ?', gf,Ah, and upon i describe kf,gf y etc. 



It is ft property of the ellipse that the sum of two lines drawn from the foci to meet in any 
point in the curve is equal to the transverse diameter, and from this the correctness of the above 
construction is evident. 



GEOMETRY. 



169 



POLYGONS. 

To Describe a Regular Polygon of any required. Number 
of Sides — Fig. 16. 

From the point o, with the distance o B, describe 
the semicircle B b A, which divide into as many 
equal parts, A a, a &, b c, etc., as the polygon is to 
have sides. 

Let a hexagon be required. 

From o to the second point of six divisions draw 
o b ; and through the other points, c, d, and e, draw 
o C, o D, etc. 

Apply the distance o B, from B to E, from E to D, 
from D to C, etc. Join these points, as b C, C D, 
etc., and the figure is described. 




C17.) 



ARCS. 
To Describe a G-otbic A.rc— - Fig. 17. 
O. ^o (18.) 





Take the line A B. 

At the points A and B draw the arcs B a and A <*, and it will describe the arc 
required. 

To Describe art Elliptic Arc on tbe Conjugate Diameter — 

Fig. 18. 

Draw the diameter A B, and in the middle, at &, erect the perpendicular k o, equal 
to the height of the arc. 

Divide the perpendicular k o into two equal parts at e ; continue the line A B on 
both sides at pleasure, and from the point A;, with the distance k o, cut A B in c and 
d ; through c e, d e, draw e e f and d e g at pleasure ; d and c are centres for the arcs 
A g and B /, and e the centre for the arc g o /, which will form the arc required. 

To Describe an Elliptic Ajtc, tbe Cbord and Heigbt being 
given — Fig. 19. 
(19.) 

Bisect A B at c ; erect the perpendicular A q, 
and draw the*linc q D equal and parallel to A c. 

Bisect A c and A q in r and n; make c I equal to 
c I), and draw the line I r q ; draw also the line 
n s D ; bisect s D with a line at right angles, and 
cutting the line cDino; draw the line o q ; make 
c p equal to c k, and draw the line o p i. 

Then from o as a centre, with the radius o D, 
describe the arc sD i; and from k and p as cen- 
tres, with the radius A A*, describe the arcs A s and 
B ?, which will complete the arc required. 




170 GEOMETRY. 



To Describe a GJ-otliic Arc — Figs. SO and. Ql. 

(20.\ i Divide the line A B (Fig. 20) into three equal parte, ef; 

from the points A and B let fall the perpendiculars A c and 
B ef, equal in length to two of the divisions of the line A B , 
draw the lines c h and d g ; from 
the points e/, with the length of 
g / B, describe the arcs A g and 
B /*, and from the points c and 
d describe the arcs g i and i h, 
and it will complete the arc re- 
quired. 







(21.) 



J 



c«— 



-*d 



Or, divide the line A B (Fig. 
21) into three equal parts at a 
and fr and on the points A, a, b, and B, with the distance 
of two divisions, make four arcs intersecting at c d. 

Through the points c, tf, and the divisions a, &, draw 
the lines cf and d e, and on the points a and b describe 
the arcs A e and B/, and on the points c d the arcs f g 
and e ef, and it will complete the arc required. 



To draw from or to the Circumference of a Circle, Lines 
leading to the Centre, When the Centre is inaccessible 
—Fig. S3, 




• 22.1 



Divide the whole or any given por- 
tion of the circumference into the de- 
sired number of parts : then, with any 
radius less than the distance of two di- 
visions, describe arcs cutting each oth- 
er, as Ar,br,cr,dr, etc. ; draw the 
lines b r, c r, etc., and they will lead to 
the centre, as required. 

To draw the end lines, as Ar, F r. 
From b describe the arc r, and with the radius b 1, from A or F as centres, cut the 
arcs A 1, etc., at r, or r, and the lines A r, F r, will lead to the centre, as required. 

To -Descrihe an Arc, or Segment of a Circle, of a large 
Radius — Fig. 23. 

Construct of suitable 
material a triangle, as 
A b C ; make A &, b C, 
each equal in length to 
the chord of the arc d c, 
aud in height twice that 
of the arc bo. At each end 
of the chord e7, c, insert a pin, and at b attach a tracer (as a pencil): move the trian- 
gle against the pins as guides, and the tracer will describe the arc required. 

Or, draw the chord AcB (Fig. 24); also draw the line h D i parallel with the 
chord, and at a distance equal to the height of the segment ; b'sect the chord in c, 

and erect the perpendicular 
c D : join A D, D B ; draw 
i A h perpendicular to A D, 
' and B i perpendicular to B D; 
erect also the perpendicular* 
Aw, B n ; divide A B and 
h i into any number of equal 
parts; draw the lines 11, 
2 2, and divide the lines A 7?, 
B ??,, each into half the num- 
ber of equal parts in A B ; 
draw lines to D from* ench division in the lines A ??, B n, and at the points of inter- 
section with the former lines describe a curve, which will be the arc or segment re 
quired. 





GEOMETRY. 



171 



To Ascertain, the Length, of* an Elliptic Curve wLxich is 
less than half of the entire Figure — Fig. 25. 



(2R; 




Let the curve of which the length is re- 
quired be A b C. 

Extend the versed sine b d to meet the 
centre of the curve in e. 

Draw the line e C, and from <% with the 
p distance e 6, describe b h; bisect h C in ?', 
A * \ and from ?, with the radius e ?*, describe k ?', 
and it is equal to half the arc A b C. 



To Ascertain the Length when the Curve is greater than 
half the entire Figure. 

Ascertain by the above problem the curve of the less portion of the figure ; sub- 
tract it from the circumference of the ellipse, and the remainder will be the length 
of the curve required. 



To Ascertain the Distance between two inaccessible Ob- 
jects, as A, IB — Fig. 26. 

(20.) 
A B 



From any point C draw a line C c, and bisect it in 
o; take any point e in the prolongation of A c, and 
draw the line e r, making o e equal to o r. 

In like manner, take any point s in the prolonga- 
tion of B C, and make o f equal to o s. Produce A o 
and C r till they meet in <?, and also B o and c r till 
they meet in b ; then a b equal A B, or the distance 
between the objects as required. 




ii^L 



To Ascertain the Distance of an inaccessible Object on a 
Level Plane — Fig. 37. 



Let it be required to ascertain the distance between 
A and B, A beiug inaccessible. 

Produce the line in the direction of A B to any point, 
as o; draw the line o d at any angle to the line A B; 
bisect the line o d, through which draw the line P. /, 
making cb equal to Be; draw the line dba; also 
through c, in the direction c A, draw the line A c or, in- 
tersecting the line dba; then b a equal A B, the dis< 
tanca required. 



(87) 


i. 


A 


=• 




===_ ig. 


C^— ^=j§§§ 


^^=r^E. 


d i\ I 




B 



bV 



a*' 



172 



AREAS OF CIRCLES. 



Areas of Circles, from -^ to 150 


—[Advancing by an Eighth.] 


Diam. 


Area. Di 


am. 


Area. 


Diam. 


Area. 


Diam. 


Area. 


«*£ 


.000192 5 




19.635 


12. 


113.098 


19. 


283.529 


64- 

1 


.000767 


"X 


20.629 


■X 


115.466 


-M 


287.272 


32 


■X 


21.6476 


•X 


117.859 


•X 


291.04 


16 


.003068 


.% 


22.6907 


.% 


120.277 


•X 


294.832 


i 


.012272 


•X 


23.7583 


M 


122.719 


X 


298.648 


•% 


24.8505 


•% 


125.185 


•X 


302.489 


3 
1.6 


.027612 


•% 


25.9673 


•X 


127.677 


•M 


306.355 


1 


.049087 


•X 


27.1086 


•X 


130.192 


•X 


310.245 


^~„~^ 6 




28.2744 


13. 


132.733 


20. 


314.16 


A 


.076699 


■X 


29.4648 


•X 


135.297 


•X 


318.099 




.110447 


-X 


30.6797 


-M 


137.887 


•X 


322.063 


.15033 


-X 


31.9191 
33.1831 


•X 


140.501 
143.139 


•X 


326.051 
330.064 


i 


.19635 


-% 


34.4717 


•X 


145.802 


•X 


334.102 


9 


. 248505 


.% 


35.7848 


'X 


148.49 


.% 


338.164 


T6 

f 


.306796 7 


% 


37.1224 

38.4846 


•X 

14. 


151.202 
153.938 


•X 

21. 


342.25 
346.361 


1* 
1 


.371224 


■X 


39.8713 


•X 


156.7 


-X 


350.497 


.441787 


•X 


41.2826 
42.7184 




159.485 
162.296 


•X 


354.657 

358.842 


i 


.518487 


X 


44.1787 


-A 


165.13 


'X 


363.051 


.601322 


.% 


45.6636 


•X 


167.99 


•% 


367.285 




% 


47.1731 


•X 


170.874 


•% 


371.543 


if 


.690292 


X 


48.7071 


•X 


173.782 


•X 


375.826 


1. 


.7854 8 




50.2656 


15. 


176.715 


22/ 


380.134 


■% 


.99402 


'x 


51.8487 


•X 


179.673 


•x 


384.466 


-X 


1.2272 


k 


53.4563 


-& 


182.655 


•x 


388.822 


•K 


1.4849 


.% 


55.0884 


•X 


185.661 


•% 


393.203 


•3* 


1.7671 


X 


56.7451 


-A 


188.692 


•x 


397.609 


.% 


2.0739 


% 


58.4264 


•X 


191.748 


-% 


402.038 


•M 


2.4053 


% 


60.1322 


.% 


194.828 i 


-K 


406.494 


.% 


2.7612 


X 


61.8625 


•X 


197.933 ' 


•X 


410.973 


2. 


3.1416 9 




63.6174 


16. 


201.062: 


23/ 


415.477 


•X 


3.5466 


X 


65.3968 


•X 


204.216! 


•y 


420.004 


.& 


3.9761 


X 


67.2008 


•X 


207.395' 


-x 


424.558 


•% 


4.4301 


% 


69.0293 


■ X 


210.598 


•X 


429.135 


•X 


4.9087 


X 


70.8823 


•A 


213.825 i 


•x 


433.737 


•% 


5.4119 


% 


72.7599 


-% 


217.077 


•% 


438.364 


•M 


5.9396 


% 


74.6621 


-% 


220.354 


-% 


443.015 


•% 


6.4918 


X 


76.5888 


•X 


223.655 


•x 


447.69 


3. 


7.0686 10 




78.54 


17. 


226.981 


24. 


452.39 


•X 


7.6699 


X 


80.5158 


x 


230.331 


•y 


457.115 


-X 


8.2958 


X 


82.5161 


•X 


233.706 


-x 


461.864 


•% 


8.9462 


% 


84.5409 


>'% 


237.105 


.% 


466.638 


-X 


9.6211 


A 


86.5903 


•A 


240.529 


-X 


471.436 


-% 


10.3206 


% 


88.6643 


•X 


243.977 j 


•x 


476.259 


•% 


11.0447 


% 


90.7628 


-% 


247.45 


-% 


481.107 


.% 


11.7933 


X 


92.8858 


•X 


250.948 


•% 


485.979 


4 


12.5664 11. 




95.0334 


18. 


254.47 


25. 


490.875 


-X 


13.3641 


y 


97.2055 


•X 


258.016 


•x 


495.796 


•x 


14.1863 


y 


99.4022 


•x 


261.587 


>x 


500.742 


•K 


15.033 


% 


101.6234 


-% 


265.183 


-% 


505.712 


•X 


15.9043 


X 


103.8691 


-A 


268.803 


•y 


510.706 


.% 


16.8002 


% 


106.1394 


•X 


272.448 


-% 


515.726 


•k 


17.7206 


% 


108.4343 


.% 


276.117 


.% 


520.769 


•x 


18.6655 


x 


110.7537 


•X 


279.811 


•X 


525.838 



AREAS OF CIRCLES. 



Area. 



Table— (Continued). 



L 



530.93 

536.048 

541.19 

546.356 

551.547 

556.763 

562.003 

567.267 

572.557 

577.87 

583.209 

588.571 

593.959 

599.371 

604.807 

610.268 

615.754 

621.264 

626.798 

632.357 

637.941 

643.549 

649.182 

654.84 

660.521 

666.228 

671.959 

677.714 

683.494 

689.299 

695.128 

700.982 

706.86 

712.763* 

718.69 

724.642 

730.618 

736.619 

742.645 

748.695 

754.769 

760.869 

766.992 

773.14 

779.313 

785.51 

791.732 

797.979 

804.25 

810.545 

816.865 

823.21 

829.579 

835.972 

842.391 

848.833 



33. 



•K 



34 



35. 



36. 



% 

% 

% 
-% 

% 

% 



37. 



7k 



38 



39 



855.301 

861.792 

868.309 

874.85 

881.415 

888.005 

894.62 

901.259 

907.922 

9X4.611 

921.323 

928.061 

934.822 

941.609 

948.42 

955.255 

962.115 

969. 

975.909 

982.842 

989.8 

996.783 

1003.79 

1010.822 

1017.878 

1024.96 

1032.065 

1039.195 

1046.349 

1053.528 

1060.732 

1067.96 

1075.213 

1082.49 

1089.792 

1097.118 

1104.469 

1111.844 

1119.244 

1126.669 

1134.118 

1141.591 

1149.089 

1156.612 

1164.159 

1171.731 

1179.327 

1186.948 

1194.593 

1202.263 

1209.958 

1217.677 

1225.42 

1233.188 

1240.981 

1248.798 

p* 



40 



41 



42 



43 



44 



45 



46 



% 



u 



% 



Area. 



1256.64 

1264.51 

1272.4 

1280.31 

1288.25 

1296.22 

1304.21 

1312.22 

1320.26 

1328.32 

1336.41 

1344.52 

1352.66 

1360.82 

1369. 

1377.21 

1385.45 

1393.7 

1401.99 

1410.3 

1418.63 

1426.99 

1435.37 

1443.77 

1452.2 

1460.66 

1469.14 

1477.64 

1486.17 

1494.73 

1503.3 

1511.91 

1520.53 

1529.19 

1537.86 

1546.56 

1555.29 

1564.04 

1572.81 

1581.61 

1590.43 

1599.28 

1608.16 

1617.05 

1625.97 

1634.92 

1643.89 

1652.89 

1661.91 

1670.95 

1680.02 

1689.11 

1698.23 

1707.37 

1716.54 

1725.73 



47. 






.% 



48. 



.y± 



49. 



50. 



-« 



51. 



•% 



52. 






53. 



:g 

H 

% 

■% 

■% 



174 



AREAS OF CIRCLE*. 



Ta"ble— (Continued). 



Piam 


Area. Tia 


m | 


Area 


Diam. 1 


Area n Dia 


m 


54. 


2290.23 61. 




2922.47 


68. 


3631.69 75. 




•% 


2300.84 


X 


2934.46 


•X 


3645.05 


X 


-}£ 


2311.48 


% 


2946.48 


-y 


3658.44 


y 


•% 


2322.15 


% 


2958.52 


•X 


3671.86 


% 


• X A 


2332.83 


X 


2970.58 


• X 


3685.29 


y 


•% 


2343.55 


% 


2982.67 


'% 


3698.76 


% 


•% 


2354.29 


% 


2994.78 


•X 


3712.24 


x 


•% 


2365.05 


X 


3006.92 


n 


3725.75 


x 


55. 


2375.83 62. 




3019.08 


69. 


3739.29 76 




•X 


2386.65 


% 


3031.26 


-X 


3752.85 


Vs 


•K 


2397.48 


y 


3043.47 


X 


3766.43 


y 


•% 


2408.34 


% 


3055.71 


• X 


3780.04 


% 


M 


2419.23 


Vi 


3067.97 


•X 


3793.68 


M 


-X 


2430.14 


% 


3080.25 


•X 


3807.34 


% 


.% 


2441.07 


% 


3092.56 


.« 


3821.02 


X 


•% 


2452.03 


X 


3104.89 


•X 


3834.73 


X 


56. 


2463.01 63. 




3117.25 


70. 


3848.46 77. 




M 


2474.02 


% 


3129.64 


•X 


3862.22 


x 


& 


2485.05 


X 


3142.04 


• X 


3876. 


x 


>% 


2496.11 


% 


3154.47 


-X 


3889.8 


x 


•X 


2507.19 


X 


3166.93 


•X 


3903.63 


y 


■% 


2518.3 


% 


3179.41 


►X 


3917.49 


% 


.% 


2529.43 


% 


3191.91 


«x 


3931.37 


X 


•% 


2540.58 


X 


3204.44 


•X 


3945.27 


x 


57. 


2551.76 64 




3217. 


71. 


3959.2 78 




■X 


2562.97 


x 


3229.58 


•X 


3973.15 


Vs 


ft 


2574.2 


H 


3242.18 


•X 


3987.13 


y 


•% 


2585.45 


% 


3254.81 


•X 


4001.13 


% 


>y 


2596.73 


X 


3267.46 


•X 


4015.16 


y 


-% 


2608.03 


X 


3280.14 


-X 


4029.21 


% 


•% 


2619.36 


% 


3292.84 


•X 


4043.29 


x 


-X 


2630.71 


X 


3305.56 


•X 


4057.39 


% 


58. 


2642.09 65 




3318.31 


72. 


4071.51 79. 




>X 


2653.49 


Vs 


3331.09 


•X 


4085.66 


y 


•X 


2664.91 


X 


3343.89 


■X 


4099.84 


X 


.% 


2676.36 


% 


3356.71 


•X 


4114.04 


X 


•X 


2687.84 


X 


3369.56 


• X 


4128.26 


y 


-X 


2699.33 


% 


3382.44 


•X 


4142.51 


% 


•X 


2710.86 


% 


3395.33 


• X 


4156.78 


x 


■X 


2722.41 


X 


3408.26 


•X 


4171 .08 


% 


59. 


2733.98 66 




3421.2 


73. 


4185.4 80 




-X 


2745.57 


Vs 


3434.17 


•X 


4199.74 


y 


•X 


2757.2 


H 


3447.17 


<x 


4214.11 


y 


J% 


2768.84 


% 


3460.19 


•X 


4228.51 


•X 


•X 


2780.51 


X 


3473.24 


<X 


4242.93 


■X 


-X 


2792.21 


•X 


3486.3 


-% 


4257.37 


• X 


-X 


2S03.93 


X 


3499.4 


•% 


4271.84 


■ X 


-X 


2815.67 


•X 


3512.52 


•X 


4286.33 


•X 


CO. 


2827.44 67 




3525.66 


74. 


4300.85 81 




•X 


2839.23 


'x 


3538.83 


•X 


4315.39 


'x 


•X 


2851.05 


•X 


3552.02 


M 


4329.96 


•y 


•X 


2862.89 


■X 


3565.24 


•X 


4344.55 


.% 


• X 


2874.76 


•X 


3578.48 


•X 


4359.17 


•y 


-X 


2886.65 


• X 


3591.74 


• X 


4373.81 


•x 


• X I 2«9«.57 


•X 


! 3605.04 


M 


43*8.47 


• X 


•X 


2910.51 


•X 


1 3618.35 


•% 


4403.16 


•X 



AREAS OP CIRCLES. 



175 



Table— (Continued).— [.Advancing by an Eighth and a Quarter.] 

I 



82. 


5281.03 89. 




6221.15 96. 




7238.25 106. 




8824.75 


-M 


5297.14 


X 


6238.64 


X 


7257.11 


X 


8866.43 


•X 


5313.28 


X 


6256.15 


X 


7275.99 


X 


8908.2 


M 


5329.44 


% 


6273.69 


X 


7294.91 


X 


8950.07 


•X 


5345. G3 


X 


6291.25 


X 


7313.84 107. 




8992.04 


•%' 


5361.84 


X 


6308.84 


X 


7332.8 


X 


9034.11 


•X 


5378.08 


X 


6326.45 


X 


7351.79 


X 


9076.28 


•% 


5394.34 


X 


6344.08 


X 


7370.79 


X 


9118.54 


83 


5410.62 90. 




6361.74 97. 




7389.83 108. 




9160.91 


-% 


5426.93 


X 


6379.42 


X 


7408.89 


X 


9203.37 


•X 


5443.26 


x 


6397.13 


X 


7427.97 


X 


9245.93 


•% 


5459.62 


x 


6414.86 


X 


7447.08 


X 


9288.58 


•X 


5476.01 


X 


6432.62 


X 


7466.21 109. 




9331.34 


•X 


5492.41 


x 


6450.4 


X 


7485.37 


X 


9374.19 


m 


5508.84 


x 


6468.21 


X 


7504.55 


X 


9417.14 


•x 


5525.3 


X 


6486.04 


X 


7523.75 


X 


9460.19 


84 


5541.78 91. 




6503.9 98. 




7542.98 110. 




9503.34 


46 


5558.29 


X 


6521.78 


X 


7562.24 


X 


9546.59 


•X 


5574.82 


X 


6539.68 


X 


7581.52 


X 


9589.93 


M 


5591.37 


x 


6557.61 


X 


7600.82 


X 


9633.37 


• X 


5607.95 


X 


6575.56 


X 


7620.15 111. 




9676.91 


-% 


5624.56 


x 


6593.54 


X 


7639.5 


X 


9720.55 


:$£ 


5641.18 


x 


6611.55 


X 


7658.88 


X 


9764.29 


•% 


5657.84 


X 


6629.57 


X 


7678.28 


X 


9808.12 


85. 


5674.51 92. 




6647.63 99. 




7697.71 112. 




9852.06 


•X 


5691.22 


X 


6665.7 


X 


7717.16 


X 


9896.09 


>K 


5707.94 


X 


6683.8 


X 


7736.63 


X 


9940.22 


•% 


5724.69 


X 


6701.93 


X 


7756.13 


X 


9984.45 


•X 


5741.47 


X 


6720.08 


X 


7775.66 113 




10028.77 


>% 


5758.27 , 


% 


6738.25 


X 


7795.21 


X 


10073.2 


.% 


5775.1 


X 


6756.45 


X 


7814.78 


X 


10117.72 


•X 


5791.94 


X 


6774.68 


X 


7834.38 


X 


10162.34 


86. 


5808.82 93. 




6792.92 100 




7854. 114 




10207.06 


•X 


5825.72 


X 


6811.2 


X 


7893.32 


X 


10251.88 


$t 


5842.64 


X 


6829.49 


X 


7932.74 


X 


10296.79 


.% 


5859.59 


X 


6847.82 


X 


7972.25 


X 


10341.8 


M 


5876.56 


X 


6866.16 101 




8011.87 115 




10386.91 


•X 


5893.55 


X 


6884.53 


X 


8051.58 


X 


10432.12 


• X 


5910.58 


X 


6902.93 


X 


8091.39 


X 


10477.43 


-X 


5927.62 


X 


6921.35 


•X 


8131.3 


X 


10522.84 


87. 


5944.69 94 




6939.79 102 




8171.3 116 




10568.34 


% 


5961.79 


X 


6958.26 


■X 


8211.41 


■X 


10613.94 


•X 


5978.91 


•X 


6976.76 


•X 


8251.61 


•X 


10659.65 


.# 


5996.05 


X 


6995.28 


X 


8291.91 


X 


10705.44 


■$8 


6013.22 


•X 


7013.82 103 




8332.31 117 




10751.34 


■H 


6030.41 


•X 


7032.39 


'x 


8372.81 


'x 


10797.34 


•X 


6047.63 


• X 


7050.98 


•X 


8413.4 


• X 


10843.43 


-H 


6064.87 


• X 


7069.59 


•X 


8454.09 


•X 


10889.62 


88. 


6082.14 95 




7088.24 104 




8494.89 118 




10935.91 


•X 


6099.43 


'•X 


7106.9 


•X 


8535.78 


'•X 


10982.3 


-% 


6116.74 


• X 


7125.59 


• X 


8576.76 


•X 


11028.78 


.% 


6134.08 


•X 


7144.31 


•X 


8617.85 


• X 


11075.37 


•X 


6151.45 


•X 


7163.04 105 




8659.03 119 




11122.05 


•X 


6168.84 


•X 


7181.81 


•X 


8700.32 


'•X 


11168.83 


•X 


6186.25 


•X 


7200.6 


•X 


8741.7 


• X 


11215.71 


•X 


6203.69 


•X 


7219.41 


•X 


8783.18 


•X 


11262. $9 



176 



AREAS OF CIRCLES. 



Tafole— (Continued).— [Advancing by a Quarter and a Hal/.] 



Diam. | Area. 


| Diam. 


Area. 


Diam. 


Area. 


Diam. 


Area. 


120. 11309.76 


•X 


12173.9 


133. 


13892.94 


142. 


15836.31 


.V 11356.93 


-K 


12222.84 


•K 


13997.6 


-& 


15948.53 


.3^ 11404.2 


125. 


12271.87 


134. 


14102.64 


143. 


16060.64 


.% ; 11451.57 


-% 


12370.25 


•X 


14208.08 


•K 


16173.15 


121. J 11499.04 


126. 


12469.01 


135. 


14313.91 


144. 


16286.05 


.yi 111546.61 


-H 


12568.17 


-X 


14420.14 


•K 


16399.35 


.y 2 |11594.27 


127. 


12667.72 


136. 


14526.76 


145. 


16513.03 


.% 111642.03 


M 


12767.66 


•Vi 


14633.77 


■ X A 


16627.11 


122. 111689.89 


128. 


12867.99 


137. 


14741.17 


146. 


16741.59 


.1^ 11737.85 


.% 


12968.72 


M 


14848.97 


•}4 


16856.45 


,y 2 11785.91 


129. 


13069.84 


138. 


14957.16 


147. 


16971.71 


.% 11834.06 


•H 


13171.35 


M 


15065.74 


•K 


17087.36 


123. (11882.32 


130. 


13273.26 


139. 


15174.71 


148. 


17203.4 


.3^! 11930.67 


• -M 


13375.56 


<H 


15284.08 


•X 


17319.84 


.J^! 11979.12 


131. 


13478.25 


140. 


15393.84 


149. 


17436.67 


.^| ,12027.66 


%i 


13581.33 


M 


15503.99 


•K 


17553.89 


124. 112076.31 


132. 


13684.81 


141. 


15614.54 


150. 


17671.5 


.3^ [12125.05 


4i 


13788.68 


-% 


15725.48, 


•K 


17789.51 



To Compute tlie Area o£ sl Diameter greater than 
any in the preceding Table: 

Rule. — Divide the dimension by two, three, four, etc., if practicable 
to do so, until it is reduced to a diameter to be found in the table. 

Take the tabular area for this diameter, multiply it by the square of 
the divisor, and the product will give the area required. 

Example. — What is the area for a diameter of 1050? 

1050^-7 = 150; tab. area, 150 = 17071.5, which x 7 2 = SG5303.5, area required. 

To Compute tlie Area of a Diameter in Feet and 
Inches, etc., "by the preceding Table. 

Rule. — Reduce the dimension to inches or eighths, as the case may 
be, and take the area in that term from the table for that number. 

Divide this number by 64 (the square of 8) if it is eighths, and the 
quotient will give the area in inches, and divide again by 144 (the square 
of 12) if it is in inches, and the quotient will give the area in feet. 

Example. — What is the area of 1 foot §J{ inches? 
1 foot 6% ins. — 18% in* = 147 eighth*. Area of 147 = 10971.11, which -hG4 = 
265. IS inches; and by 144= 1M feet. 



To Compute the Area of* an Integer and. a Fraction 
not given in the Table. 

JffuLR, — Double, treble, or quadruple the dimension given, until the fraction is in- 
creased to a whole number, or to one of those in the table, as > s , *£, etc., provided 
it is practicable to do so. 

Take the area for this diameter ; and if it is double of that for which the area ia 
required, take one fourth of it; if treble, take one sixteenth of it, etc., etc. 

ExAMri-K. — Required the area for a circle of ?.% inches. 

2.% X 2 = 4.%, area for which — 15.033, which 4- 4 = 3.75S ins. 



CIRCUMFERENCES OF CIRCLES. 



177 





Circumferences of Circles, 


from ^ to ISO. 


Diam. | 


Circum. | Di? 


m. | 


Circum. 1 Dia 


m. 


Circum. Dis 


m. 


Circum. 


i 


.04909 o 




15.708 12. 




37.6992 19 




59.6904 


.09818 


H 


16.1007 


y 


38.0919 


H 


60.0831 


S3 


y 


16.4934 


y 


38.4846 


y 


60.4758 


ft 


.19635 


% 


16.8861 


% 


38.8773 


% 


60.8685 


i 


.3927 


8 


17.2788 


y 


39.27 


y 


61.2612 




% 


17.6715 


% 


39.6627 


% 


61.6539 


ft 


.589 


% 


18.0642 


% 


40.0554 


% 


62.0466 


1 


.7854 


% 


18.4569 


% 


40.4481 


% 


62.4393 


noir ,^ 6. 




18.8496 13. 




40.8408 20 




62.832 


A 


.981/0 


Vs 


19.2423 


y 


41.2335 


y 


63.2247 


3 


1.1781 ! 


y 


19.635 


y 


41.6262 


y 


63.6174 


JL 


1.37445 


% 


20.0277 


% 


42.0189 


% 


64.0101 


16 




H 


20.4204 


y 


42.4116 


y 


64.4028 


i 


1.5708 • 


% 


20.8131 


% 


42.8043 


% 


64.7955 


ft 

5 


1.76715 


% 


21.2058 


% 


43.197 i 


% 


65.1882 


1.9635 ? ; 


% 


21.5985 


% 


43.5897 


% 


65.5809 


8 




21.9912 14. 




43.9824 21. 




65.9736 


3 


2.15985 ! 


H 


22.3839 


y 


44.3751 


% 


66.3663 


2.3562 


y 


22.7766 


y 


44.7678 


y 


66.759 


4r 


2.55255 


% 


23.1693 
23.562 


y 


45.1605 
45.5532 


y 


67.1517 
67.5444 


£ 


2.7489 


% 


23.9547 


% 


45.9459 


% 


67.9371 


8 


2.94525 


% 


24.3474 


% 


46.3386 


% 


68.3298 


16 


% 


24.7401 


% 


46.7313 


% 


68.7225 


1. 


3.1416 8* 




25.1328 15 




47.124 22 




69.1152 




% 


3.5343 


% 


25.5255 


X 


47.5167 


y 


69.5079 




y 


3.927 


y 


25.9182 


y 


47.9094 


y 


69.9006 




% 


4.3197 


% 


26.3109 


% 


48.3021 


% 


70.2933 




y 


4.7124 


y 


26.7036 


y 


48.6948 


y 


70.686 




% 


5.1051 


% 


27.0963 


% 


49.0875 


% 


71.0787 




% 


5.4978 


% 


27.489 


% 


49.4802 


% 


71.4714 




% 


5.8905 


% 


27.8817 


% 


49.8729 


% 


71.8641 


2 




6.2832 9 




28.2744 16 




50.2656 23 




72.2568 




y 


6.6759 


y 


28.6671 


y 


50.6583 


% 


72.6495 




y 


7.0686 


% 


29.0598 


y 


51.051 


y 


73.0422 




% 


7.4613 


•% 


29.4525 


% 


51.4437 


% 


73.4349 




h 


7.854 


y 


29.8452 


y 


51.8364 


y 


73.8276 




% 


8.2467 


-% 


30.2379 


% 


52.2291 


% 


74.2203 




% 


8.6394 


-% 


30.6306 


% 


52.6218 


% 


74.613 




% 


9.0321 


% 


31.0233 


% 


53.0145 


% 


75.0057 


3 




9.4248 10 




31.416 17 




53.4072 24 




75.3984 




'y 


9.8175 


~.y* 


31.8087 


y 


53.7999 


'-y 


75.7911 




-H 


10.2102 


.y 


32.2014 


>y 


54.1926 


•k 


76.1838 




% 


10.6029 


.% 


32.5941 


% 


54.5853 


.% 


76.5765 




-H 


10.9956 


-y 


32.9868 


•y 


54.978 ! 


M 


76.9692 




-% 


11.3883 


.% 


33.3795 


-% 


55.3707 


•% 


77.3619 




>% 


11.781 


.% 


33.7722 


X 


55.7634 


.% 


77.7546 




-% 


12.1737 


•% 


34.1649 


-% 


56.1561 


•y 


78.1473 


4 




12.5664 11 




34.5576 18 




56.5488 25 




78.54 




'•% 


12.9591 


'•y 


34.9503 


'•y 


56.9415 


'.y 


78.9327 




y 


13.3518 


>y 


35.343 


•y 


57.3342 


-y 


79.3254 




>% 


13.7445 


•96 


35.7357 


.% 


57.7269 


-% 


79.7181 




M 


14.1372 


~X 


36.1284 


-y 


58.1196 


-M 


SO. 1108 




•% 


14.5299 


•% 


36.5211 


•% 


58.5123 


•% 


80.5035 




•k 


14.9226 


M 


36.9138 


-K 


58.905 


•% 


80.8962 




.% 


15.3153 


•% 


37.3065 


>X 


59.2977 


.% 


81.2889 



178 



CIRCUMFERENCES OF CIRCLES. 



Ta~ble— (Continued).— [.Advancing by an Eighth.] 



•% 



■% 



■Vs 



'H 



■% 



■% 

■M 
■% 






■% 



% 






Circum. Dia 


m. 


Circum. Die 


m. 


Circum. 


81.6816 33. 




103.673 40. 




125.664 


82.0743 


% 


104.065 


yk 


126.057 


82.467 


% 


104.458 


u 


126.449 


82.8597 


% 


104.851 


% 


126.842 


83.2524 


% 


105.244 


% 


127.235 


83,6451 


% 


105.636 


% 


127.627 


84.0378 


% 


106.029 


% 


128.02 


84.4305 


% 


106.422 


% 


128.413 


84.8232 34. 




106.814 41 




128.806 


85.2159 


% 


107.207 


y$ 


129.198 


85.6086 


% 


107.6 


% 


129.591 


86.0013 


% 


107.992 


% 


129.984 


86.394 


% 


108.385 


y 


130.376 


86.7867 


%. 


108.778 


% 


130.769 


87.1794 


% 


109.171 


% 


131 J62 


87.5721 


% 


109.563 


% 


131.554 


87.9648 35 




109.956 42 


/o 


131.947 


88.3575 


% 


110.349 


% 


132.34 


88.7502 


¥ 


110.741 


y 


132.733 


89.1429 


% 


111.134 




133.125 


89.5356 


K 


111.527 


133.518 


89.9283 


% 


111.919 


133.911 


90.321 


% 


112.312 


134.303 


90.7137 


% 


112.705 


% 


134.696 


91.1064 36 




113.098 43 




135.089 


91.4991 


% 


113.49 


-Vs 


135.481 


91.8918 


% 


113.883 


% 


135.874 


92.2845 


% 


114.276 


•% 


136.267 


92.6772 


H 


114.668 


3*> 


136.66 


93.0699 


% 


115.061 


-% 


137.052 


93.4626 


% 


115.454 




137.445 


93.8553 


% 


115.846 


137.838 


94.248 37 




116.239 44 


138.23 


94.6407 


Vs 


116.632 


'•H 


138.623 


95.0334 


K 


117.025 


139.016 


95.4261 


% 


117.417 




139.408 


95.8188 


M 


117.81 


139.801 


96.2115 


•% 


118.203 




140.194 


96.6042 


% 


118.595 


140.587 


96.9969 


% 


118.988 


% 


140.979 


97.3896 38 




119.381 45 




141.372 


97.7823 


% 


119.773 


Vs 


141.765 


98.175 


M 


120.166 


y± 


142.157 


98.5677 


% 


120.559 


% 


142.55 


98.9604 


X 


120.952 


k 


142.943 


99.3531 


% 


121.344 


% 


143.335 


99.7458 


% 


121.737 


-K 


143.728 


100.1385" 


-% 


122.13 


X 


144.121 


100.5312 39 




122.522 46 




144.514 


100.923-9 


'•% 


122.915 


Vs 


144.906 


101.3166 


•X 


123.308 


>x 


145.299 


101.7093 


-% 


123.7 


•% 


145.692 


102.102 


»K 


124.093 


>K 


146.084 


102.4947 


-% 


124.486 


>% 


146.477 


102.8874 


>% 


124.879 


% 


146.87 


103.2801 


•% 


125.271 


.% 


147.262 



47 



48 



49 



50 



51 



52 



53 



k 

% 



H 



Vs 



CIRCUMFERENCES OF CIRCLES. 



179 



Table— (Continued).— lAdvancing by an Eighth.] 



Piam. 


Circum. Dii 


im. 


Circum. Dh 


im. 


Circum. Dii 


im. 


Circum. 


54. 


169.646 61 




191.638 68 




213.629 75 




235.62 




■K 


170.039 


'% 


192.03 


H 


214.021 


y 


236.013 




k 


170.432 


y 


192.423 


-y 


214.414 


-y 


236.405 




% 


170.824 


% 


192.816 


•% 


214.807 


% 


236.798 




% 


171.217 


>H 


193.208 


H 


215.2 


y 


237.191 




•% 


171.61 


% 


193.601 


% 


215.592 


% 


237.583 




>% 


172.003 


% 


193.994 


% 


215.985 


% 


237.976 




% 


172.395 


% 


194.386 


% 


216.378 


% 


238.369 


55 




172.788 62 




194.779 69 




216.77 76 




238.762 




'Vs 


173.181 


% 


195.172 


y 


217.163 


•y 


239.154 




-y 


173.573 


y 


195.565 


y 


217.556 


•y 


239.547 




.% 


173.966 


% 


195.957 


% 


217.948 


% 


239.94 




y 


174.359 


% 


196.35 


H 


218.341 


y 


240.332 




% 


174.751 


% 


196.743 


% 


218.734 


% 


240.725 




% 


175.144 


% 


197.135 


% 


219.127 


% 


241.118 




.% 


175.537 


% 


197.528 


% 


219.519 


% 


241.51 


56 




175.93 63 




197.921 70 




219.912 77 




241.903 




y 


176.322 


Vs 


198.313 


y 


220.305 


y 


242.296 




y 


176.715 


y 


198.706 


y 


220.697 


y 


242.689 




% 


177.108 


% 


199.099 


% 


221.09 


% 


243.081 




y 


177.5 


y 


199.492 


y 


221.483 


y 


243.474 




% 


177.893 


% 


199.884 


% 


221.875 


% 


243.867 




% 


178.286 


% 


200.277 


k 


222.268 


% 


244.259 




% 


178.678 


% 


200.67 


% 


222.661 


% 


244.652 


57 




179.071 64 




201.062 71 ' 




223.054 78 




245.045 




•y 


179.464 


k 


201.455 


Vs 


223.446 


y 


245.437 




-y 


179.857 


y 


201.848 


y 


223.839 


y 


245.83 




% 


180.249 


% 


202.24 


% 


224.232 


% 


246.223 




% 


180.642 


H 


202.633 


y 


224.624 


y 


246.616 




% 


•181.035 


% 


203.026 


% 


225.017 


% 


247.008 




% 


181.427 


M 


203.419 


% 


225.41 


% 


247.401 




% 


181.82 


% 


203.811 


% 


225.802 


% 


247.794 


58 




182.213 , 65 




204.204 72 




226.195 79 




248.186 




>y 


182.605 


y 


204.597 


y 


226.588 


y 


248.579 




•y 


182.998 


y 


204.989 


y 


226.981 


y 


248.972 




% 


183.391 


% 


205.382 


% 


227.373 


% 


249.364 




y* 


183.784 


H 


205.775 


y 


227.766 


y 


249.757 




% 


184.176 


% 


206.167 


% 


228.159 


% 


250.15 




% 


184.569 


% 


206.56 


% 


228.551 


% 


250.543 




% 


184.962 


% 


206.953 


% 


228.944 


% 


250.935 


59 




185.354 66 




207.346 73 




229.337 80 




251.328 




•y 


185.747 


y% 


207.738 


y 


229.729 


y 


251.721 




•y 


186.14 


y 


208.131 


y 


230.122 


y 


252.113 




-% 


186.532 


% 


208.524 


% 


230.515 


% 


252.506 




>K 


186.925 


a 


208.916 


y 


230.908 


y 


252.899 




% 


187.318 


% 


209.309 


% 


231.3 


% 


253.291 




% 


187.711 


% 


209.702 


% 
% 


231.693 


% 


253.684 




% 


188.103 


% 


210.094 


232.086 


% 


254.077 


GO 




188.496 67 




210.487 74 




232.478 81 




254.47 




•K 


188.889 


% 


210.88 


y 


232.871 


y 


254.862 




.k 


189.281 


y 


211.273 


y 


233.264 


y 


255.255 




% 


189.674 


% 


211.665 


% 


233.656 


% 


255.648 




k 


190.067 


H 


212.058 


y 


234.049 


y 


256.04 




-% 


190.459 


% 


212.451 


% 


234.442 


% 


256.433 




-X 


190.852 


% 


212.843 


% 


234.835 


H 


2M.S26 




% 


191.245 


% 


213.236 


K 


235.227 


% 


257.218 



180 



CIRCUMFERENCES OF CIRCLES. 



Ta"ble— (Continued). — [Advancing by an Eighth and a Quarter.] 



Diam. 


Circura. II Diam. 


Circum. 1 Di 


am. 


82. 


257.611 89 




279.602 96 






■K 


258.004 


•y 


279.995 


>% 




U 


258.397 


-y 


280.388 


>y 




.% 


258.789 


.% 


280.78 


•% 




.& 


259.182 


y 


281.173 


>y 




-% 


259 . 575 


.% 


281.566 


•y 




•% 


259.967 


' 6 A 


281.959 


>% 




% 


260.36 


% 


282.351 


-% 


83 




260.753 90 




282.744 97 






Vs 


261.145 


y 


283.137 


•x 




y 


261.538 


y 


283.529 


•y 




% 


261.931 


% 


283.922 


>% 




y 


262.324 


y 


284.315 


•y \ 




% 


262.716 


% 


284.707 


•% 




% 


263.109 


% 


285.1 


•% 




% 


263.502 


% 


285.493 


•% 


84] 




263.894 91 




285.886 98 






y 


264.287 


y 


286.278 


'-% 




y 


264.68 


y 


286.671 


•y 




% 


265.072 


% 


287.064 


% 




H 


265.465 


y 


287.456 


-y 




% 


265.858 


•% 


287.849 


-% 




% 


266.251 


% 


288.242 


•% 




% 


266.643 


% 


288.634 


-% 


85* 




267.036 92 




289.027 99 






% 


267.429 


y 


289.42 


y 




H 


267.821 


y 


289.813 


>y 




% 


268.214 


% 


290.205 


•% 




H 


268.607 


y 


290.598 


>y 




% 


268.999 


% 


290.991 


% 




% 


269.392 


% 


291.383 


>% 




% 


269.785 


% 


291.776 


-% 


86! 




270.178 93 




292.169 100 






y 


270.57 


y 


292.562 


y 




K 


270.963 


y 


292.954 


y 




% 


271.356 


% 


293.347 


% 




y 


271.748 


y 


293.74 101 






% 


272.141 


% 


294.132 


y 




% 


272.534 


% 


294.525 


y 




% 


272.926 


% 


294.918 


% 


87! 




273.319 94. 




295.31 102 






X 


273.712 


y 


295.703 


y 




H 


274.105 


y 


296.096 


y 




% 


274.497 


% 


296.488 


% 




y 


274.89 


y 


296.881 103. 






% 


275.283 


% 


297.274 


y 




% 


275.675 


% 


297.667 


y 




% 


276.068 


% 


298.059 


% 


88. 




276.461 95. 




298.452 104. 






% 


276.853 


y 


298.845 


y 




X 


277.246 


y 


299.237 


y 




% 


277.629 


% 


299.63 


% 




y 


278.032 


y 


300.023 105. 






% 


278.424 


% 


300.415 


y 




% 
% 


278.817 
279.21 


% 


300.808 
301.201 | . 


y 

% \ 



Circum. 


| Diam. 


i Circum. 


301.594 


106 




333.01 


301.986 




-y 


333.795 


302.379 




■y 


334.58 


302.772 




-% 


335.366 


303.164 


107 




336.151 


303.557 




'•y 


336.937 


303.95 




-y 


337.722 


304.342 




•M 


338.507 


304.735 


108 




339.293 


305.128 




'•y 


340.078 


305.521 




.y 


340.864 


305.913 




•% 


341.649 


306.306 


109 




342 434 


306.699 




>u 


343.22 


307.091 




•y 


344.005 


307.484 




•M 


344.791 


307.877 


110 




345.576 


308.27 




'-y 


346.361 


308.662 




■y 


347.147 


309.055 




•% 


347.932 


309.448 


111 




348.718 


309.84 




-H 


349.503 


310.233 




>y 


350.288 


310.626 




-% 


351.074 


311.018 


112 




351.859 


311.411 




'.y 


352.645 


311.804 




-y 


353.43 


312.196 




•K 


354.215 


312.589 


113 




355.001 


312.982 




'-y 


355.786 


313.375 




-y 


356.572 


313.767 




>% 


357.357 


314.16 


114 




358.142 


314.945 




-y 


358.928 


315.731 




•y 


359.713 


316.516 




•X 


360.499 


317.302 


115 




361.284 


318.087 




'•y 


362.069 


318.872 




-y 


362.855 


319.658 




•H 


363.64 


320.443 


116 




364.426 


321.229 




'-y 


365.211 


322.014 




•y 


365.996 


322.799 




.% 


366.782 


323.585 


117 




367.567 


324.37 




'•y 


368.353 


325.156 




-y 


369.138 


325.941 




•H 


369.923 


326.726 


118 




370.709 


327.512 




•y 


371.494 


328.297 




•y 


372.28 


329.083 




-K 


373.065 


329.868 


119 




373.85 


330.653 




'•y 


374.636 


331.439 




-y 


375.421 


332.224 




•X 


376.207 



CIRCUMFERENCES OF CIRCLES. 



181 



Table— {Continued).— [Advancing by a Quarter and a IlaJ/.J 



Diam. 


Circum. 


Diam. 


120. 


376.992 


•X 


-K 


377.777 


-% 


- X A 


378.563 


125. 


.% 


379.348 


M 


121. 


380.134 


126. 


-% 


380.919 


•X 


>M 


381.704 


127. 


•X 


382.49 


M 


122. 


383.275 


128. 


•« 


384.061 


•X 


•% 


384.846 


129. 


•H 


385.631 


•X 


123. 


386.417 


130. 


# 


387.202 


-H 


•M 


387.988 


131. 


•M 


388.773 


•X 


124. 


389.558 


132. 


• ik 


390.344 


•K 



Circum. 



391.129 

391.915 

392.7 

394.271 

395.842 

397.412 

398.983 

400.554 

402.125 

403.696 

405.266 

406.837 

408.408 

409.979 

411.55 

413.12 

414.691 

416.262 



133. 

135. 

rJ 
136. 

! •> 
137. 

•> 

138. 
i 

139.'* 

# i 

140*/ 
141 !" 



Circum. r 


Diam. | 


Cireum. 


417.833 


142. 


446.107 


419.404 


•X 


447.678 


420.974 


143. 


449:249 


422.545 


M 


450:82 


424.116 


144. 


452.39 


425.687 


1 >x 


453:961 


427.258 


Ub. 


455:532 


428.828 


•K 


457.103 


430.399 


146. 


458\674 


431.97 


' -H 


460.244 


433.541 


147. 


461.815 


435.112 


■ -yt 


463,386 


436.682 


148. 


464,957 


438.253 


•K 


466,528 


439.824 


149. 


468,098 


441.395 


[•X 


469,669 


442.966 


150. 


471,24 


444.536 


1 M 


1 472,811 



To Compute the Circninference of a Diameter 
greater tlian any in tlie preceding Table. 

Rule. — Divide the dimension by two, three, four, etc., if practicable 
to do so, until it is reduced to a diameter to be found in the table. 

Take the tabular circumference for this dimension, multiply it by 
2, 3, 4, 5, etc., according as it was divided, and the product will give 
the circumference required. ■ ■. 

Example — What is the circumference for a diameter of 1050 ? 
1050 -r- 7 = 150 ; tab. circum., 150 = 471.239, which x7 = 3298.673, circum. required. 

To Compute tlie Circumference for an Integer and 
Fraction not given in. the Table. 

Rule. — Double, treble, or quadruple the dimension given, until the 
fraction is increased to a whole number or to one of those in the ta- 
ble, as %, %, etc., provided it is practicable to do so. 

Take the circumference for this diameter ; and if it is double of 
that for which the circumference is required, take one half of it ; if 
treble, take one third of it ; and if quadruple, one fourth of it. 

Example. — Required the circumference of 2.21875 inches. 
2.21S75 x2 = 4.4375 = 4j&, which x2 = 8.%; tab. circum. = 27.8817, which -h 4 
= 6.9704 ms. 



To Compute the Circumference of a Diameter in 
Feet and. Inches, etc.* by the preceding Table. 

Rule. — Reduce the dimension to inches or eighths, as the case may he, and take 
the circumference in that term from the table for that number. 

Divide this number by 8 if it is in eighths, and by 12 if in inches, and the quotient 
will give the area in feet. 

Example. — Required the circumference of a circle of 1 foot 6% inches. 
1 foot b% ins. == 18% ins. = 147 eighths. Circum. of 147 = 461. S15, which ~ 8 = 
57.727 inches ; and b / 12 = 4 SI feet. 



182 AREAS AND CIRCUMFERENCES OF CIRCLES. 



Areas and. Circumferences, from rfr to 100. 

[Advancing by Tenths.] 



Diam. 


Area. 


Circum. 


Diam. 


Area. 


Circum. 


.1 


.007854 


.31416 


.4 


22.9023 


16.9646 


.2 


.031416 


.62832 


.5 


23.7583 


17.2788 


.3 


.070686 


.94248 


.6 


24.6301 


17.593 


.4 


.125664 


1.2566 


.7 


25.5176 


17.9071 


.5 


.19635 


1.5708 


.8 


26.4209 


18.2213 


.6 


.282744 


1.885 


.9 


27.3398 


18.5354 


.7 


.384846 


2.1991 


6. 


28.2744 


18.8496 


.8 


.502656 


2.5133 


.1 


29.2247 


19.1638 


.9 


.636174 


2.8274 


.2 


30.1908 


19.4779 


1. 


.7854 


3.1416 


.3 


31.1725 


19.7921 


.1 


.9503 


3.4558 


.4 


32.17 


20.1062 


.2 


1.131 


3.7699 


.5 


33.1831 


20.4204 


.3 


1.3273 


4.0841 


.6 


34.212 


20.7346 


.4 


1.5394 


4.3982 


.7 


35.2566 


21.0487 


.5 


1.7671 


4.7124 


.8 


36.3169 


21.3629 


.6 


2.0106 


5.0266 


.9 


37.8929 


21.677 


.7 


2.2698 


5.3407 


7. 


38.4846 


21.9912 


.8 


2.5447 


5.6549 


.1 


39.592 


22.3054 


.9 


2.8353 


5.969 


.2 


40.7151 


22.6195 


2. 


3.1416 


6.2832 


.3 


41.854 


22.9337 


.1 


3.4636 


6.5974 


.4 


43.0085 


23.2473 


.2 


3.8013 


6.9115 


.5 


44.1787 


23.562 


.3 


4.1548 


7.2257 


.6 


45.3647 


23.8762 


.4 


4.5239 


7.5398 


.7 


46.5664 


24.1903 


.5 


4.9087 


7.854 


.8 


47.7837 


24.5045 


.6 


5.3093 


8.1682 


.9 


49.0168 


24.8186 


.7 


5.7256 


8.4823 


8. 


50.2656 


25.1328 


.8 


6.1575 


8.7965 


.1 


51.5301 


25.447 


.9 


6.6052 


9.1106 


.2 


52.8103 


25.7611 


3. 


7.0686 


9.4248 


.3 


54.1062 


26.0753 


.1 


7.5477 


9.739 


.4 


55.4178 


26.3894 


.2 


8.0425 


10.0531 


.5 


56.7451 


26.7036 


.3 


8.553 


10.3673 


.6 


58.0882 


27.0178 


.4 


9.0792 


10.6814 


.7 


59.4469 


27.3319 


.5 


9.6211 


10.9956 


.8 


60.8214 


27.6461 


.6 


10.1788 


11.3098 


.9 


62.2115 


27.9602 


.7 


10.7521 


11.6239 


9. 


63.6174 


28.2744 


.8 


11.3412 


11.9381 


.1 


65.039 


28.5886 


.9 


11.9459 


12.2522 


.2 


66.4763 


28.9027 


4. 


12.5664 


12.5664 


.3 


67.9292 


29.2169 


.1 


13.2026 


12.8806 


.4 


69.3979 


29.531 


.2 


13.8545 


13.1917 


.5 


70.8823 


29.8452 


.3 


14.522 


13.5089 


.6 


72 .-3825 


30.1594 


.4 


15.2053 


13.823 


.7 


73.8983 


30.4735 


.5 


15.9043 


14.1372 


.8 


75.4298 


30.7877 


.6 


16.6191 


14.4514 


.9 


76.9771 


31.1018 


.7 


17-3495 


14.7655 


10. 


78.54 


31.416 


.8 


18.0956 


15.0797 


.1 


80.1187 


31.7302 


.9 


18.8575 


15.3938 


.2 


81.713 


32.0443 


5. 


19.635 


15.708 


.3 


83.3231 


32.3585 


.1 


20.4283 


16.0222 


.4 


84.9489 


32.6726 


.2 


21.2372 


16.3363 


.5 


86.5903 


32.9868 


.3 


22.0619 


16.6505 


.6 1 


88.2475 


33.301 



AREAS AND CIRCUMFERENCES OF CIRCLES. 



183 







Ta"ble— (Contin 


ued). 






Diam. 


Area. 


i Circum. Dia 


m. 


Area. i 


Circum. 


.7 


89.9204 


33.6151 


.2 


206.1204 


50.8939 


.8 


91.6091 


33.9293 


.3 


208.6729 


51.2081 


.9 


93.3134 


34.2434 


.4 


211.2412 


51.5222 


11. 


95.0334 


34.5576 


.5 


213.8251 


51.8364 


.1 


96.7691 


34.8718 i 


.6 


216.4248 


52.1505 


.2 


98.5206 


35.1859 


.7 


219.0402 


52.4647 


.3 


100.2877 


35.5001 


.8 


221.6713 


52.7789 


.4 


102.0706 


35.8142 


.9 


224.3181 


53.093 


.5 


103.8691 


36.1284 ! 17 




226.9806 


53.4072 


.6 


105.6834 


36.4426 


!l 


229.6588 


53.7214 


.7 


107.5134 


36.7567 


,2 


232.3527 


54.0355 


.8 


109.3591 


37.0709 


.3 


235.0624 


54.3497 


.9 


111.2205 


37.385 


.4 


237.7877 


54.6638 


12. 


113.0976 


37.6992 


.5 


240.5287 


54.978 


.1 


114.9904 


38/0134 1 


.6 


243.2855 


55.2922 


.2 


116.8989 


38.3275 


.7 


246.058 


55.6063 


.3 


118.8232 


38.6417 


8 


248.8461 


55.9205 


.4 


120.7631 


38.9558 ; 


.9 


251.65 


56.2346 


.5 


122.7187 


39.27 1 18 




254.4696 


56.5488 


.6 


124.6901 


39.5842 


.1 


257.3049 


56.863 


.7 


126.6772 


39.8983 


.2 


260.1559 


57.1771 


.8 


128.6799 


40.2125 


3 


263.0226 


57.4913 


.9 


130.6984 


40.5266 


4 


265.905 


57.8054 


13. 


132.7326 


40.8408 


.5 


268.8031 


58.1196 


.1 


134.7825 


41.155 


.6 


271.717 


58.4338 


.2 


136.8481 


41.4691 


7 


274.6465 


58.7479 


.3 


138.9294 


41.7833 


8 


277.5918 


59.0621 


.4 


141.0264 


42.0974 


9 


280.5527 


59.3762 


.5 


143.1391 


42.4116 J 19 




283.5294 


59.6904 


.6 


145.2676 


42.7258 


1 


286.5218 


60.0046 


.7 


147 .4117 


43.0399 


2 


289.5299 


60.3187 


.8 


149.5716 


43.3541 


3 


292.5536 


60.6329 


.9 


151.7471 


43.6682 


4 


295.5931 


60.947 


14. 


153.9384 


43.9824 


5 


298.6483 


61.2612 


.1 


156.1454 


44.2966 


6 


301.7193 


61.5754 


.2 


158.3681 


44.6107 


7 


304.806 


61.8895 


.3 


160.6064 


44.9249 


8 


307.9082 


62.2037 


.4 


162.8605 


45.239 


9 


311.0263 


62.5178 


.5 


165.1303 


45.5532 20 




314.16 


62.832 


.6 


167.4159 


45.8674 


1 


317.3094 


63.1462 


.7 


169.7171 


46.1815 


2 


320.4746 


63.4603 


.8 


172.034 


46.4957 ! 


3 


323.6555 


63.7745 


.9 


174.3667 


46.8098 


4 


326.8521 


64.0886 


15. 


176.715 


47.124 


5 


330.0643 


64.4028 


.1 


179.0791 


47.4382 


6 


333.2923 


64.717 


.2 


181.4,588 


47.7523 


7 


336.536 


65.0311 


.3 


183.8543 


48.0665 


8 


339.7955 


65.3453 


.4 


186.2655 


48.3806 


9 


343.0706 


65.6594 


.5 


188.6924 


48.6948 21 




346.3614 


65.9736 


.6 


191.1349 


49.009 


1 


349.6679 


66.2878 


.7 


193.5932 


49.3231 


2 


352.9902 


66.6019 


.8 


196.0673 


49.6373 


3 


356.3281 


66.9161 


.9 


198.557 


49.9514 [ 


4 


359.6818 


67.2302 


16. 


201.0624 


50.2656 


5 


363.0511 


67.5444 


.1 


203.5835 


50.5797 


6 


866.4362 


67.8586 



184 AREAS AND CIRCUMFERENCES OF CIRCLES. 



Ta"ble— {Continued). 



Diam. 


i Area. 


Circum. 


Diam. 


Area. 


Carcofli. 


.7 


369.837 


68.1727 


•2 


581.0703 


85.4515 


.8 


373.2535 


68.4869 


.3 


585.3508 


85.7657 


.9 


376.6857 


68.801 


■ i 


589.6469 


86.0798 


22. 


380.1336 


69.1152 


.5 


593.9587 


86.394 


.1 


383.5972 


69.4294 


.6 


598.2863 


86.7082 


.2 


387.0765 


69.7435 


.7 


602.6296 


87.0223 


.3 


390.5716 


70.0577 


.8 


606.9885 


87.3365 


.4 


394.0823 


70.3718 


.9 


611.3632 


87/6506 


.5 


397.6087 


70.686 


28. 


615.7536 


87.9648 


.6 


401.1509 


71.0002 


.1 


620.1597 


88.79 


.7 


404.7088 


71.3143 


.2 


624.5815 


88.5931 


.8 


408.2823 


71.6285 


.3 


629.019 


88.9073 


.9 


411.8716 


71.9426 


.4 


633.4722 


89.2214 


23. 


415.4766 


72.2568 


.5 


637.9411 


89.5356 


.1 


419.0973 


72.571 


.6 


642.4258 


89.8498 


.2 


422.7337 


72.8851 


.7 


646.9261 


90.1639 


.3 


426.3858 


73.1993 


.8 


651.4422 


90.4781 


.4 


430.0536 


73.5134 


.9 


655.9739 


90.7922 


.5 


433.7371 


73.8276 


29. 


660.5214 


91.1064 


.6 


437.4364 


74.1418 


.1 


665.0846 


91.4206 


.7 


441.1513 


74.4559 


.2 


669.6635 


91.7347 


.8 


444.882 


74.7701 


.3 


674.258 


92.0489 


.9 


448.6283 


75.0842 


.4 


678.8683 


92.363 


24. 


452.3904 


75.3984 


.5 


683.4943 


92.6772 


.1 


456.1682 


75.7126 


.6 


688.1361 


92.9914 


.2 


459.9617 


76.0267 


.7 


692.7935 


93.3055 


.3 


463.7708 


76.3409 


.8 


697.4666 


93.6197 


. .4 


467.5957 


76.655 


.9 


702.1555 


93.9338 


.5 


471.4363 


76.9692 


30. 


706.86 


94.248 


.6 


475.2927 


77.2834 


.1 


711.5803 


94.5622 


.7 


479.1647 


77.5975 


.2 


716.3162 


94.8763 


.8 


483.0524 


77.9117 


.3 


721.0679 


95.1905 


.9 


486.9559 


78.2258 


.4 


725.8353 


95.5046 


25. 


490.875 


78.54 


.5 


730.6183 


95.8188 


.1 


494.8099 


78.8542 


.6 


735.4171 


96.133 


.2 


498.7604 


78.1683 


.7 


740.2316 


96.4471 


.3 


502.7267 


79.4825 


.8 


745.0619 


96.7613 


.4 


506.7087 


79.7966 


.9 


749.9078 


97.5754 


.5 


510.7063 


80.1108 


31. 


754.7694 


97.3896 


.6 


514.7106 


80.425 


.1 


759.6467 


97.7038 


.7 


518.7488 


80.7391 


.2 


764.5398 


98.0179 


.8 


522.7937 


81.0583 


.3 


769.4485 


98.3321 


.9 


526.8542 


81.3674 


.4 


774.373 


98.6462 


26. 


530.9304 


81.6816 


.5 


779.3131 


98.9604 


.1 


535.0223 


81.9958 


.6 


784.269 


99.^746 


.2 


539.13 


82.3099 


. t 


789.2406 


99.0887 


.3 


543.2533 


82.6241 


.8 


794.2279 


99.9029 


.4 


547.3924 


82.9382 


.9 


799.2309 


100.217 


.5 


551.5471 


83.2524 


32. 


804.2496 


100.5312 


.6 


555.7176 


83.5666 


.1 


809.284 


100.8454 


.7 


559.9038 


83.8807 


.2 


814.3341 


101.1595 


.8 


564.1057 


84.1949 


.3 


819.4 


101.4737 


.9 


568.3233 


84.509 


.4 


824.4815 


101.7878 


27. 


572.5566 


84.82;V2 


.5 


829.5787 


102.102 


.1 


576.8056 


85.1374 


.6 


834.6917 


102.4162 



AREAS AND CIRCUMFERENCES OF CIRCLES. 



185 







Ta"ble— {Continued) 






Diam. 


Area. 


Circum. 


Diam. 


Area. 


i Circum. 


.7 


839.8204 


102.7303 


.2 


1146.0871 


120.0091 


.8 


844.9647 


103.0445 


.3 


1152.0954 


120.3233 


.9 


850.1248 


103.3586 


.4 


1158.1194 


120.6374 


33. 


855.3006 


103.6728 


.5 


1164.1591 


120.9516 


.1 


860.4921 


103.987 


i .6 


1170.2146 


121.2658 


.2 


865.6993 


104.3011 


.7 


1176.2857 


121.5799 


.3 


870.9222 


104.6153 


1 .8 


1182.3726 


121.8941 


.4 


876.1608 


104.9294 


.9 


1188.4751 


122.2082 


.5 


881.4151 


105.2436 


39. 


1194.5934 


122.5224 


.6 


886.6852 


105.5578 


.1 


1200.7274 


122.8366 


.7 


891.9709 


105.8719 


.2 


1206.8771 


123.1507 


.8 


897.2724 


106.1861 


.3 


1213.0424 


123.4649 


.9 


902.5895 


106.5002 


.4 


1219.2235 


123.779 


34. 


907.9224 


106.8144 


.5 


1225.4203 


124.0932 


.1 


913.271 


107.1286 


.6 


1231.6329 


124.4074 


.2 


918.6353 


107.4427 


.7 


1237.8611 


124.7215 


.3 


924.0152 


107.7569 


.8 


1244.105 


125.0357 


.4 


929.4109 


108.071 


.9 


1250.3647 


125.3498 


.5 


934.8223 


108.3852 


40. 


1256.64 


125.664 


.6 


940.2495 


108.6994 


.1 


1262.9311 


125.9782 


.7 


945.6923 


109.0135 


.2 


1269.2378 


126.2923 


.8 


951.1508 


109.3277 


! .3 


1275.5603 


126.6065 


.9 


956.6251 


109.6418 


.4 


1281.8985 


126.9206 


35. 


962.115 


109.956 


.5 


1288.2523 


127.2348 


.1 


967.6207 


110.2702 


i .6 


1294.6219 


127.549 


.2 


973.142 


110.5843 


.7 


1301.0072 


127.8631 


.3 


978.6791 


110.8985 


.8 


1307.4083 


128.1773 


.4 


984.2319 


111.2126 


.9 


1313.825 


128.4914 


.5 


989.8003 


111.5268 


41. 


1320.2574 


128.8056 


.6 


995.3845 


111.841 


.1 


1326.7055 


129.1198 


.7 


1000.9844 


112.1551 


.2 


1333.1694 


129.4339 


.8 


1006.6001 


112.4693 


.3 


1339.6489 


129.7481 


.9 


1012.2314 


112.7834 


.4 


1346.1442 


130.0622 


36. 


1017.8784 


113.0976 


1 .5 


1352.6551 


130.3764 


.1 


1023.5411 


113.4118 


.6 


1359.1818 


130.6906 


.2 


1029.2196 


113.7259 


.7 


1365.7242 


131.0047 


.3 


1034.9137 


114.0401 


.8 


1372.2823 


131.3189 


.4 


1040.6236 


114.3542 


.9 


1378.8561 


131.633 


.5 


1016.3491 


114.6684 


42. 


1385.4456 


131.9472 


.6 


1052.0904 


114.9826 


.1 


1392.0508 


132.2614 


.7 


1057.8474 


115.2967 


.2 


1398.6717 


132.5755 


.8 


1063.6201 


115.6109 


.3 


1405.3084 


132.8897 


.9 


1069.4085 


115.925 


.4 


1411.9607 


133.2038 


37. 


1075.2126 


116.2392 


.5 


1418.6287 


133.518 


.1 


1081.0324 


116.5534 


.6 


1425.3125 


133.8322 


.2 


1086.8879 


116.8675 


.7 


1432.012 


134.1463 


.3 


1092.7192 


117.1817 


! .8 


1438.7271 


134.4605 


.4 


1098.5861 


117.4958 


; .9 


1445.458 


134.7746 


.5 


1104.4687 


117.81 


43. 


1452.2046 


135.0888 


.6 


1110.3671 


118.1242 


.1 


1458.9669 


135.403 


.7 


1116.2812 


118.4383 


.2 


1465.7449 


135.7171 


.8 


1122.2109 


118.7525 


.3 


1472.5386 


136.0313 


.9 


1128.1564 


119.0666 


.4 


1479.348 


136.3454 


38. 


1134.1176 


119.3808 


.5 


. 1486.1731 


136.6596 


.1 


1140.0945 


119.695 


.6 


1493.014 


136.9738 



186 



AREAS AND CIRCUMFERENCES OF CIRCLES. 







Table — {Continued) 






Diam. 


Area. 


Circum. 


Diam. 


Area. 


Circum. 


.7 


1499.8705 


137.2879 


.2 


1901.1707 


154.5667 


.8 


1506.7428 


137.6021 


.3 


1908.9068 


154.8809 


.9 


1513.6307 


137.9162 


.4 


1916.6587 


155.195 


44. 


1520.5344 


138.2304 


.5 


1924.4263 


155.5092 


.1 


1527.4538 


138.5446 


.6 


1932.2097 


155.8234 


.2 


1534.3889 


138.8587 


..7 


1940.0087 


156.1375 


.3 


1541.3396 


139.1729 


.8 


1947.8234 


156.4517 


.4 


1548.3061 


139.487 


.9 


1955.6539 


156.7658 


.5 


1555.2883 


139.8012 


50. 


1963.5 


157.08 


.6 


1562.2863 


140.1154 


.1 


1971.3619 


157.3942 


.7 


1569.2999 


140.4295 


.2 


1979.2394 


157.7083 


.8 


1576.3292 


140.7437 


.3 


1987.1327 


158.0225 


.9 


1583.3743 


141.0578 


.4 


1995.0417 


158.3366 


45. 


1590.435 


141.372 


.5 


2002.9663 


158.6509 


.1 


1597.5115 


141.6862 


.6 


2010.9067 


158.965 


.2 


1604.6036 


142.0003 


.7 


2018.8628 


159.2791 


.3 


1611.7115 


142.3145 


.8 


2026.8347 


159.5933 


.4 


1613.8351 


142.6286 


.9 


2034.8222 


159.9074 


.5 


1625.9743 


142.9428 


51. 


2042.8254 


160.2216 


.6 


1633.1293 


143.257 


.1 


2050.8443 


160.5358 


.7 


164Q.3 


143.5711 


.2 


2058.879 


160.8499 


.8 


1647.4865 


143.8853 


.3 


2066.9293 


161.1641 


.9 


1654.6886 


144.1994 


.4 


2074.9954 


161.4782 


46. 


1661.9064 


144.5136 


.5 


2083.0771 


161.7924 


.1 


1669.1399 


144.8278 


.6 


2091.1746 


162.1066 


.2 


1676.3892 


145.1419 


.7 


2099.2878 


162.4207 


.3 


1683.6541 


145.4561 


.8 


2107.4167 


162.7349 


.4 


1690.9348 


145.7702 


.9 


2115.5613 


163.Q49 


.5 


1698.2311 


146.0844 


52. 


2123.7216 


163.3632 


.6 


1705.5432 


146.3986 


.1 


2131.8976 


163.6774 


.7 


1712.871 


146.7127 


.2 


2140.0893 


163.9915 


.8 


1720 2145 


147.0269 


.3 


2148.2968 


164.3057 


.9 


1727.5737 


147.341 


.4 


2156.5199 


164.6198 


47. 


1734.9486 


147.6552 


.5 


2164.7587 


164.934 


.1 


1742.3392 


147.9694 


.6 


2173.0133 


165.2482 


.2 


1749.7455 


148.2835 


.7 


2181.2836 


165.5623 


.3 


1757.1676 


148.5977 


.8 


2189.5695 


165.8765 


.4 


1764.6053 


148.9118 


.9 


2197.8712 


166.1906 


.5 


1772.0587 


149.226 


53. 


2206.1886 


166.5048 


.6 


1779.5279 


149.5402 


.1 


2214.5217 


166.819 


.7 


1787.0128 


149.8543 


.2 


2222.8705 


167.1331 


.8 


1794.5133 


150.1685 


.3 


2231.235 


167.4473 


.9 


1802.0296 


150.4826 


.4 


2239.6152 


167.7614 


48. 


1809.5616 


150.7968 


.5 


2248.0111 


168.0756 


.1 


1817.1093 


151.111 


.6 


2256.4228 


168.3898 


.2 


1824.6727 


151.4251 


.7 


2264.8501 


168.7039 


.3 


1832.2518 


151.7393 


.8 


2273.2932 


169.0181 


.4 


1839.8466 


152.0534 


.9 


2281.7519 


169.3322 


.5 


1847.4571 


152.3676 


54. 


2290.2264 


169.6464 


.6 


1855.0834 


152.6818 


.1 


2298.7166 


169.9606 


.7 


1862.7253 


152.9959 


.2 


2307.2225 


170.2747 


.8 


1870.383 


153.3101 


.3 


2315.744 


170.5889 


.9 


1878.0563 


153.6242 


.4 


2321.2813 


170.903 


49. 


1885.7454 


153.9384 


.5 


2332.8343 


171.2172 


.1 


1893.4502 


154.2526 


.6 


2341.4031 


171.5314 



AREAS AND CIRCUMFERENCES OF CIRCLES. 



187 



Ta~ble — {Continued). 



Diam. 


Area. 


Circum. 


Diam. 


Area. 


Circum. 


.7 


2349.9875 


171.8455 


.2 


2846.321 


189.1243 


.8 


2358.5876 


172.1597 


.3 


2855.7851 


189.4385 


.9 


2367.2035 


172.4738 


.4 


2865.2649 


189.7526 


55. 
.1 


2375.835 

2384.4823 


172.788 
173.1022 


.5 
.6 


2874.7603 
2884.2715 


190.0668 
190.381 


.2 


2393.1452 


173.4163 


.7 


2893.7984 


190.6951 


.3 

.4 


2401.8239 
2410.5183 


173.7305 
174.0446 


.8 
.9 


2903.3411 
2912.8994 


191.0093 
191.3234 


.5 


2419.2283 


174.3588 


61. 


2922.4734 


191.6376 


.6 


2427.9541 


174.673 


.1 


2932.0631 


191.9518 


.7 


2436.6957 


174.9871 


.2 


2941.6686 


192.2659 


.8 


2445.4529 


175.3013 


.3 


2951.2897 


192.5801 


.9 


2454.2258 


175.6154 


.4 


2960.9266 


192.8942 


56. 


2463.0144 


175.9296 


.5 


2970.5791 


193.2084 


.1 


2471.8187 


176.2438 


.6 


2980.2474 


193.5226 


.2 


2480.6388 


176.5579 


.7 


2989.9314 


193.8367 


.3 


2489.4745 


176.8721 


.8 


2999.6311 


194.1509 


.4 


2498.326 


177.1862 


.9 


3009.3465 


194.465 


.5 


2507.1931 


177.5004 


62. 


3019.0776 


194.7792 


.6 


2516.076 


177.8146 


.1 


3028.8244 


195.0934 


.7 


2524.9736 


178.1287 


.2 


3038.5869 


195.4075 


.8 


2533.8889 


178.4429 


.3 


3048.3652 


195.7217 


.9 


2542.8189 


178.757 


.4 


3058.1591 


196.0358 


57. 


2551.7646 


179.0712 


.5 


3067.9687 


196.35 


.1 


2560.726 


179.3854 


.6 


3077.7941 


196.6642 


.2 


2569.7031 


179.6995 


.7 


3087.6341 


196.9783 


.3 


2578.696 


180.0137 


.8 


3097.4919 


197.2925 


.4 


2587.7045 


180.3278 


.9 


3107.3644 


197.6066 


.5 


2596.7287 


180.642 


63. 


3117.2526 


197.9208 


.6 


2605.7687 


180.9562 


.1 


3127.1565 


198.235 


.7 


2614.8244 


181.2703 


.2 


3137.0761 


198.5491 


.8 


2623.8957 


181.5845 


.3 


3147.0114 


198.8633 


.9 


2632.9828 


181.8986 


.4 


3156.9624 


199.1774 


58. 


2642.0856 


182.2128 


.5 


3166.9291 


199.4916 


.1 


2651.2041 


182.527 


.6 


3176.9116 


199.8058 


.2 


2660.3383 


182.8411 


.7 


3186.9097 


200.1199 


.3 


2669.4882 


183.1553 


.8 


3196.9236 


200.4341 


.4 


2678.6538 


183.4694 


.9 


3206.9531 


200.7482 


.5 


2687.8351 


183.7836 


64. 


3216.9984 


201.0624 


.6 


2697.0322 


184.0978 


.1 


3227.0594 


201.3766 


.7 


2703.2449 


184.4119 


.2 


3237.1361 


201.6907 


.8 


2715.4734 


184.7261 


.3 


3247.2284 


202.0049 


.9 


2724.7175 


185.0402 


.4 


3257.3365 


202.319 


59. 


2733.9774 


185.3544 


.5 


3267.4603 


202.6332 


.1 


2743.253 


185.6686 


.6 


3277.5999 


202.9474 


.2 


2752.5443 


185.9827 


.7 


3287.7551 


203.2615 


.3 


2761.8512 


186.2969 


.8 


3297.9261 


203.5757 


.4 


2771.1739 


186.611 


.9 


3308.1127 


203.8898 


.5 


2780.5123 


186.9252 


65. 


3318.315 


204.204 


.6 


2789.8665 


187.2394 


.1 


3328.5331 


204.5182 


.7 


2799.2363 


187.5535 


.2 


3338.7668 


204.8323 


.8 


2808.6218 


187.8677 


.3 


3349.0163 


205.1465 


.9 


2818.0231 


188.1818 


.4 


3359.2815 


205.4606 


60. 


2827.44 


188.496 


.5 


3369.5623 


205.7748 


.1 


2836.8727 


188.8102 


.6 


3379.8589 


206.089 



188 



AREAS AND CIRCUMFERENCES OF CIRCLES. 







Table— {Continued) 






Diam. 


Area. 


Circum. 


Diam. 


Area. 


Circnm. 


.7 


3390.1712 


206.4031 


.2 


3981.5382 


223.6819 


.8 


3400.4993 


206.7173 


.3 


3992.7301 


223.9961 


.9 


3410.843 


207.0314 


.4 


4003.9378 


224.3102 


66. 


3421.2024 


207.3456 


.5 


4015.1611 


224.6244 


.1 


3131.5775 


207.6598 


.6 


4026.4002 


224.9386 


.2 


3441.9684 


207.9739 


.7 


4037.655 


225.2527 


.3 


3452.3749 


208.2881 


.8 


4048.9255 


225.5669 


.4 


3162.7972 


208.6022 


.9 


4060.2117 


225.881 


.5 


3473.2351 


208.9164 


72. 


4071.5136 


226.1952 


.6 


3483.6888 


209.2306 


.1 


4082.8312 


226.5094 


.7 


3494.15.82 


209.5447 


.2 


4094.1645 


226.8235 


.8 


3504.6433 


209.8589 


.3 


4105.5136 


227.1377 


.9 


3515.1441 


210.173 


.4 


4116.8783 


227.4518 


67. 


3525.6606 


210.4872 


.5 


4128.2587 


227.766 


.1 


3536.1928 


210.8014 


.6 


4139.655 


228.0802 


.2 


3546.7407 


211.1155 


.7 


4151.0668 


228.3943 


.3 


3557.3044 


211.4297 


.8 


4162.4943 


228.7085 


.4 


3567.8837 


211.7438 


.9 


4173.9376 


229.0226 


.5 


3578.4787 


212.058 


73. 


4185.3966 


229.3368 


.6 


3589.0895 


212.3722 


.1 


4196.8713 


229.651 


.7 


3599.716 


212.6863 


.2 


4208.3617 


229.9651 


.8 


3610.3581 


213.0005 


.3 


4219.8678 


230.2793 


.9 


3621.016 


213.3146 


.4 


4231.3896 


230.5934 


68. 


3631.6896 


213.6288 


.5 


4242.9271 


230.9076 


.1 


3642.3789 


213.943 


.6 


4254.4804 


231.2218 


.2 


3653.0839 


214.2571 


.7 


4266.0493 


231.5359 


.3 


3663.805 


214.5713 


.8 


4277.634 


231.8501 


.4 


3674.541 


214.8454 


.9 


4289.2343 


232.1642 


.5 


3685.2931 


215.1996 


74. 


4300.8504 


232.4784 


.6 


3696.061 


215.5138 


.1 


4312.4822 


232.7926 


.7 


3706.8445 


215.8279 


.2 


4324.1297 


233.1067 


.8 


3717.6438 


216.1421 


.3 


4335.7928 


233.4209 


.9 


3728.4587 


216.4562 


.4 


4347.4717 


233.735 


69. 


3739.2894 


216.7704 


.5 


4359.1663 


234.0492 


.1 


3750.1358 


217.0846 


.6 


4370.8767 


234.3634 


.2 


3760.9979 


217.3987 


.7 


4382.6027 


234.6775 


.3 


3771.8756 


217.7129 


.8 


4394.3444 


234.9917 


.4 


3782.7691 


218.027 


.9 


4406.1019 


235.3058 


.5 


3793.6783 


218.3412 


75. 


4417.875 


235.62 


.6 


3804.6033 


218.6554 


.1 


4429.6639 


235.9342 


.7 


3815.5439 


218.9695 


.2 


4441.4684 


236.2483 


.8 


3*26-5002 


219.2837 


.3 


4453.2887 


236.5625 


.9 


3847.4722 


219.5978 


.4 


4465.1247 


236.8766 


70. 


3848.46 


219.912 


.5 


4476.9763 


237.1908 


.1 


3859.4635 


220.2262 


.6 


4488.8437 


237.505 


.2 


8870.4826 


220.5403 


.7 


4500.7268 


237.8191 


.3 


3881.5175 


220.8545 


.8 


4512.6257 


238.1333 


.4 


38:'2.56*1 


221.1686 


.9 


4524.5402 


238.4474 


.5 


39d3.6343 


221.4828 


76. 


4536.4704 


238.7616 


.6 


3914.7163 


221.797 


.1 


4548.4163 


239.0758 


.7 
.8 


3925.814 
3936.9275 


222.1111 

222.4253 


.2 
.3 


4560.378 
4572.3553 


239.3899 
239.7041 


.9 


3948.9566 


222.7394 


.4 


4584.3484 


240.0182 


71. 


3959.2014 


223.0536 


.5 


4596.3571 


240.3324 


.1 


3970.3619 


223.3678 


.6 


4608.3816 


240.6466 



AREAS AND CIRCUMFERENCES OP CIRCLES. 



189 



Diam. 



77. 



.7 
.8 
.9 

!l 

.2 
.3 
.4 
.5 
.6 
.7 



.9 

78. 
.1 
.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 

79. 
.1 
.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 

80. 
.1 
.2 
.3 
.4 
.5 
.6 
.7 



81. 
.1 
.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 

82. 
.1 



Area. 



4620.4218 

4632.4777 

4644.5493 

4656.6366 

4668.7396 

4680.8583 

4692.9928 

4705.1429 

4717.3087 

4729.4903 

4741.6876 

4753.9005 

4766.1292 

4778.3736 

4790.6337 

4802.9095 

4815.201 

4827.5082 

4839.8311 

4852.1698 

4864.5241 

4876.8942 

4889.2799 

4901.6814 

4914.0986 

4926.5315 

4938.98 

4951.4443 

4963.9243 

4976.4201 

4988.9315 

5001.4586 

5014.0015 

5026.56 

5039.1343 

5051.7242 

5064.3299 

5076.9513 

5089.5883 

5102.2411 

5114.9096 

5127.5939 

5140.2938 

5153.0094 

5165.7407 

5178.4878 

5191.2505 

5204.0289 

5216.8231 

5229.633 

5242.4586 

5255.2999 

5268.1569 

5281.0296 

5293.918 



Ta"ble— ( Continued). 

Diam. I 



Circum. 



240 ..9607 

241.2749 

241.589 

241.9032 

242.2174 

242,5315 

242,8457 

243,1598 

243.474 

243.7882 

244.1023 

244.4165 

244.7306 

245.0448 

245.359 

215.6731 

245.9873 

246.3014 

246.6156 

246.9298 

247.2439 

247.5581 

247.8722 

248.1864 

248.5006 

248.8147 

249.1289 

249.443 

249.7572 

250.0714 

250.3855 

250.6997 

251.0138 

251 .328 

251.6422 

251.9563 

252.2705 

252.5846 

252.8988 

253.213 

253.5271 

253.8413 

254.1554 

254.4696 

254.7838 

255.0979 

255.4121 

255.7262 

256.0404 

256.3546 

256.6687 

256.9829 

257.297 

257.6112 

257.9254 



Area. 



Circum. 



.2 
.3 

.4 
.5 
.6 

.7 



83. 



.1 

.2 
.3 
.4 
.5 

.6 

.7 



84. 



85. 



.1 
.2 
.3 
.4 
.5 
.6 
.7 
.8 
.9 

5. 
.1 
.2 
.3 

.4 
.5 
.6 



5306.8221 

5319.742 

5332.6775 

5345.6287 

5358.5957 

5371.5784 

5384.5767 

5397.5908 

5410.6206 

5423.6661 

5436.7273 

5449.8042 

5462.8968 

5476.0051 

5489.1292 

5502.2689 

5515.4244 

5528.5955 

5541.7824 

5554.985 

5568.2033 

5581.4372 

5594.6869 

5607.9523 

5621.2335 

5634.5303 

5647.8428 

5661.1711 

5674.515 
- 5687.8747 
5701.25 
5714.6411 
5728.0479 
5741.4703 
5754.9085 
5768.3624 
5781.8321 
5795.3174 



86. 


5808.8184 


.1 


5822.3351 


.2 


5835.8676 


.3 


5849.4157 


.4 


5862.9796 


.5 


5876.5591 


.6 


5890.1544 


.7 


5903.7654 


.8 


5917.3921 


.9 


5931.0345 


87. 


5944.6926 


.1 


5958.3644 


.2 


5972.0559 


.3 


5985.7612 


.4 


5999.4821 


.5 


6013.2187 


.6 


6026.9711 



258.2395 

258.5537 

258.8678 

259.182 

259.4962 

259/8103 

260.1245 

260.'4386 



190 



AREAS AND CIRCUMFERENCES OF CIRCLES. 



Ta"ble — ( Continued ) . 



Diam. 


Area. 


Circum. 


Diam. 


Area. 


Circum. 


.7 


6040.7392 


275.5183 


.2 


6822.1729 


292.797] 


.8 


6054.5229 


275.8325 


.3 


6836.8206 


293.111,- 


.9 


6068.3224 


275.1466 


.4 


6851.484 


293.425- 


88. 


6082.1376 


276.4608 


.5 


6866.1631 


293.739* 


.1 


6095.9685 


276.775 


.6 


6880.858 


294.053* 


.2 


6109.8151 


277.0891 


.7 


6895.5685 


294. 367 J 


.3 


6123.6774 


277.4033 


.8 


6910.2948 


294.6821 


.4 


6137.5554 


277.7174 


.9 


6925.0367 


294.996: 


.5 


6151.4491 


278.0316 


94. 


6939.7944 


295.310- 


.6 


61Q5.3586 


278.3458 


.1 


6954.5678 


295.624( 


.7 


6179.2837 


278.6599 


' .2 


6969.3569 


295.938- 


.8 


6193.2246 


278.9741 


.3 


6984.1616 


296.252! 


.9 


6207.1811 


279.2882 


.4 


6998.9821 


296.567 


89. 


6221.1534 


279.6024 


.5 


7013.8183 


296.881$ 


.1 


6235.1414 


279.9166 


.6 


7028.6703 


297.195- 


.2 


6249.1451 


280.2307 


.7 


7043.5379 


297.509, 


.3 


6263.1644 


280.5449 


.8 


7058.4212 


297.823' 


.4 


6277.1995 


280.859 


.9 


7073.3203 


298.137* 


.5 


6291.2503 


281.1732 


95. 


7088.235 


298.452 


.6 


6305.3169 


281.4874 


.1 


7103.1655 


298. 766', 


.7 


6319.3991 


281.8015 


.2 


7118.1116 


299.080: 


.8 


6333.497 


282.1157 


.3 


7133.0735 


299.394; 


.9 


6347.6107 


282.4298 


.4 


7148.0511 


299.708( 


90. 


6361.74 


282.744 


.5 


7163.0443 


300.022* 


.1 


6375.8851 


283.0582 


.6 


7178.0533 


300.337 


.2 


6390.0458 


283.3723 


.7 


7193.078 


300. 651 1 


.3 


6404.2223 


283.6865 


.8 


7208.1185 


300.965: 


.4 


6418.4144 


284.0006 


.9 


7223.1746 


301.279- 


.5 


6432.6223 


284.3148 


; 96. 


7238.2464 


301.593( 


.6 


6446.8459 


284.629 


.1 


7253.3339 


301.907* 


.7 


6461.0852 


284.9431 


.2 


7268.4372 


302.2211 


.8 


6475.3403 


285.2573 


.3 


7283.5561 


302.536" 


.9 


6489.611 


285.5714 


.4 


7298.6908 


302.850 


91. 


6503.8974 


285.8856 


.5 


7313.8411 


303.164- 


.1 


6518.1995 


286.1998 


.6 


7329.0072 


303.478( 


.2 


6532.5174 


286.5139 


.7 


7344.189 


303.792' 


.3 


6546.8909 


286.8281 


.8 


7359.3865 


304.1061 


.4 


6561.2002 


287,1422 


.9 


7374.5997 


304.421 


.5 


6575.5651 


287.4564 


97. 


7389.8286 


304.7355 


.6 


6589.9458 


287.7706 


.1 


7405.0732 


305.049- 


.7 


6604.3422 


288.0847 


.2 


7420.3335 


305.363. 


.8 


6618.7543 


288.3989 


.3 


7435.6096 


305.677' 


.9 


6633.1821 


288.713 


.4 


7450.9013 


305.991* 


92. 


6647.6256 


289.0272 


.5 


7466.2087 


306.306 


.1 


6662.0848 


289.3414 


.6 


7481.5319 


306.6205 


.2 


6676 . 5598 


289.6555 


.7 


7496.8708 


306.934, 


.3 


6691.0504 


289.9697 


.8 


7512.2253 


307.248, 


.4 


6705.5567 


290.2838 


.9 


7527.5956 


307.562* 


.5 


6720.0787 


290.598 


98. 


7542.9816 


307.876* 


.6 


6734.6165 


290.9121 


.1 


7558.3833 


308.191 


.7 


6749.17 


291.2263 


.2 


7573.8007 


308.505 


.8 


6763.7391 


291.5405 


.3 


7589.2338 


308.819: 


.9 


6778.324 


291.8546 


.4 


7604.6826 


300.133' 


93. 


6792.9246 


292.1688 


.5 


7620.1471 


389.447* 


.1 


6807.5409 


292.483 


.6 


7635.6274 


309.761* 



AEEAS AND CIRCUMFERENCES OF CIRCLES. 



191 



Taole— (Continued). 



Diam. 


Area. 


Circum. 


Diam. 


Area. . 


Circum. 


.7 


7651.1233 


310.0759 


.4 


7760.0347 


312.275 


.8 


7666.635 


310.3901 


.5 


7775.6563 


312.5892 


.9 


7682.1623 


310.7042 


.6 


7791.2937 


312.9034 


99. 


7697.7054 


311.0184 


.7 


7806.9467 


313.2175 


.1 


7713.2642 


311.3326 


.8 


7822.6154 


313.5317 


.2 


7728.8337 


311.6467 


.9 


7838.2999 


313.8458 


.3 


7744.4288 


311.9609 


100. 


7854. 


314.16 



To Compote the Area or Circumference of a Di- 
ameter greater than any in. the preceding Table, 

See Rules, pages 176 and 181. 

Or, If the Diameter exceeds 100 and is less than 1001, 
Remove the decimal point, and take out the area or circumference 
as for a Whole Number by removing the decimal point, if for the area, 
two places to the right ; and if for the circumference, one place. 

Illustration.— The area of 96.7 is 7344.189 ; hence for 967 it is 731418.9; and 
the circumference of 96.7 is 303.7927, and for 967 it is 3037.927. 



Areas and. Circumferences of Circles. 



From 1 to 50 Feet [advancing by an Inch'], or from 1 to 50 Inches ladvancing 






by a Twelfth], 






Diam. 


Area. 


Circum. 


Diam. 


Area. 


Circum. 




Feet. 


Feet. 




Feet. 


Feet. 


1ft. 


.7854 


3.1416 


sft. 


7.0686 


9.4248 


1 


.9217 


3.4034 


1 


7.4668 


9.6866 


2 


1.069 


3.6652 


2 


7.8758 


9.9484 


3 


1.2272 


3.927 


3 


8.2958 


10.2102 


4 


1.3963 


4.1888 


4 


8.7267 


10.472 


5 


1.5763 


4.4506 


5 


9.1685 


10.7338 


6 


1.7671 


4.7124 


6 


9.6211 


10.9956 


7 


1.969 


4.9742 


7 


10.0848 


11.2574 


8 


2.1817 


5.236 


8 


10.5593 


11.5192 


9 


2.4053 


5.4978 


9 


11.0447 


11.781 


10 


2.6398 


5.7596 


10 


11.541 


12.0428 


11 


2.8853 


6.0214 


11 


12.0483 


12.3046 


2 ft. 


3.1416 


6.2832 


ifc 


12.5664 


12.5664 


1 


3.4088 


6.545 


i 


13.0955 


12.8282 


2 


3.687 


6.8068 


2 


13.6354 


13.09 


3 


3.9761 


7.0686 


3 


14.1863 


13.3518 


4 


4.2761 


7.3304 


4 


14.7481 


13.6136 


5 


4.5869 


7.5922 


5 


15.3208 


13.8754 


6 


4.9087 


7.854 


6 


15.9043 


14.1372 


7 


5.2415 


8.1158 


7 


16.4989 


14.499 


8 


5.5852 


8.3776 


8 


17.1043 


14.6608 


9 


5.9396 


8.6394 


9 


17.7206 


14.9226 


10 


6.305 


8.9012 


10 


18.3478 


15.1844 


11 


6.6814 


9.163 


11 


18.9859 


15.4462 



192 



AREAS AND CIRCUMFERENCES OF CIRCLES, 







Ta"ble— {Continued) 






Diam. 


Area. 


Circum. 


Diam. 


Area. 


Circum, 




Feet. 


Feet. 




Feet. 


Feet. 


5 ft. 


19.635 


15.708 


6 


70.8823 


29.8452 


1 


20.2949 


15.9698 


7 


72.1314 


30.107 


2 


20.9658 


16.2316 


8 


73.3913 


30.3688 


3 


21.6476 


16.4934 


9 


74.6621 


30.6306 


4 


22.3403 


16.7552 


10 


75.9439 


30.8924 


5 


23.0439 


17.017 


11 


77.2365 


31.1542 


6 


23.7583 


17.2788 


10/*. 


78.54 


31.416 


7 


24.4837 


17.5406 


79.8545 


31.6778 


8 


25.201 


17.8024 


2 


81.1798 


31.9396 


9 


25.9673 


18.0642 


3 


82.5161 


32.2014 


10 


26.7254 


18.326 


4 


83.8633 


32.4632 


11 


27.4944 


18.5878 


5 


85.2214 


32.725 


eft 


28.2744 


18.8496 


6 


86.5903 


32.9868 


i 


29.0653 


19.1114 


7 


87.9703 


33.2486 


2 


29.867 


19.3732 


8 


89.3611 


33.5104 


3 


30.6797 


19.635 


9 


90.7628 


33.7722 


4 


31.5033 


19.8968 


10 


92.1754 


34.034 


5 


32.3378 


20.1586 


11 


93.599 


34.2958 


6 


33.1831 


20.4204 


11 ft 
1 


95.0334 


34.5576 


7 


34.0394 


20.6822 


96.4787 


34.8194 


8 


34.9067 


20.944 


2 


97.935 


35.0812 


9 


35.7848 


21.2058 


3 


99.4022 


35.343 


10 


36.6738 


21.4676 


4 


100.8803 


35.6048 


11 


37.5738 


21.7294 


5 


102.3693 


35.8666 


7ft 


38.4846 


21.9912 


6 


103.8691 


36.1284 


1 


39.4064 


22.253 


7 


105.38 


36.3902 


2 


40.339 


22.5148 


8 


106.9017 


36.652 


3 


41.2826 


22.7766 


9 


108.4343 


36.9138 


A 


A9 9Q71 


23.0384 
23.3002 


10 
11 


109.9778 
111.5323 


37.1756 
37.4374 


4: 

5 


<±6. Lot X 

43.2025 


6 


44.1787 


23.562 


12/*. 


113.0976 


37.6992 


7 


45.1659 


23.8238 


114.6739 


37.961 


8 


46.1641 


24.0856 


2 


116.261 


38.2228 


9 


47.1731 


24.3474 


3 


117.8591 


38.4846 


10 


48.193 


24.6092 


4 


119.468 


38.7464 


11 


49.2238 


24.871 


5 


121.088 


39.0082 


8 ft 


50.2656 


25.1328 


6 


122.7187 


39.27 


1 


51.3183 


25.3946 


7 


124.3605 


39.5318 


2 


52.3818 


25.6564 


8 


126.0131 


39.7936 


3 


53.4563 


25.9182 


9 


127.6766 


40.0554 


4 


54.5417 


26.18 


10 


129.351 


40.3172 


5 


55.638 


26.4418 


11 


131.0366 


40.579 


6 


56.7451 


26.7036 


13/*. 


132.7326 


40.8408 


7 


57.8632 


26.9654 


134.4398 


41.1026 


8 


58.9923 


27.2272 


2 


136.1578 


41.3644 


9 


60.1322 


27.489 


3 


137.8868 


41.6262 


10 


61.283 


27.7508 


4 


139.6267 


41.888 


11 


62.4448 


28.0126 


5 


141.3774 


42.1498 


9/*. 


63.6174 


28.2744 


6 


143.1391 


42.4116 


1 


64.801 


28.5362 


7 


144.9117 


42.6734 


2 


65.9954 


28.798 


8 


146.6953 


42.9352 


3 


67.2008 


29.0598 


9 


148.4897 


43.197 


4 


68.417 


29.3216 


10 


150.295 


43.4588 


5 


69.6442 


29.5834 


11 


152.1113 


43.7206 



AREAS AND CIRCUMFERENCES OF CIRCLES. 



193 



Ta"ble — ( Continued) . 



Diam. 


Area. 


Circum. 


Diam. 


Area. 


Circum. 




Feet. 


Feet. 




Feet. 


Feet. 


14/*. 


153.9384 


43.9824 


6 


268.8031 


58.1196 


1 


155.7764 


44.2442 


7 


271.2302 


58.3814 


2 


157.6254 


44.506 


8 


273.6683 


58.6432 


3 


159.4853 


44.7678 


9 


276.1172 


58.905 


4 


161.3561 


45.0296 


10 


278.577 


59.1668 


5 


163.2378 


45.2914 


11 


281.0477 


59.4286 


6 


165.1303 


45.5532 


19/*. 


283.5294 


59.6904 


7 


167.0338 


45.815 


1 


286.0219 


59.9522 


8 


168.9483 


46.0768 


2 


288.5255 


60.214 


9 


170.8736 


46.3386 


3 


291.0398 


60.4758 


10 


172.8098 


46.6004 


4 


293.5651 


60.7376 


11 


174.7569 


46.8622 


5 


296.1012 


60.9994 


15/*. 


176.715 


47.124 


6 


298.6483 


61.2612 


1 


178.684 


47.3858 


7 


301.2064 


61.523 


2 


180.6638 


47.6476 


8 


303.7753 


61.7848 


3 


182.6546 


47.9094 


9 


306.3551 


62.0466 


4 


184.6563 


48.1712 


10 


308.9458 


62.3084 


5 


186.6689 


48.433 


11 


311.5475 


62.5702 


6 


188.6924 


48.6948 


20/*. 
1 


314.16 


62.832 


7 


190.7267 


48.9566 


316.7834 


63.0938 


8 


192.7721 


49.2184 


2 


319.4178 


63.3556 


9 


194.8283 


49.4802 


3 


322.0631 


63.6174 


10 


196.8954 


49.742 


4 


324.7193 


63.8792 


11 


198.9734 


50.0038 


5 


327.3864 


64.141 


16/*. 


201.0624 


50.2656 


6 


330.0643 


64.4028 


1 


203.1622 


50.5274 


7 


332.7532 


64.6646 


2 


205.273 


50.7892 


8 


335.4531 


64.9264 


3 


207.3947 


51.051 


9 


338.1638 


65.1882 


4 


209.5273 


51.3128 


10 


340.8854 


65.45 


. 5 


211.6707 


51.5746 


11 


343.618 


65.7118 


6 


213.8252 


51.8364 


21/*. 


346.3614 


65.9736 


7 


215.9904 


52.0982 


349.1157 


66.2354 


8 


218.1667 


52.36 


2 


351.881 


66.4972 


9 


220.3538 


52.6218 


3 


354.6572 


66.759 


10 


222.5518 


52.8836 


4 


357.4442 


67.0208 


11 


224.7607 


53.1454 


5 


360.2422 


67.2826 


17/*. 


226.9806 


53.4072 


6 


363.0511 


67.5444 


1 


229.2113 


53.669 


7 


365.8709 


67.8062 


2 


231.453 


53.9308 


8 


368.7017 


68.068 


3 


233.7056 


54.1926 


9 


371.5433 


68.3298 


4 


235.9691 


54.4544 


10 


374.3958 


68.5916 


5 


238.2434 


54.7162 


11 


377.2592 


68.8534 


6 


240.5287 


54.978 


22/*. 
1 


380.1336 


69.1152 


7 


242.8249 


55.2398 


383.0188 


69.377 


8 


245.1321 


55.5016 


2 


385.915 


69.6388 


9 


247.4501 


55.7634 


3 


388.8221 


69.9006 


10 


249.779 


56.0252 


4 


391.74 


70.1624 


11 


252.1188 


56.287 


5 


394.6089 


70.4242 


18/*. 


254.4696 


56.5488 


6 


397.6087 


70.686 


1 


256.8312 


56.8106 


7 


400.5594 


70.9478 


2 


259.2038 


57.0724 


8 


403.5211 


71.2096 


3 


261.5873 


57.3342 


9 


406.4936 


71.4714 


4 


263».9817 


57.596 


10 


409.177 


71 .7332 


5 


266.3869 


57.8578 


11 
I 


412.4713 


71.995 



194 



AREAS AND CIRCUMFERENCES OF CIRCLES. 



Table— ( Continued) . 



Diam. 


Area. 


1 Circum. 


1 1 Diam. 


I Area. 


Circum. 




Feet. 


Feet. 




Feet. 


Feet. 


23/*. 


415.4766 


72.2568 


6 


593.9587 


86.394 


1 


418.4927 


72.5186 


7 


597.5639 


86.6558 


2 


421.5198 


72.7804 


8 


601.18 


86.9176 


3 


424.5578 


73.0422 


9 


604.8071 


87.1794 


4 


427.6067 


73.304 


10 


608.445 


87.4412 


5 


430.6664 


73.5658 


11 


612.0938 


87.703 


6 


433.7371 


73.8276 


28 ft. 


615.7536 


87.9648 


7 


436.8187 


74.0894 


1 


619.4242 


88.2266 


8 


439.917 


74.3512 


2 


623.1058 


88.4884 


9 


443.0147 


74.613 


3 


626.7983 


88.7502 


10 


446.129 


74.8748 


4 


630.5016 


89.012 


11 


449.2542 


75.1366 


5 


634.2159 


89.2738 


%ift 


452,3904 


75.3984 


6 


637.9411 


89.5356 


1 


455.5374 


75.6602 


7 


641.6772 


89.7974 


2 


458.6954 


75.922 


8 


645.4243 


90.0592 


3 


461.8613 


76.1838 


9 


649.1822 


90.321 


4 


465.044 


76.4456 


10 


652.951 


90.5828 


5 


468.2347 


76.7074 


11 


656.7307 


90.8446 


6 


471.4363 


76.9692 


29/*. 


660.5214 


91.1064 


7 


474.6488 


77.231 


664.3229 


91.3682 


8 


477.8723 


77.4928 


2 


668.1354 


91.63 


9 


481.1066 


77.7546 


3 


671.9588 


91.8918 


10 


484.3518 


78.0164 


4 


675.7931 


92.1536 


11 


487.6076 


78.2782 


5 


679.6382 


92.4154 


25/*. 


490.875 


78.54 


6 


683.4943 


92.6772 


1 


494.1529 


78.8018 


7 


687.3613 


92.939 


2 


497.4418 


79.0636 


8 


691.2393 


93.2008 


3 


500.7416 


79.3254 


9 


695.1281 


93.4626 


4 


504.0523 


79.5872 


10 


699.0278 


93.7244 


5 


507.3738 


79.849 


11 


702.9384 


93.9862 


6 


510.7063 


80.1108 


30/*. 


706.86 


94.248 


7 


514.0413 


80.3726 


1 


710.7924 


94.5098 


8 


517.404 


80.6344 


2 


714.7358 


94.7716 


9 

10 


520.7693 
524.1454 


80.8962 
81.158 


3 
4 


718.6901 
722.6553 


95.0334 
95.2952 


11 


527.5324 


81.4198 


5 


726.6313 


95.557 


23/^. 


530.9304 


81.6816 


6 


730.6183 


95.8188 


1 


534.3313 


81.9434 


7 


734.6162 


96.0806 


2 


537.759 


82.2052 


8 


738.6251 


96.3424 


3 


541.1897 


82.467 


9 


742.6448 


96.6042 


4 


514.6313 


82.7288 


10 


746.6754 


96.866 


5 


548.0837 


82.9906 


11 


750.7164 


97.1278 


6 


551.5471 


83.2524 


31/*. 


754.7694 


97.3896 


7 


555.0214 


83.5142 


1 


758.8327 


97.6514 


8 


558.5066 


83.776 


2 


762.907 


97.9132 


9 


562.0028 


84.0378 


3 


766.9022 


98.175 


10 


565.5098 


84.2996 


4 


771.0883 


98.4368 


11 


569.0277 


84.5614 


5 


775.1952 


98.6986 


27/*. 


572.5566 


84.8232 


6 


779.3131 


98.9604 


1 


576.0963 


85.085 


7 


783.4419 


99.2222 


2 


579.6467 


85.3468 


8 


787.5817 


99.484 


3 


583.2086 


85.6086 


9 


791.7323 


99.7458 


4 


586.781 


85.8704 


10 


795.8938 


100.0076 


5 


590.3644 


86.1322 l| 


11 


800.0662 1 


100.2694 



AREAS AND CIRCUMFERENCES OF CIRCLES. 



195 



Ta"ble— {Continued). 



Area. 



Feet. 

804.2496 
808.4439 
812.649 
816.8651 
821.092 
825.3299 
829.5787 
833.8384 
838.1091 
842.3906 
846.683 
850.9863 
855.3006 
859.6257 
863.9618 
868.3088 
872.6667 
877.0354 
881.4151 
885.8057 
890.2073 
894.6197 
899.043 
903.4772 
907.9224 
912.3784 
916.8454 
921.3233 
925.812 
930.3117 
934.8223 
939.3439 
943.8763 
948.4196 
952.9738 
957.5392 
962.115 
966.7019 
971.2998 
975.9086 
980.5287 
985.1588 
989.8005 
994.4527 
999.116 
1003.7903 
1008.4754 
1013.1714 
1017.8784 
1022.5962 
1027.325 
1032.0647 
1036.8153 
1041.5767 



Circum. 



Feet 

100.5312 

100.793 

101.0548 

101.3166 

101.5784 

101.8402 

102.102 

102.3638 

102.6256 

102.8874 

103.1492 

103.411 

103.6728 

103.9346 

104.1964 

104.4582 

104.72 

104.9818 

105.2436 

105.5054 

105.7672 

106.029 

106.2908 

106.5526 

106.8144 

107.0762 

107.338 

107.5998 

107.8616 

108.1234 

108.3852 

108.647 

108.9088 

109.1706 

109.4324 

109.6942 

109.956 

110.2178 

110.4796 

110.7414 

111.0032 

111.265 

111.5268 

111.7886 

112.0504 

112.3122 

112.574 

112.8358 

113.0976 

113.3594 

113.6212 

113.883 

114.1448 

114.4066 





Feet. 


6 


1046.3491 


' 7 


1051.1324 


8 


1055.9266 


9 


1060.7318 


10 


1065.5478 


11 


1070.3747 


37 ft. 


1075.2126 


1 


1080.0613 


2 


1084.921 


3 


1089.7916 


4 


1094.6731 


5 


1099.5654 


6 


1104.4687 


7 


1109.3839 


8 


1114.308 


9 


1119.2441 


10 


1124.191 


11 


1129.1489 


38 ft. 


1134.1176 


1 


1139.0972 


2 


1144.0878 


3 


1149.0893 


4 


1154.1017 


5 


1159.1249 


6 


1164.1591 


7 


1169.2042 


8 


1174.2603 


9 


1179.3272 


10 


1184.405 


11 


1189.4937 


39//. 


1194.5934 


1 


1199.7039 


2 


1204.8254 


3 


1209.9578 


4 


1215.101 


5 


1220.2552 ' 


6 


1225.4203 


7 


1230.5963 


8 


1235.7833 


9 


1240.9811 


10 


1246.1898 


11 


1251.4094 


40//. 


1256.64 


1 


1261.8814 


2 


1267.1338 


3 


1272.3971 


4 


1277.6712 


5 


1282.9563 


6 


1288.2523 


7 


1293.5592 


8 


1298.877 


9 


1304.2058 


10 


l:-W).5!f)4 


11 1 


1.314.8959 



196 



AREAS AND CIRCUMFERENCES OF CIRCLES. 







T a"bl e— ( Continued) . 






Diam. | 


Area. 


Circum. 


Diam. 


Area. 


Circum. 




Feet. 


Feet. 




Feet. 


Feet. 


tifl. 


1320.2574 


128.8056 


6 


1625.9743 


142.9428 


1325.6297 


129.0674 


7 


1631.9357 


143.2046 


2 


1331.013 


129.3292 


.8 


1637.9081 


143.4664 


3 


1336.4072 


129.591 


9 


1643.8913 


143.7282 


4 


1341.8123 


129.8528 


10 


1649.8854 


143.99 


5 


1347.2282 


130.1146 


11 


1655.8904 


144.251S 


6 


1352.6551 


130.3764 


46/*. 


1661.9064 


144.51CC 


7 


1358.0929 


130.6382 


1 


1667.9332 


144.775^ 


8 


1363.5416 


130.9 


2 


1673.971 


145.037* 


9 


1369.0013 


131.1618 


3 


1680.0197 


145.299 


10 


1374.4718 


131.4236 


4 


1686.0792 


1451560* 


11 


1379.9532 


131.6854 


5 


1692.1497 


145.822( 


42./*. 


1385.4456 


131.9472 


6 


1698.2311 


146.0844 


1 


1390.9488 


132.209 


7 


1704.3334 


146.3461 


2 


1396.463 


132.4708 


8 


1710.4267 


146.608 


3 


1401.9881 


132.7326 


9 


1716.5408 


146.869$ 


4 


1407.5241 


132.9944 


10 


1722.6658 


147. 13K 


5 


1413.0709 


133.2562 


11 


1728.8017 


147.393- 


6 


1418.6287 


133.518 


47/*. 


1734.9486 


147.655; 


7 


1424.1974 


133.7798 


1 


1741.1063 


147.917 


8 


1429.777 


134.0416 


2 


1747.275 


148.178* 


9 


1435.3676 


134.3034 


3 


1753.4546 


148.440( 


10 


1440.969 


134.5652 


4 


1759.6451 


148.702< 


11 


1446.5813 


134.827 


5 


1765.8464 


148.964 


43/*. 
1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 


1452.2046 

1457.8387 

1463.4838 

1469.1398 

1474.8066 

1480.4844 

1486.1731 

1491.8717 

1497.5833 

1503.3047 

1509.037 

1514.7802 


135.0888 

135.3506 

135.6124 

135.8742 

136.136 

136.3978 

136.6596 

136.9214 

137.1832 

137.445 

137.7068 

137.9786 


6 

7 

8 

9 

10 

11 

48/*. 

2 

d 

4 
5 

6 


1772.0587 

1778.2819 

1784.516 

1790.7611 

1797.017 

1803.2838 

1809.5616 

1815.8502 

1822.1498 

1828.4603 

1834.7817 

1841.1139 

1847.4571 


149.226 
149.4871 
149.749 
150.011 
150.273 
150.535 
150.796 
151.058 
151.320 
151.582 
151.844 
152.105 
152.367 


44/*. 


1520.5344 


138.2304 


7 


1853.8112 


152.629 


1 


1526.2994 


138.4922 


8 


1860.1763 


152.891 


2 


1532.0754 


138.754 


9 


1866.5522 


153.153 


3 


1537.8623 


139.0158 


10 


1872.939 


153.414 


4 


1543.66 


139.2776 


11 


1879.3367 


153.676 


5 


1549.4687 


139.5394 


49/*. 
1 


1885.7454 


153.9^8 


6 


1555.2883 


139.8012 


1892.1649 


154.200 


7 


1561.1188 


140.063 


2 


1898.5954 


154.462 


8 


1566.9603 


140.3248 


3 


1905.0368 


154.723 


9 


1572.8126 


140.5866 


4 


1911.4897 


154.985 


10 


1578.6756 


140.8484 


5 


1917.9522 


155.247 


11 


1584.5499 


141.1102 


6 


1924.4263 


155.509 


45/*. 


1590.435 


141.372 


7 


1930.9113 


155.771 


1 


1596.3309 


141.6338 


8 


1937.4073 


156.032 


2 


1602.2378 


141.8956 


9 


1943.9142 


156.294 


3 


1608.1556 


142.1574 


10 


1950.4318 


156.55f 


4 


1614.0843 


142.4192 


11 


1956.9604 


156.818 


5 


1620.0238 


142.681 


50/*. 


1963.5 


157.08 



SIDES OF EQUAL SQUAKES. 



197 



Table of tlie Sides of Squares- equal in Area to a 
Circle of any Diameter. 









Fkom 1 


. TC 100. 








Diara. 


Side of Sq. Pi 


am. 


Side of Sq. 


Diam. 


Side of Sq. 


] Diam. | Side of Sq. 


1. 


.8862 14 




12.4072 


27. 


23.9281 


1 40. 


35.4491 


;M 


1.1078 


'hi 


12.6287 


H 


24.1497 


■A 


35.6706 


-a- 


1.3293 


■x 


12.8503 


-A 


24.3712 


-A 


35.8922 


•% 


1.5509 


■% 


13.0718 


M 


24.5928 


& 


36.1137 


2. 


1.7724 15 




13.2934 


28. 


24.8144 


41. 


36.3353 


-M 


1.994 


'■k 


13.515 


'U 


25.0359 


-A 


36.5569 


*Vi 


2.2156 


■H 


13.7365 


-A 


25.2575 


-A 


36.7784 


•K 


2.4371 


■ K 


13.9581 


•X 


25.479 


>% 


37. 


3. 


2.6587 16 




14.1796 


29. 


25.7006 


42. 


37.2215 


-A 


2.8802 


'■X 


14.4012 


$L 


25.9221 


•A 


37.4431 


-A 


3.1018 


..% 


14.6227 


• X A 


26.1437 


-A 


37.6646 


•X 


3.3233 


■% 


14.8443 


M 


26.3653 


•X 


37.8862 


4. 


3.5449 17 




15.0659 


30. 


26.5868 


43. 


38.1078 


-A 


3.7665 


'•% 


15.2874 


M 


26.8084 


'$i 


38.8293 


-A 


3.988 


■■% 


15.509 


•& 


27.0299 


•A* 


38.5509 


•X 


4.2096 


.% 


15.7305 


•X 


27.2515 


•X 


38.7724 


5. 


4.4311 18 




15.9521 


31. 


27.473 


44. 


38.994 


rX 


4.6527 


'■% 


16.1736 


& 


27.6946 


•A 


39.2155 


r& 


4.8742 


■ l A 


16.3952 


-A 


27.9161 


•sl 


39.4371 


•X 


5.0958 


■ % 


16.6168 


•k 


28.1377 


•% 


39.6587 


6. 


5.3174 19 




16.8383 


32. 


28.3593 


45. 


39.8802 


•U 


5.5389 


'x 


17.0599 


•t 


28.5808 


-A 


40.1018 


4* 


5.7605 


% 


17.2814 


•S 


28.8024 


. -A 


40.3233 


■X 


5.982 


% 


17.503 


•X 


29.0239 


M 


40.5449 


7. 


6.2036 20 




17.7245 


33. 


29.2455 


46. 


40.7664 


•X 


6.4251 


x 


17.9461 


•U 


29.467 


-A 


40.988 


♦X 


6.6467 


x 


12.1677 


-si 


29.6886 


•A 


41.2096 


•X 


6.8683 


% 


18.3892 


>% 


29.9102 


m 


41 .4311 


8. 


7.0898 21 




18.6108 


34. 


30.1317 


47. 


41.6527 


-X 


7.3114 


X 


18.8323 


$t 


30.3533 


W 


41.8742 


*i? 


7.5329 


X 


19.0539 


•X 


30.5748 


-A 


42.0958 


-Ik 


7.7545 


% 


19.2754 


>X 


30.7964 


• X 


42.3173 


9. 


7.976 22 




19.497 


35. 


31.0179 


48. 


42.5839 


•V 


8.1976 


X 


19.7185 


>1 


31.2395 


-A 


42.7604 


•X 


8.4192 


X 


19.9401 


- X A 


31.4611 


-A 


42.982 


•X 


8.6407 


% 


20.1617 


M 


31.6826 


X 


43.2036 


10. 


8.8623 23. 




20.3832 


36. 


31.9042 


49. 


43.4251 


•X 


9.0838 


X 


20.6048 


-A 


32.1257 


.% 


43.6467 


..k 


9.3054 


% 


20.8263 


-A 


32.3473 


-A 


43.8682 


•X 


9.5269 


X 


21.0479 


•S 


32.5688 


$L 


44.0898 


Li. 


9.7485 24. 




21.2694 


37. 


32.7904 


50. 


44.3113 


•X 


9.97 


X 


21.491 


•A 


33.0112 


ik 


44.5329 


•X 


10.1916 


X, 


21.7126 


-K 


33.2335 


-A 


44.7545 


-X 


10.4132 


% 


21.9341 


W 


33.4551 


•X 


44.976 


L2. 


10.6347 25. 




22.1557 


38. 


33.6766 


51. 


45.1976 


•K 


10.8563 


X 


22.3772 


-A 


33.8982 


-A 


45.4191 


•X 


11.0778 


X 


22.5988 


'K 


84.1197 


-A 


45.6407 


•X 


11.2994 


% 


22.8203 


•X 


34.3413 


•X 


45.8622 


13. 


11.5209 26. 




23.0419 


39. 


34.5628 


52. 


46.0838 


•X 


11.7425 


X 


23.2634 


x 


34.7884 , 


.% 


46.3064 


•X 


11.9641 


X 


23.485 


-A 


35.006 


.', 


46.5269 


•X 


12.1856 


% 


23.7066 


- -X 


35.2275 1 


• X 


46.7485 



R* 



198 



SIDES OF EQUAL SQUARES. 



Side of Sq. || Diam. 



Table— (Continued). 

Side of Sq. Diam. 



46.97 

47.1916 

47.4131 

47.6347 

47.8562 

48.0778 

48.2994 

48.5209 

48.7425 

48.964 

49.1856 

49.4071 

49.6287 

49.8503 

50.0718 

50.2934 

50.5149 

50.7365 

50.958 

51.1796 

51.4012 

51.6227 

51.8443 

52.0658 

52.2874 

52.5089 

52.7305 

52.9521 

53.1736 

53.3952 

53.6167 

53.8383 

54.0598 

54.2814 

54.503 

54.7245 

54.9461 

55.1676 

55.3892 

55.6107 

55.8323 

56.0538 

56.2754 

56.497 

56.7185 

56.9401 

57.1616 

57.3832 



6b. 



66. 



■Y 
% 



67. 



68. 



69. 



M 

% 



70. 



J* 

■.§ 

H 



71. 



72. 






■8 

M. 



-7± 



76. 






57.6047 

57.8263 

58.0479 

58.2694 

58.491 

58.7125 

58.9341 

59.1556 

59.3772 

59.5988 

59.8203 

60.0419 

60.2634 

60.485 

60.7065 

60.9281 

61.1497 

61.3712 

61.5928 

61.8143 

62.0359 

62.2574 

62.479 

62.7006 

62.9221 

63.1437 

63.3652 

63.5868 

63.8083 

64.0299 

64.2514 

64.4730 

64.6946 

64.9161 

65.1377 

65.3592 

65.5808 

65.8023 

66.0239 

66.2455 

66.467 

66.6886 

66.9104 

67.1317 

67.3532 

67.5748 

67.7964 

68.0179 



80 



81 



82 



83 



84 



85 



86 



8fc 



Side of Sq. Di 


*m. 


Side of Sq. 


68.2395 89 




78.8742 


68.461 . 


•3< 


79.0957 


68.6826 


Y 


79.3173 


68.9041 


% 


79.5389 


69.1257 90 




79.7604 


69.3473 


\k 


79.982 


69.5688 


>k 


80.2035 


69.7904 


% 


80". 4251 


70.0119 91 




80.6467 


70.2335 


y± 


80.8682 


70.455 


Yi 


81.0898 


70.6766 


% 


81.3113 


70.8981 92 




81.5329 


71.1197 


•H 


81.7544 


71.3413 


-y 


81.976 


71.5628 


% 


82.1975 


71.7844 | 93 




82.4191 


72.0059 


x 


82.6407 


72.2275 


■Y 


82.8622 


72.4491 


• 3 A 


83.0838 


72.6706 94 




83.3053 


72.8921 


Y 


83.5269 


73.1137 


Y 


83.7484 


73.3353 


-% 


83.970 


73.5568 95 




84.1916 


73.7784 


'U 


84.4131 


73.9999 


-Y 


84.6347 


74.2215 


% 


84.8562 


74.4431 96 




85.0778 


74.6647 


Y 


85.2993 


74.8862 


Y 


85.5209 


75.1077 


% 


85.7425 


75.3293 97 




85.9646 


75.5508 


'y 


86.185 


75.7724 


Y 


86.4071 


75.9934 


% 


86.6289 


76.2155 98 




86.8502 


76.4371 


'% 


87.07j8 


76.6586 


Y 


87,2933 


76.8802 


% 


87.5449 


77.1017 99 




87.7364 


77.3233 


X 


87.958 


77.5449 


Y 


88.1796 


77.7664 


% 


87.4011 


77.988 100 




88.6227 


78.2095 


Y 


88.8442 


78.4316 


Y 


89.0658 


78.6526 


% 


89.2874 



APPLICATION OF THE TABLE. 



To Ascertain, a Square that sliall have the same Area as 
a Given Circle. 

Ilt/ustration. — What is the side of a square that has the same area as a circle of 
73/ 5 inches? 

By table of Area?, page 174, opposite to 73J^, is 4214.11 , and in this table i- G4.9161, 
the si-le of a square having the sama area as a circle of T3j^ inches in diameter. 



LENGTHS OF CIRCULAR ARCS 



199 



Taole of tlie Lengths of Circular Arcs. 

Hie Diameter of a Circle assumed to be Unity, and divided into 1000 

equal Parts. 

H'ght. | Lensth. H'ght j Length. | H'ght. Length. H'ght. Length, 



.1 


1.02645 


.101 


1.02698 


.102 


1.02752 


.103 


1.02806 


.101 


1.0286 


.105 


1.02914 


.106 


1.0297 


.107 


1.03026 


.108 


1.03082 


.109 


1.03139 


.110 


1.03196 


.111 


1.03254 


.112 


1.03312 


.113 


1.03371 


.114 


1.0343 


.115 


1.0349 


.116 


1.03551 


.117 


1.03611 


.118 


1.03672 


.119 


1.03734 


.12 


1.03797 


.121 


1.0386 


.122 


1.03923 


.123 


1.03987 


.124 


1.04051 


.125 


1.04116 


.126 


1.04181 


.127 


1.04247 


.128 


1.04313 


.129 


1.0438 • 


.13 


1.04447 


.131 


1.04515 


.132 


1.04584 


.133 


1.04652 


.131 


1.04722 


.135 


1.04792 


.136 


1.04862 


.137 


1.04932 


.138 


1.05003 


.139 


1.05075 


.11 


1.05147 


.141 


1.0522 


.142 


1.05293 


.143 


1.05367 


.144 


1.05441 


.145 


1.05516 


.146 


1.05591 


.147 


1.05667 


.148 


1.05743 


,149 


1.05819 


.15 


1.05896 


.151 


1.05973 



.152(1 
. 153 1 1 



.154 
.155 
.156 
.157 
.158 
.159 
.16 
161 
.162 
163 
,164 
,165 
,166 
.167 
,168 
.169 
.17 
171 
,172 
.173 
174 
.175 
.176 
.177 
.178 
.179 
.18 
.181 
.182 
.183 
.184 
.185 
.186 
.187 
.188 
.189 
.19 
.191 
.192 
.193 
.194 
.195 
.196 
.197 
.198 
.199 
.2 

.201 
.202 
.203 



.06051 
0613 
.06209 
.06288 
06368 
06449 
0653 
06611 
06693 
08775 
06858 
06941 
07025 
07109 
07194 
07279 
07365 
07451 
07537 
07624 
07711 
07799 
07888 
07977 
08066 
08156 
08246 
08337 
08428 
08519 
08611 
08704 
08797 
0889 
08984 
09079 
09174 
09269 
09365 
09461 
09557 
09654 
09752 
.0985 
09949 
10048 
10147 
10247 
10348 
.10447 
.10548 
1065 



.204 

.205 

.206 

.207 

.208 

.209 

.21 

.211 

.212 

.213 

.214 

.215 

.216 

.217 

.218 

.219 

.22 

.221 

.222 

.223 

.224 

.225 

.226 

.227 

.228 

.229 

.23 

.231 

.232 

.233 

.234 

.235 

.236 

.237 

.238 

.239 

.24 

.241 

.242 

.243 

.244 

.245 

.246 

.247 

.248 

.249 

.25 

.251 

.252 

.253 

.254 

.255 



1.10752 

1.10855 

1.10958 

1.11062 

1.11165 

1.11269 

1.11374 

1.11479 

1.11584 

1.11692 

1.11796 

1.11904 

1.12011 

1.12118 

1.12225 

1.12334 

1.12445 

1.12556 

1.12663 

1.12774 

1.12885 

1.12997 

1.13108 

1.13219 

1.13331 

1.13444 

1.13557 

1.13671 

1.13786 

1.13903 

1.1402 

1.14136 

1.14247 

1.14363 

1.1448 

1.14597 

1.14714 

1.14831 

1.14949 

1.15067 

1.15186 

1.15308 

1.15429 

1.15549 

1.1567 

1.15791 

1.15912 

1.16033 

1.16157 

1.16279 

1.16-102 

1.16526 



.256 
.257 
.258 
.259 
.26 
.261 
.262 
.263 
.264 
.265 
.266 
.267 
.268 
.269 
.27 
.271 
.272 
.273 
.274 
.275 
.276 
.277 
.278 
.279 
.28 ■ 
.281 
282 
283 
.284 
.285 
.286 
.287 
.288 
.289 
.29 
.291 
.292 
.293 
.294 
.295 
.296 
.297 
.298 
.299 
.3 

.301 
.302 
.303 
.304 
.305 
.306 
.307 



1.16649 

1.16774 

1.16899 

1.17024 

1.1715 

1.17275 

1.17401 

1.17527 

1.17655 

1.17784 

1.17912 

1.1804 

1.18162 

1.18294 

1.18428 

1.18557 

1.18688 

1.18819 

1.18969 

1.19082 

1.19214 

1.19345 

1.19477 

1.1961 

1.19743 

1.19887 

1.20011 

1.20146 

1.20282 

1.20419 

1.20558 

1.20696 

1.20828 

1.20967 

1.21202 

1.21239 

1.21381 

1.2152 

1.21658 

1.21794 

1.21926 

1 : . 22061 

1.22203 

1.22347 

1.22495 

1.22635 

1 .22776 

1.22918 

1.33061 

1.23205 

1.&3849 

1.23494 



H'ght. Length. 



.308 1.23636 



.309 
.31 
.311 
.312 



1.2378 
1.23925 
1.2407 
1.24216 

.313 | 1.2436 

.31411.24506 

.315 

.316 

.317 

.318 

.319 

.32 

.321 

.322 

.323 

.324 

.325 

.326 

.327 

.328 

.329 

.33 

.331 

.332 

.333 

.334 

.335 

.336 

.337 

.338 

.339 

.34 

.341 

.342 

.343 

.344 

.345 

,346 

.347 

.348 

.349 

.35 

.351 

.352 

.353 

,354 

.355 

,356 

,357 

.358 

.359 



200 



LENGTHS OF CIECULAE ARCS. 



Ta"ble— (Continued). 



H»ght. Length. 


H'ght 


I Length. 


H'ght. 


Length 


H'ght. 

.445 


Length. 


H'ght. 


Length. 


.36 1.31599 


.389 


1.36425 


.417 1.41324 


1.46441 


.473 1.51764 


.361 1.31761 


.39 


1.36596 


.4J8 1.41503 


.446 


1.46628 


.474 1.51958 


.362 1.31923 


.391 


1.36767 


.419 1.41682 


.447 


1.46815 


.475 1.52152 


.363 1.32086 


.392 


1.36939 


.42 1.41861 


.448 


1.47002 


.476 1.52346 


.364 1.32249 


.393 


1.37111 


.421 1.42041 


.449 


1.47189 


.477 


1.52541 


.365 1.32413 


.394 


1.37283 


.422 1.42222 


.45 


1.47377 


.478 


1.52736 


.366 1.32577 


.395 


1.37455 


.423 1.42402 


.451 


1.47565 


.479 


1.52931 


.367 1.32741 


.396 


1.37628 


.424 1.42583 


.452 


1.47753 


.48 |1.53126 


.368 1.32905 


.397 


1.37801 


.425 1.42764 


.453 


1.47942 


.481 1.53322 


.369 1.33069 


.398 


1.37974 


.426 1.42945 


.454 


1.48131 


.482 1.53518 


.37 1.33234 


.399 


1.38148 


.427 1.43127 


.455 


1.4832 


.483 1.53714 


.371 1.33399 


.4 


1.38322 


.428 1.43309 


.456 


1.48509 


.484 1.5391 


.372 1.33564 


.401 


1.38496 


.429 


1.43491 


.457 


1.48699 


.485 1.54106 


.373 1.3373 


.402 


1.38671 


.43 


1.43673 


.458 


1.48889 


.486 1.54302 


.374 1.33896 


.403 


1.38846 


.431 


1.43856 


.459 


1.49079 


.487 


1.54499 


.375 1.34063 


.404 


1.39021 


.432 


1.44039 


.46 


1.49269 


.488 


1.54696 


.376 1.34229 


.405 


1.39196 


.433 1.44222 


.461 


1.4946 


.489 


1.54893 


.377 1.34396 


.406 


1.39372 


.434 1.44405 


.462 


1.49651 


.49 


1.5509 


.378 1.34563 


.407 


1.39548 


.435 1.44589 


.463 


1.49842 


.491 


1.55288 


.379 1.34731 


.408 


1.39724 


.436 1.44773 


.464 


1.50033 


.492 1.55486 


.38 1.34899 


.409 


1.399 


.437 1.44957 


.465 


1.50224 


.493 1.55685 


.381 1.35068 


.41 


1.40077 


.438 1 1.45142 


466 


1.50416 


.494 1.55854 


.382 1.35237 


.411 


1.40254 


.439 


1.45327 


.467 


1,50608 


.495 1.56083 


.383 1.35406 


.412 


1.40432 


.44 


1.45512 


.468 


1.508 


.496 1.56282 


.384 1.35575 


.413 


1.406 


.441 


1.45697 


..469 


1.50992 


.497 1.56481 


.385 1.35744 


.414 


1.40788 


.442 i 1.45883 


.47 


1.51185 


.498 1.5668 


.386 1.35914 


.415 


1.40966 


.443 1.46069 


.471 1.51378 


-499 1.56879 


.387 1.36084 


.416 


1.41145 


.444 1.46255 


: .472 


1.51571 


.5 1.57079 


.388,1.36254 






















To Ascertain, tlie Length, of an J^.to of a Circle "by- 
the preceding Table. 

Eule. — Divide the height by the base, find the quotient in the col- 
umn of heights, and take the length of that height from the next right- 
hand column. Multiply the length thus obtained by the base of the 
arc, and the product will give the length of the arc. 

Example —What is the length of an arc of a circle, the base or span of it being 
100 feet, and the height 25 feet? 

25 -f- 100 =.25 ; and .25, per table, =± 1.15912, the length of the base, ichich, being 
multiplied by 100 — 115.912 feet. 

Note.— When, in the division of a height by the base, the quotient has a remain- 
der after the third pbice of decimals, and great accuracy is required, 

Take the length for the first three figures, subtract" it from the next following 
length; multiply the remainder by the said fractional remainder, add the product 
to the first length, and the sum will be the length for the whole quotient. 

Exa.\tple. — What is the length of an arc of a circle, the base of which is 35 feet, 
and the height or versed sine S feet ? 

S -H 35 = .22S5714 ; the tabular length for .22$ = 1.13331, and for .229 = 1.13144, the 
difference between which is .00113. Then .5714 X .00113 = .000645682. 

Hence .228 = 1.18331. 

and .0005714= 0006456S2 

1.1331155682, the sum 
by which the base of the arc is to be multiplied, and 1.133C55682 X 35=39.68845 
feiL 



LENGTHS OF SEMI-ELLIPTIC ARCS. 



201 



Table of the Lengths of Semi-Elliptic .A^rcss. 

The Transverse Diameter of an Ellipse assumed to be Unity, and divided 
into 1000 equal Parts. 

H'ght. Length. I H'ght.! Length. H'ght. Length. H'ght. | Length. 



.1 1 

.10111 
[l 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1. 
1. 
1. 
1 
1 
1 
1 

1. 
1. 
1. 
1 

1. 
1. 
1. 
1. 
1. 
1 
1 

1. 
1. 
1 
1. 
1. 
1 
1 
1 
1 



.04162 
.04262 
.04362 
.04462 
.04562 
.04662 
.04762 
.04862 
.04962 
.05063 
.05164 
.05265 
.05366 
.05467 
.05568 
.05669 
.0577 
.05872 
.05974 
.06076 
.06178 
.0628 
.06382 
.06484 
.06586 
.06689 
06792 
06895 
06998 
07001 
07204 
07308 
07412 
07516 
07621 
07726 
07831 
07937 
08043 
08149 
08255 
08362 
08469 
08576 
08684 
08792 
08901 
0901 
09119 
09228 
0933 
09448 
09558 



.153 
.154 
.155 
.156 
.157 
.158 
.159 
.16 
.161 
.162 
.163 
.164 
.165 
.166 
.167 
.168 
.169 
.17 
.171 
.172 
.173 
.174 
.175 
.176 
.177 
.178 
.179 
.18 
.181 
.182 
.183 
.184 
.185 
,186 
,187 
188 
189 
19 
191 
192 
193 
194 
195 
,196 
,197 
.198 
.199 
,2 
201 
.202 
.203 
,204 
,205 



1.09669 

1.0978 

1.09891 

1.10002 

1.10113 

1.10224 

1.10335 

1.10447 

1.1056 

1.10672 

1.10784 

1.10896 

1.11008 

1.1112 

1.11232 

1.11344 

1.11456 

1.11569 

1.11682 

1.11795 

1.11908; 

1.12021 1 

1.12134! 

1.12247 

1.1236 j 

1.12473 j 

1.12586' 

1.12699 

1.12813 

1.12927 

1.13041 

1.13155 

1.13269 

1.13383 

1.13497 

1.13611 

1.13726 

1.13841 

1.13956 

1.14071 

1.14186 

1.14301 

1.14416 

14531 
1.14646 
1.14762 
1.14888 
1.15014 

15131 
1.15248 
1.15366 
1.15484 
1.15602 



.206 

.207 

.208 

.209 

.21 

.211 

.212 

.213 

.214 

.215 

.216 

.217 

.218 

.219 

.22 

.221 

.222 



1.1572 

1.15838 

1.15957 

1.16076 

1.16196 

1.16316 

1.16436 

1.16557 

1.16678 

1.16799 

1.1692 

1.17041 

1.17163 

1.17285 

1.17407 

1 . 1 7529 

1.17651 



.223! 1.17774 



.224 


1.17897 


.225 


1.1802 


.226 


1.18143 


.227 


1.18266 


.228 


1.1839 


.229 


1.18514 


.23 


1.18638 


.231 


1.18762 


.232 


1.18886 


.233 


1.1901 


.234 


1.19134 


.235 


1.19258 


.236 


1.19382 


.237 


1.19506 


.238 


1.1963 


.239 


1.19755 


.24 


1.1988 


.241 


1.20005 


.242 


1.2013 


.243 


1.20255 


.244 


1.2038 


.245 


1.20506 


.246 


1.20632 


.247 


1.20758 


.248 


1.20884 


.249 


1.2101 


.25 


1.21136 


.251 


1.21263 



252 ; 1.2139 



.253 

.254 
.255 
.256 
.257 



1.21517 

1.21644 

1.21772 

1.219 

1.22028 



.258:1.22156 



.259 

.26 

.261 

.262 

.263 

.264 

.265 

.266 

.267 

.268 

.269 

.27 

.271 

.272 

.273 

.274 

.275 

.276 

.277 

.278 

.279 

.28 

.281 

.282 

.283 

.284 

.285 

.286 

.287 

.288 

.289 

.29 

.291 

.292 

.293 

.294 

.295 

.296 

.297 

.298 

.299 

.3 

.301 

.302 

.303 

.304 

.305 



1.22284 
1.22412 
1.22541 
1.2267 
1.22799 
1.22928 
1.23057 
1.23186 
1.23315 
1.23445 
1.23575 
1.23705 
1.23835 
1.23966 
1.24097 
1.24228 
1.24359 
1.2448 
1.24612 
1.24744 
1.24876 
1.2501 
1.25142 
1.25274 
1.25406 
1.25538 
1.2567 
1.25803 
1.25936 
1.26069 
1.26202 
1.26335 
1.26468 
1.26601 
1.26734 
1.26867 
1.27 
1.27133 
1.27267 
1.27401 
1.27535 
1.27669 
1.27803 
1.27937 
1.28071 
1.28205 
1.28339 
306 1.28474 
307 : 1.28609 
308 1.28744 
309 ' 1.28879 
31 1 1.29014 
31111.29149 



H'ght. ; Length. 

.312 1.29285 



1.29421 
1.29557 
1.29603 
1.29829 
1.29965 
1.30102 
.319 1.302S9 
.32 1.30376 
.321 1.30513 
.322 1.3065 
1.30787 
1.30924 
1.31061 
1.31198 
1.31335 
1.31472 
1.3161 
1.31748 
1.31886 
1.32024 
1.32162 
1.323 
1.32438 
1.32576 
1.32715 
1.32854 
1.32993 
1.33132 
1.33272 
1.33412 
1.33552 
1.33692 
1.33833 
1.33974 
1.34115 
1.34256 
1.34397 
1.34539 
1.34681 
1.34823 
1.34965 
1.35108 
1.35251 
1.35394 
1.35537 
1.3568 
1.35823 
1.35967 
1.36111 
1.36255 
363 1.36399 
364;1.36:)i:> 



313 
.314 
.315 
.316 
.317 
,318 



.323 

.324 
.325 
.326 
.327 
.328 
.329 
.33 
.331 
.332 
.333 
.334 
.335 
336 
.337 
.338 
.339 
.34 
.341 
.342 
.343 
.344 
.345 
.346 
.347 
.348 
.349 
.35 
.351 
.352 
.353 
.354 
.355 
.356 
.357 
.358 
.359 
.36 
.361 
.362 



202 



LENGTHS OF SEMI-ELLIPTIC ARCS. 









Table 


—(Continued), 








H'ght. 

.365 


' Length. 


1 H'ght. | Length. 


j H'ght 


Length. 


1 H'ght 


Length. 


1 H'ght 


Length. 


1.36688 


.421 1.44913 


.477 


1.53469 


.533 


1.62216 


, .589 


1.71065 


.366 1.36833 


.422 


1.45064 


.478 


1.53625 


1 .534 


1.62372 


.59 


1.71225 


.367 1.36978 


.423 


1.45214 


.479 


1.53781 


1 .535 


1.62528 


.591 


1.71286 


.368 1.37123 


.424 


1.45364 


.48 


1.53937 


.536 


1.62684 


.592 


1.71546 


.369 1.37268 


.425 


1.45515 


.481 


1.54093 


| .537 


1.6284 


.593 


1.71707 


.37 1.37414 


.426 


1.45665 


.482 


1.54249 


! .538 


1.62996 


.594 


1.71868 


.371 1.37662 


.427 


1.15815 


.483 


1.54405 


.539 


1.63152 


.595 


1.72029 


.372 1.37708 


.428 


1.45966 


.484 


1.54561 


.54 


1.63309 


.596 


1.7219 


.373 1.37854 


.429 


1.46167 


.485 


1.54718 


.541 


1.63465 


.597 


1.7235 


.374 1.38 


.43 


1.46268 


.486 


1.54875 


.542 


1.63623 


.598 


1.72511 


.375 1.38146 


.431 


1.46419 


.487 


1.55032 


.543 


1.6378 


.599 


1.72672 


.376 1.38292 


.432 


1.4657 


.488 


1.55189 


.544 


1.63937 


.6 


1.72833 


.377 1.38439 


.433 


1.46721 


.489 


1.55346 


.545 


1.64094 


.601 


1.72994 


.378 1.38585 


.434 


1.46872 ! 


.49 


1.55503 


.546 


1.64251 


.602 


1.73155 


.379 1.38732 


.435 


1.47023 


.491 


1.5566 


.547 


1.64408 


.603 


1.73316 


.38 ■1.38879 


.436 


1.47174 


.492 


1.55817 


.548 


1.64565 


.604 


1.73477 


.331 1.39024 


.437 


1.47326 


.493 


1.55974 


.549 


1.64722 


.605 


1.73638 


.332 1.39169 


.438 


1.47478 


.494 


1.56131, 


.55 


1.64879 


.606 


1.73799 


.383 1.39314 


.439 


1.4763 ! 


.495 


1.56289 


.551 


1.65036 


.607 


1.7396 


.384 1.39459 


.44 


1.47782 


.496 


1.56447 


.552 


1.65193 


.608 


1.74121 


.385 1.39605 


.441 


1.47934 


.497 


1.56605 


.553 


1.6535 


.609 


1.74283 


.386 1.39751 


.442 


1.48086 


.498 


1.56763 


.554 


1.65507 


.61 


1.74444 


.387 1.39897 


.443 


1.48238 


.499 


1.56921: 


.555 


1.65665 


.611 


1.74605 


.388 1.40043 


.444 


1.48391 


.5 


1.57089 


.556 


1.65823 


.612 


1.74767 


.389 1.40189 


.445 


1.48544 


.501 


1.57234 


.557 


1.65981 


.613 


1.74929 


.39 1.40335 


.446 


1.48697 


.502 


1.57389 


.558 


1.66139 


.614 


1.75091 


.391 1.40181 


.447 


1.4885 


.503 


1.57544 


.559 


1.66297 


.615 


1.75252 


.392 1.40627 


.448 


1.49003 


.504 1.57699 


.56 


1.66455 


.616 


1.75414 


.393 1.40773 


.449 


1.4-3157 


.5)5 


1.57854 


.561 


1.66613 


.617 


1.75576 


.394 1.40919 


.45 


1.49311. 


.506 


1.58009, 


.562 


1.66771 


.618 


1.75738 


.395 1.41065 


.451 


1.49465 ' 


.507 


1.58164! 


.563 


1.66929 


.619 


1.759 


.396 1.41211 


.452 


1.49618 


.508 


1.58319! 


.564 


1.67087 


.62 


1.76062 


.397 1.41357 


.453 


1.497711 


.509 


1. 58474 ' 


.565 


1.67245 


.621 


1.76224 


.398 1.41504 


.454 


1.49924 


.51 


1.58629 


.m 


1.67403 


.622 


1.76386 


.399 1,41651 


.455 


1.50077! 


.511 


1.58784; 


.567 


1.67561 


.623 


1.76548 


.4 1.41793 


.456 


1.5023 ! 


.512 


1.5894 


.568 


1.67719 


.624 


1.7671 


.401 1.41945 


.457 


1.50383: 


.513 


1.59090 


.569 


1.67877 


.625 


1.76872 


.402 1.42092! 


.458 


1.50536; 


.514 


1.59252 


.570 


1.68036 


.626 


1.77034 


.403 1.42239, 


.459 


1.50689: 


.515 


1.59408: 


.571 


1.68195 


.627; 


1.77197 


.404 1.42386; 


.46 


1.50842 


.516 


1.59564 ( 


.572 


1.68354 


.628) 


1.77359 


.405 1.42533, 


.461 


1.50996 


.517 


1.5972 


.573 


1.68513 


.629 


1.77521 


.406 1.42681: 


.462 


1.5115 I 


.518 


1.59876 


.574! 


1.68672 


.630 


1.77684 


.407 1.42829 


.463 


1.51304! 


.519 


1.60032 


.575 | 


1.68831 


.631 


1.77847 


.408 1.42977 


.464 


1.51458 


.52 


1.60188 


.576; 


1.6899 


.632 


1.78009 


.409 1.43125 


.465 


1.51612 


.521 


1.60344 


.577 ; 


1.69149 


.6331 


1.78172 


.41 1.43273 


.466 


1.51766 


.522 


1.605 


.578 ; 


1.69308 


.634| 


1.78335 


.411 1.43421 


.467 


1.5192 ; 


.523 


1.60656 


.579' 


1.69467 


.6351 


1.78498 


.412 1.43-)6.) 


.468 


1.52074 


.524 


1.60812 


.580' 


1.69626 


.636 


1.7866 


.413 1.43718 


.469 


1.52229 


.525 


1.60968' 


.581' 


1.69785 


.637, 


1.78823 


.414 1.43867 


.47 


1.52384 


.526 


1.61124 


.582 


1.69945 


.638 


1.78986 


.415 


1.44016 


.471 


1.52539: 


.527 


1.6128 


.583 


1.70105 


.639 


1.79149 


.416 


1.4416.5 


.472 


1.52691 


.528 


1. 61436 ' 


.584 


1.70264 


.64 


1.79312 


.417 


1.44314 


.473 


1.52849 


.529 


1.61592: 


.585 1.70424 


.641 


1.79475 


.418 


1.44463 


.474 


1.53004 




1.61748 


.586 1.70584 


.642 


1.70038 


.419 


1.44613 


.475 


1.53159 


.531 


1.61904 


.587 1.70745 


.643 


1.79801 


.42 


1.44763 


.476 


1.53314! 


.532 


1.6206 1 


.588 


1.70905 


.644, 


1.79964 



LENGTHS OF SEMI-ELLIPTIC AKCS. 



203 



Ta"ble— (.Continued). 



H'gh 


t Length. 


H'ght. | Length. 


H'gh 


t. Length. 


H'ght 


.i Length. 


1 H'ght 


Length. 


Tm 


1.80127 


.70111.89352 .757 


' 1.98794 


.813 


2.0848 


.869 


2.18475 


.646 


1.8029 


.702! 1.89519 .75£ 


S 1.98964 


.814 


2.08656 


.87 


2.18656 


.647 


1.80454 


.70; 


1 ; 1.8968c 


.75£ 


1.99134 


.815 


2.08832 


.871 


2.18837 


.648 


1.80617 


1.704 


1.89851 


.76 


1.99305 


.816 


2.09008 


.872 


2.19018 


.649 


1.8078 


.705 


1.90017 


.761 


1.99476 


.817 


2.09198 


.873 


2.192 


.65 


1.80943 


|.706 


1.90184 


.762 


1.99647 


.818 


2.0936 


.874 


2.19382 


.651 


1.81107 


1.707 


1.9035 


.763 


1.99818 


.819 


2.09536 


.875 


2.19564 


.652 


1.81271 


.708 


1 1.90517 


.764 


1.99989 


.82 


2.09712 


.876 


2.19746 


.653 


1.81435 


.709 


11.90684 


.765 


2.0016 


.821 


2.09888 


.877 


2.19928 


.654 


1.81599 


.71 


1.90852 


.766 


2.00331 


.822 


2.10065 


.878 


2.2011 


.655 


1.81763 


.711 


1.91019 


.767 


2.00502 


.823 


2.10242 


.879 


2.20292 


.656 


1.81928 


.712 


1.91187 


.768 


2.00673 


.824 


2.10419 


.88 


2.20474 


.657 


1.82091 


.713 


1.91355 


.769 


2.00844 


.825 


2.10596 


.881 


2.20656 


.658 


1.82255 


.714 


1.91523 


.77 


2.01016 


.826 


2.10773 


.882 


2.20839 


.659 


1.82419 


.715 


1.91691 


.771 


2.01187 


.827 


2.1095 


.883 


2.21022 


.66 


1.82583 


.716 


1.91859 


.772 


2.01359 


.828 


2.11127 


.884 


2.21205 


.661 


1.82747 


.717 


1.92027 


.773 


2.01531 


.829 


2.11304 


.885 


2.21388 


.662 


1.82911 


.718 


1.92195 


.774 


2.01702 


.83 


2.11481 


.886 


2.21571 


.663 


1.83075 


.719 


1.92363 


.775 


2.01874 


.831 


2.11659 


.887 


2.21754 


.664 


1.8324 


.72 


1.92531 


.776 


2.02045 


.832 


2.11837 


.888 


2.21937 


.665 


1.83404 


.721 


1.927 


.777 


2.02217 


.833 


2.12015 


.889 


2.2212 


.666 


1.83568 


.722 


1.92868 


.778 


2.02389 


.834 


2.12193 


.89 


2.22303 


.667 


1.83733 


.723 


1.93036 


.779 


2.02561 


.835 


2.12371 


.891 


2.22486 


.668 


1.83897 


.724 


1.93204 


.78 


2.02733 


.836 


2.12549 


.892 


2.2267 


,669 


1.84061 


.725 


1.93373 


.781 


2.02907 


.837 


2.12727 


.893 


2.22854 


.67 


1.84226 


.726 


1.93541 


.782 


2.0308 


.838 


2.12905 


.894 


2.23038 


.671 


1.84391 


.727 


1.9371 


.783 


2.03252 


.839 


2.13083 


.895 


2.23222 


.672 


1.84556 


.728 


1.93878 


.784 


2.03425 


.84 


2.13261 


.896 


2.23406 


.673 


1.8472 


.729 


1.94046 


.785 


2.03598 


.841 


2.13439 


.897 


2.2359 


.674 


1.84885 


.73 


1.94215 


.786 


2.03771 


.842 


2.13618 


.898 


2.23774 


.675 


1.8505 


.731 


1.94383 


.787 


2.03944 


.843 


2.13797 


.899 


2.23958 


.676 


1.85215 


.732 


1.94552 


.788 


2.04117 


.844 


2.13976 


.9 


2.24142 


.677 


1.85379 


.733 


1.94721 


.789 


2.0429 


.845 


2.14155 


.901 


2.24325 


.678 


1.85544 


.734 


1.9489 


.79 


2.04462 


.846 


2.14334 


.902 


2.24508 


.679 


1.85709 


.735 


1.95059 


.791 


2.04635 


.847 


2.14513 


.903 


2.24691 


.68 


1.85874 


.736 


1.95228 


.792 


2.04809 


.848 


2.14692 


.904 


2.24874 


.681 


1.86039 


.737 


1.95397 


.793 


'•2.04983 


.849 


2.14871 


.905 


2.25057 


.682 


1.86205 


.738 


1.95566 


.794 


2.05157 


.85 


2.1505 


.906 


2.2524 


.683 


1.8637 


.739 


1.95735 


.795 


2.05331 


.851 


2.15229 


.907 


2.25423 


.684 


1.86535 


.74 


1.95994 


.796 


2.05505 


.852 


2.15409 


.908 


2.25606 


.685 


1.867 


.741 


1.96074 


.797 


2.05679 


.853 


2.15589 


.909 


2.25789 


.686 


1.86866 


.742 


1.96244 


.798 


2.05853 


.854 


2.1577 


.91 2.25972 


.687 


1.87031 


.743 


1.96414 


.799 


2.06027 


.855 


2.1595 


.911 | 


2.26155 


,688 


1.87196 


.744 


1.96583 


.8 


2.06202 


.856 


2.1613 


.912 


2.26338 


,689 


1.87362 


.745 


1.96753 


.801 


2.06377 


.857 


2.16309 


.913 


2.26521 


.69 


1.87527 


.746 


1.96923 


.802 


2.06552 


.858 


2.16489 


.914 


2.26704 


.691 


1.87693 


.747 


1.97093 


.803 


2.06727 


.859 


2.16668 


.915 


2.26888 


.692 


1.87859 


.748 


1.97262 


.804 


2.06901 


.86 


2.16848 


.916 


2.27071 


.693 


1.88024 


.749 


1.97432 


.805 


2.07076 


.861 


2.17028 


.917 


2.27254 


694 


1.8819 


.75 


1.97602 


.806 


2.07251 


.862 


2.17209 


.918 


2.27437 


695 


1.88356 


.751 


1.97772 


.807 


2.07427 


.863 


2.17389 


.919 


2.2762 


.696 


1.88522 


.752 


1.97943 


.808 


2.07602 


.864 


2.1757 


.92 


2.27803 


697 


1.88688 


.753 


1.98113 


.809 


2.07777 


.865 


2.17751 


.921 


2.27987 


.698 


1.88854 


.754 


1.98283 


.81 


2.07953 


.866 


2.17932 


.922 


2.2817 


.699 


1.8902 


.755 


1.98453 


.811 


2.08128 


.867 


2.18113 


.92;; 


2.28354 


7 


1.89186 


.756 


1.98623 


.812 


2.08304 


.868 


2.182941 


.924 


2.28537 



204 



LENGTHS OF SEMI-ELLIPTIC ARCS. 



Table— {Continued). 



H'ght. 


Length. 1 


.925 


2.2872 


.926 


2.28903 


.927 


2.29086 


.928 


2.2927 


.929 


2.29453 


,93 


2.29636 


.931 


2.2982 


.932 


2.30004 


.933 


2.30188 


.934 


2.30373 


.935 


2.30557 


.936 


2.30741 


.937 


2.30926 


.938 


2.31111 


.939 


2.31295 


.94 


2.31479 



H'ght. 


Length 


~.M1 


2.31666 


.942 


2.31852 


.943 


2.32038 


.944 


2.32224 


.945 


2.32411 


.946 


2.32598 


.947 


2.32785 


.948 


2.32972 


.949 


2.3316 


.95 


2.33348 


.951 


2.33537 


.952 


2.33726 


.953 


2.33915 


.954 


2.34104 


.955 


2.34293 



H'ght. 


1 Length. 


H'ght. 


Length 


H'ght. Length. 


.956 2.34483 


.971 


2.37334 


.986 2.40208 


.957 2.34673 


.972 


2.37525 


.987 2.404 


.958 2.34862 


.973 


2.37716 


.988 2.40592 


.959 2.35051 


.974 


2.37908 


.989! 2.40784 


.96 


2.35241 


.975 2.381 


.99 ; 2.40976 


.961 


2.35431 


.976 


2.38291 


.991 2.41169 


.962 


2.35621 


.977 


2.38482 


.992 2.41362 


.963 


2.3581 


.978 


2.38673 


.993 2.41556 


.964 


2.36 


.979 


2.38864 


.994 2.41749 


.965 2.36191 


.98 


2.39055 


.995 2.41943 


.966 2.36381 


.981 


2.39247 


.996 2.42136 


.967' 2.36571 


.982 


2.39439 


.997 2.42329 


.968,2.36762 


.983 


2.39631 


.998 2.42522 


.969 2.36952 


.984 


2.39823 


.999' 2.42715 


.,: 


2.37143 


.985 


2.40016 


1. 2.42L-08 



To 

( 



Ascertain tlie Length, of a Semi-Elliptic Arc 
riglit Semi-Ellipse) toy tlie preceding Table. 






Rule. — Divide the height by the base, find the quotient in the col- 
umn of heights, and take the length of that height from the next right- 
hand column. Multiply the length thus obtained by the base of the 
arc, and the product will be the length of the arc. 

Example. — What is the length of the arc of a semi-ellipse, the base being 70 feet, 
and the height 30.10 feet? 

30.10^-70 = .43 ; and .43 ; per table, = 1.4626S. 
Then 1.4326S X 70 — 102.3S76/<^. 

When the Curve is not that of a Right Semi-Ellipse, the Height being 
half of the. Transverse Diameter. 

Rule. — Divide half the base by twice the height, then proceed as 
in the preceding example ; multiply the tabular length by twice the 
height, and the product will be the length required. 

KxAMrLE.— What is the length of the arc of a semi-ellipse, the height being 35 feet, 
and the base 60 feet? 

6 r > -^ 2 = 30, and 30 -^- 35x2 = .42S, the tabular length of which is 1.45966. 
Then 1.45966x35x2 = 102.1762 feet. 

Note. — if in the division of a height by the base there is a remainder, proceed in 
the manner given for the Lengths of Circular Arcs, page 200. 






AREAS OF THE SEGMENTS OF A CIRCLE. 



205 



Table of the Areas of the Segments of a Circle. 

The Diameter of a Circle assumed to be Unity, and divided into 1000 
equal Parts. 



Versed 
Sine. 


Seg. Area. 


1 Versed 
1 Sine. 


TooT 


.00004 


.053 


.002 


.00012 


.054 


.003 


.00022 


.055 


.001 


.00034 


.056 


.005 


.00047 


.057 


.006 


.00062 


.058 


.007 


.00078 


.059 


.008 


.00095 


.06 


.009 


.00113 . 


.061 


.01 


.00133 


.062 


.011 


.00153 


.063 


.012 


.00175 


.064 


.013 


.00197 


.065 


.014 


.0022 


.066 


.015 


.00244 


.067 


.016 


.00268 


.068 


.017 


.00294 


.069 


.018 


.0032 


.07 


.019 


.00347 


.071 


.02 


.00375 


.072 


.021 


.00403 


.073 


.022 


.00432 


.074 


.023 


.00462 


.075 


.024 


.00492 


.076 


.025 


.00523 


.077 


.026 


.00555 


.078 


.027 


.00587 


.079 


.028 


.00619 


.08 


.029 


.00653 


.081 


.03 


.00686 


.082 


.031 


.00721 


.083 


.032 


.00756 


.084 


.033 


.00791 


.085 


.034 


.00827 


.086 


.035 


.00364 


.087 


.036 


.00901 


.088 


.037 


.00938 


.089 


.038 


.00976 


.09 


.039 


.01015 


.091 


.04 


.01054 


.092 


.041 


.01093 


.093 


.042 


.01133 


.094 


.043 


.01173 


.095 


.044 


.01214 


.096 


.045 


.01255 


.097 


.046 


.01297 


.098 


.047 


.01339 


.099 


.048 


.01382 


.1 


.049 


.01425 


.101 


.05 


.01468 


.102 


.051 


.01512 


.103 


.052 


.01556 ; 


.104 



Seg. Area. 


Versed 
Sine. 


Seg. Area. 


Versed 


Seg. Area 


Versed 

Sine. 


Seg. Atm. 


.01601 


.105 


.04391 


.157 


.07892 


.209 


.11908 


.01646 


.106 


.04452 


.158 


.07965 


.21 


.1199 


.01691 


.107 


.04514 


.159 


.08038 


.211 


.12071 


.01737 


.108 


.04575 


.16 


.08111 


.212 


.12153 


.01783 


.109 


.04638 


.161 


.08185 


.213 


.12235 


.0183 


.11 


.047 


.162 


.08258 


.214 


.12317 


.01877 


.111 


.04763 


.163 


.08332 


.215 


.12399 


.01924 


.112 


.04826 


.164 


.08406 


.216 


.12481 


.01972 


.113 


.04889 


.165 


.0848 


.217 


.12563 


.0202 


.114 


.04953 


.166 


.08554 


.218 


.12646 


.02068 


.115 


.05016 


.167 


.08629 


.219 


.12728 


.02117 


.116 


.0508 


.168 


.08704 


.22 


.12811 


.02165 


.117 


.05145 


.169 


.08779 


.221 


.12894 


.02215 


.118 


.05209 


.17 


.08853 


.222 


.12977 


.02265 


.119 


.05274 


.171 


.08929 


.223 


.1306 


.02315 


.12 


.05338 


.172 


.09004 


.224 


.13144 


.02336 


.121 


.05404 


.173 


.0908 


.225 


.13227 


.02417 


.122 


.05469 


.174 


.09155 


.226 


.13311 


.02468 


.123 


.05534 


.175 


.09231 


.227 


.13394 


.02519 


.124 


.056 


.176 


.09307 


.228 


.13478 


.02571 


.125 


.05666 


.177 


.09384 


.229 


.13562 


.02624 


.126 


.05733 


.178 


.0946 


.23 


.13646 


.02676 


.127 


.05799 


.179 


.09537 


.231 


.13731 


.02729 


.128 


.05866 


.18 


.09613 


.232 


.13815 


.02782 


.129 


.05933 


.181 


.0969 


.233 


.139 


.02835 


.13 


.06 


.182 


.00767 


.234 


.13984 


.02889 


.131 


.06037 


.183 


.09845 


.235 


.14069 


.02943 


.132 


.06135 


.184 


.09922 


.236 


.14154 


.02997 


.133 


.06203 


.185 


.1 


.237 


.14239 


.03052 


.134 


.06271 


.186 


.10077 


.238 


.14324 


.03107 


.135 


.06339 


.187 


.10155 


.239 


.14409 


.03162 


.136 


.06407 


.188 


.10233 


.24 


.14494 


.03218 


.137 


.06476 


.189 


.10312 


.241 


.1458 


.03274 


.138 


.06545 


.19 


.1039 


.242 


.14665 


.0333 


.139 


.06614 


.191 


.10468 


.243 


.14751 


.03387 


.14 


.06683 


.192 


.10547 


.244 


.14837 


.03444 


.141 


.06753 


.193 


.10626 


.245 


.14923 


.03501 


.142 


.06822 


.194 


.10705 


.246 


.15009 


.03558 


.143 


.06892 


.195 


.10784 


.247 


.16095 


.03616 


.144 


.06962 


.196 


.10864 


.248 


.15182 


.03674 


.145 


.07033 


.197 


.10943 


.249 


.15268 


.03732 


.146 


.07103 


.198 


.11023 


.25 


.15355 


.0379 


.147 


.07174 


.199 


.11102 


.251 


.15441 


.03849 


.148 


.07245 


.2 


.11182 


.252 


.15528 


.03J08 


.149 


.07316 


.201 


.11262 


.253 


.15615 


.03968 


.15 


.07387 


.202 


.11343 


.254 


.15702 


.04027 


.151 


.07459 


.203 


.11423 


.255 


.15789 


.04087 


.152 


.07531 


.204 


.11503 


.256 


.15876 


.04148 


.153 


.07603 


.205 


.11584 


.257 


.15964 


.04208 


.154 


.07675 


.206 


.11665 


.258 


.16051 


.04269 


.155 


.07747 


.207 


.11746 


.259 


.16139 


.0431 


.156 


.0782 


.208 


.11827 


.26 


.16226 



206 



AREAS OF THE SEGMENTS OF A CIRCLE. 



Table— (Continued). 



Versed',-. „ . „ 
Sine. Se »- Area 


1 Versed 
1 Sine. 


Seg. Area 


] Versed 
) Sine. 


Seg. Area. 


Versed 
Sine. 


Seg. Area 


! Versed 
: Sine. 


iSeg. Area 


.261 


i .16314 


.309 


.20645 


.357 


.25167 


.405 


.29827 


.453 


.34557 


.262 


! .16402 


.31 


.20738 


.358 


.25263 


.406 


.29925 


.454 


.34676 


.263 


.1649 


.311 


.2083 


.359 


.25359 


.407 


.30024 


.455 


.34776 


.264 


.16578 


.312 


.20923 


.36 


.25455 


.408 


.30122 


.456 


.34875 


.265 


.16666 


.313 


.21015 


.361 


.25551 


.409 


.3022 


.457 


.34975 


.266 


.16755 


.314 


.21108 


.362 


.25647 


.41 


.30319 


.458 


. 35075 


.267 


.16844 


.315 


.21201 


.363 


.25743 


.411 


.30417 


.459 


.35174 


.268 


.16931 


.316 


.21294 


.364 


.25839 


.412 


.30515 


.46 


.35274 


.269 


.1702 


.317 


.21387 


.365 


.25936 


.413 


.30614 


.461 


.35374 


.27 


.17109 


.318 


.2148 


.366 


.26032 


.414 


.30712 


.462 


.35474 


.271 


.17197 


.319 


.21573 


.367 


.26128 


.415 


.30811 


.463 


. 35573 


.272 


.17287 


.32 


.21667 


.368 


.26225 


.416 


.30909 


.164 


.356/3 


.273 


.17376 


.321 


.2176 


.369 


.26321 


.417 


.31008 


.465 
1 .466 


. 35773 


.274 


.17465 


.322 


.21853 


.37 


.26418 


.418 


.31107 


.35872 


.275 


.17554 


.323 


.21947 


.371 


.26514 


.419 


.31205 


.467 


.35972 


.276 


.17643 


.324 


.2204 


.372 


.26611 


.42 


.31304 


.468 


.36072 


.277 


.17733 


.325 


.22134 


.373 


.26708 


.421 


.31403 


.469 


.36172 


.278 


.17822 


.326 


.22228 


.374 


.26804 


.422 


.31502 


.47 


.36272 


.279 


.17912 


.327 


.22321 


.375 


.26901 


.423 


.316 


.471 


.36371 


.28 


. 18002 


.328 


.22415 


.376 


.26998 


.424 


.31699 


.472 


.36471 


.281 


.18092 


.329 


.22509 


.377 


.27095 


.425 


.31798 


.473 


.36571 


.282 


.18182 


.33 


.22603 


.378 


.27192 


.426 


.31897 


.474 


.36671 


.283 


.18272 


.331 


.22697 


.379 


.27289 


.427 


.31996 


.475 


.36771 


284 


.18361 


.332 


.22791 


.38 


.27386 


.428 


.32095 


.476 


.36871 


285 


.18452 


.333 


.22886 


.381 


.27483 


.429 


.32194 


.477 


.36971 


286 


.18542 


.334 


.2298 


.382 


.27580 


.43 


.32293 


.478 


.37071 


287 


.18633 


.335 


.23074 


.383 


.27677 


.431 


.32391 


.479 


.3717 


288 


.18723 


.336 


.23169 


.384 


.27/75 


.432 


.3249 


.48 


.3727 


289 


.18814 


.337 


.23263 


.385 


.27872 


.433 


.3259 


.481 


.3737 


29 


.18905 


.338 


.23358 


.386 


.27969 


.434 


.32689 


.482 


.3747 


291 


.18995 


.339 


.23453 


.387 


.28067 


.435 


.32788 


.483 


.3757 


292 


.19086 


.34 


.23547 


.388 


.28164 


.436 


.32887 


.484 


.3767 


293 


.19177 


.341 


.23642 


.389 


.28262 


.437 


.32987 


.485 


.3777 


294 


.19268 


.342 


.23737 


.39 


.28359 


.438 


.33086 


.486 


.3787 


295 


.1936 


.343 


.23832 


.391 


.28457 


.439 


.33185 


.487 


.3797 


296 


.19451 


.344 


.23927 


.392 


.28554 


.44 


.33284 


.488 


.3807 


297 


.19542 


.345 


.24022 


.393 


.28652 


.441 


.33384 


.489 


.3817 


298 


.19634 


.346 


.24117 


.394 


.2875 


.442 


.33483 


.49 


.3827 


299 


.19725 


.347 


.24212 


.395 


.28848 


.443 


.33582 


.491 


.3837 


3 


.19817 


.348 


.24307 


.396 


.28945 


.444 


.33682 


.492 


.3847 


301 


.19908 


.349 


.24403 


.397 


.29043 


.445 


.33781 


.493 


.3857 


302 


.2 


.35 


.24498 


.398 


.29141 


.446 


.3388 


.494 


.3867 


303 


! 20092 


.351 


.24593 


.399 


.29239 


.447 


.3398 


.495 


.3877 


304 


.20184 


.352 


.24689 


.4 


.29337 


.448 


.34079 


.496 


.3887 


305 


.20276 


.353 


.24784 


.401 


.29435 


.449 


.34179 


.497 


.389; 


306 


.20368 


.354 


.2488 


.402 


.29533 


.45 


.34278 


.498 


.3907 


307 


.2046 


.355 


.24976 


.403 


.29631 


.451 


.34378 


.499 


.3917 


308| 


.20553 II 


.356 


.25071 


.404 | 


.29729 1 


.452 ; 


.34477 


.5 


.3927 



To Ascertain tlie Area ofa Segment of* a Circle by 

the preceding Table. 

Rule. — Divide the height or versed sine by the diameter of the cir- 
cle; find the quotient in the column of versed sines. Take the area 
noted in the next column, multiply it by the square of the diameter, 
and it will give the area. 



AREAS OF THE ZONES OF A CIRCLE. 



2or 



Example. — Required the area of a segment, its he'ght being 10, and the diameter 
of the circle 50 feet. 

10 -=-50 = .2, and .2, per table, = .111S2; then .11182 X 502 = 219. 55 feet. 

Note If, in the division of a height by the base, the quotient has a remainder 

after the third place of decimals, and great accuracy is required, 

Take the area for the first three figures, subtract it from the next following area, 
multiply the remainder by the said fraction, and add the product to the first area ; 
the sum will be the area for the whole quotient. 

• 2. What is the area of a segment of a circle, the diameter of which is 10 feet, and 
the height of it 1575 feet? 

1.575^-10 = .1575; the tabular area for .157 = .07892, and for .15S = .07965, the 
difference between which is .0C073. 
Then .5 X. 00073 = .000365. 
Hence .157 =.07S92 

.01)05 = .000365 

.079285, the sum by which the square of 
the diameter of the circle is to be multiplied; and .0792S5xl02 = 7.9L85/^. 



Table of th.e Areas of tlie Zones of a Circle. 

The Diameter of a Circle assumed to be Unity, and divided into 1000 
equal Parts. 



H'ght. 


Area. 


(Height. 


Area. 


(Height. 


Area. 


'Height. 


Area. 


Height. 


Area. 


.001 


.001 


.035 


.03497 


.069 


.06878 


.103 


.10227 


.137 


.13527 


.002 


.002 


.036 


.03597 


.07 


.06977 


.104 


.10325 


.138 


.13623 


.003 


.003 


.037 


.03697 


.071 


.07076 


.105 


.10422 


.139 


.13719 


.004 


.004 


.038 


.03796 


.072 


.07175 


.106 


.1052 


.14 


.13815 


.005 


.005 


.039 


.03896 


.073 


.07274 


.107 


.10618 


.141 


.13911 


.006 


.006 


.04 


.03996 


.074 


.07373 


.108 


.10715 


.142 


.14007 


.007 


..007 


.041 


.04095 


.075 


.07472 


.109 


.10813 


.143 


.14103 


.008 


.008 


.042 


.04195 


.076 


.0755 


.11 


.10911 


.144 


.14198 


.009 


.009 


.043 


.04295 


.077 


.07669 


.111 


.11008 


.145 


.14294 


.01 


.01 


.044 


.04394 


.078 


.07768 


.112 


.11106 


.146 


.1439 


.011 


.011 


.045 


.04494 


.079 


.07867 


.113 


.11203 


.147 


.14485 


.012 


.012 


.046 


.04593 


.08 


.07966 


.114 


.113 


.148 


.14581 


013 


.013 


.047 


.04693 


.081 


.08064 


.115 


.11398 


.149 


.14677 


.014 


.014 


.048 


.04793 


.082 


.08163 


.116 


.11495 


.15 


.14772 


.015 


.015 


.049 


.01892 


.083 


.08262 


.117 


.11592 


.151 


.14867 


.016 


.016 


.05 


.04992 


.084 


.0836 


.118 


.1169 


.152 


.14962 


.017 


.017 


.051 


.05091 


.085 


.08459 


.119 


.11787 


.153 


.15058 


.018 


.018 


.052 


.0519 


.086 


.08557 


.12 


.11884 


.154 


.15153 


.019 


.019 


.053 


.0529 


.087 


.08656 


.121 


.11981 


.155 


.15248 


.02 


.02 


.054 


.05389 


.088 


.08754 


.122 


.12078 


.156 


.15343 


.021 


.021 


.055 


.05489 


.089 


.08853 


.123 


.12175 


.157 


.15438 


.022 


.022 


.056 


.05588 


.09 


.08951 


.124 


.12272 


.158 


.15533 


.023 


.023 


.057 


.05688 


.091 


.0905 


.125 


.12369 


.159 


.15628 


.024 


.024 


.058 


.05787 


.092 


.09148 


.126 


.12469 


.16 


.15723 


o025 


.025 


.059 


.05886 


.093 


.09246 


.127 


.12562 


.161 


.15817 


.026 


.02599 


.06 


.05986 


.094 


.09344 


.128 


.12659 


.162 


.15912 


.027 


.02699 


.061 


.06085 


.095 


.09443 


.129 


.12755 


.163 


.16006 


.028 


.02799. 


.062 


.06184 


.096 


.0954 


.13 


.12852 


.164 


.16101 


.029 


.02898 


.063 


.06283 


.097 


.09639 


.131 


.12949 


.165 


.16195 


.03 


.02998 


.064 


.06382 


.098 


.09737 


.132 


.13045 


.166 


.1629 


.031 


.03098 ! 


.065 


.06482 


.099 


.09835 


.133 


.13141 


.167 


.16384 


.032 


.03198 \ 


.066 


.0658 


.1 


.09933 


.134 


.13238 


.108 


.16478 


.063 


.03298 


.067 


.0668 


.101 


. 10031 


.135 


.13334 


.169 


.16572 


.034 


.03397 , 


.068 


.0678 


.102 


.10129 


.136 


.1343 


.17 


.16667 



208 



AREAS OF THE ZONES OF A CIRCLE. 



Table— (Continued ). 



H'ght. 


Area. 


jHeight. 


Area. 


Height. 


Area. 


Height 


I Area. 


j (Height, j Area. 


.171 


.16761 


.227 


.21894 


.283 


.26706 


.339 


. 31085 


.395 


.34879 


.172 


.16855 


.228 


.21983 


.284 


.26789 


.34 


.31159 


.396 


.3494 


.173 


.16948 


.229 


.22072 


.285 


.26871 


.341 


.31232 


.397 


.35001 


.171 


.17042 


.23 


.22161 


.286 


.26953 


.342 


.31305 


.398 


.35062 


.175 


.17136 


.231 


.2225 


.287 


.27035 


.343 


.31378 


.399 


.35122 


.176 


.1723 


.232 


.22335 


.288 


.27117 


.344 


.3145 


.4 


.35182 


.177 


.17323 


.233 


.22427 


.289 


.27199 


.345 


.31523 


.401 


.35242 


.178 


.17417 


.234 


.22515 


.29 


.2728 


.346 


.31595 


.402 


.35302 


.179 


.1751 


.235 


.22604 


.291 


.27362 


.347 


.31667 


.403 


! .35361 


.18 


.17603 


.236 


.22692 


.292 


.27443 


.348 


.31739 


.404 


.3542 


.181 


.17697 


.237 


.2278 


.293 


.27524 


.349 


.31811 


.405 


.35479 


.182 


.1779 


.238 


.22868 


.294 


.27605 


.35 


.31882 


.406 


. 35538 


.183 


.17883 


.239 


.22956 


.295 


.27686 


.351 


.31954 


.407 


.35596 


.181 


. 17976 


.24 


.23044 


.296 


.27766 


.352 


.32025 


.408 


.35654 


.185 


.18069 


.241 


.23131 


.297 


.27847 


.353 


.32096 


.409 


.35711 


.186 


.18162 


.242 


.23219 


.298 


.27927 


.354 


.32167 


.41 


.35769 


.187 


.18254 


.243 


.23306 


.299 


.28007 


.355 


.32237 


.411 


.35826 


.188 


.18347 


.244 


.23394 


.3 


.28088 


.356 


.32307 


.412 


.35883 


.189 


.1844 


.245 


.23481 


.301 


.28167 


.357 


.32377 


.413 


.35939 


.19 


.18532 


.246 


.23568 


.302 


.28247 


.358 


.32447 


.414 


. 35995 


.191 


.18325 


.247 


.23655 


.303 


.28327 


.359 


.32517 


.415 


.36051 


.192 


.18717 


.248 


.23742 


.304 


.28403 


.36 


.32587 


.416 


.36107 


.193 


.18809 


.249 


.23829 


.305 


.28486 


.361 


.32656 


.417 


.36162 


,191 


.18902 


.25 


.23915 


.306 


.28565 


.362 


.32725 


.418 


.36217 


.195 


.18994 


.251 


.24002 


.307 


.28644 


.363 


.32794 


.419 


.36272 


.196 


.19086 


.252 


.24089 


.308 


.28723 


.364 


.32862 


.42 


.36326 


.197 


.19178 


.253 


.24175 


.309 


.28801 


.365 


.32931 


.421 


.3638 


.198 


.1927 


.254 


.24261 


.31 


.2888 


.366 


.32999 


.422 


.36434 


.199 


.19361 


.255 


.24347 


.311 


.28958 


.367 


.33067 


.423 


.36488 


.2 


.19453 


.256 


.24433 


.312 


.29036 


.368 


.33135 


.424 


.36541 


.201 


.19545 


.257 


.24519 


.313 


.29115 


.369 


.33203 


.425 


.36594 


.202 


.19636 


.258 


.24604 


.314 


.29192 


.37 


.3327 


.426 


.36646 


.203 


.19728 


.259 


.2469 


.315 


.2927 


.371 


.33337 


.427 


.36698 


.201 


.19819 


.26 


.24775 


.316 


.29348 


.372 


.33404 


.428 


.3675 


.205 


.1901 


.261 


.24861 


.317 


.29425 


.373 


.33471 


.429 


.36802 


.206 


.20001 


.262 


.24946 


.318 


.29502 


.374 


.33537 


.43 


.36853 


.207 


.20092 


.263 


.25021 


.319 


.2958 


.375 


.33604 


.431 


.36904 


.208 


.20183 


.264 


.25116 


.32 


.29656 


.376 


.3367 


.432 


.36954 


.209 


.20274 


.265 


.25201 


.321 


.29733 


.377 I 


.33735 


.433 


.37005 


.21 


.20365 i 


.266 


.25285 


.322 


.2981 


.378 


.33801 


.434 


, 37054 


.211 


.20156 I 


.267 


.2537 


.323 


.29886 


.379 


.33866 


.435 


.37104 


.212 


.20546 j 


.268 


.25455 


.324 


.29962 


.38 


.33931 


.436 


.37153 


.213 


.20637 


.269 


.25539 


.325 


.30039 


.381 


.33996 


.437 


.37202 


.214 


.20727 


.27 


.25623 


.326 


.30114 


.382 


.34061 


.438 


.3725 


.215 


.20818 


.271 


.25707 


.327 


.3019 


.383 : 


.34125 


.439 


.37298 


.216 


.20908 


.272 


.25791 


.328 


.30266 


.384 


.3419 


.44 


.37346 


.217 


.20998 


.273 


. 25875 


.329 


.30341 


.885 


.34253 


.441 


.37393 


.218 


.21088 


.274 


.25959 


.33 


.30416 


.386 


.34317 


.442 


.3744 


.219 


.21178 


.275 


.26043 


.331 


.30491 


.387 ! 


.3138 


.443 


.37487 


.22 


.21288 


.276 


.26126 


.332 


.30566 


.388 ! 


.34444 


.444 


.37533 


.221 


.21858 


.277 


.26209 


.333 


.30641 


.389 1 


.34507 


.445 


.37579 


.222 


.21447 


.278 


.26293 


.334 


.30715 


.39 ! 


.34S69 


.446 


.37624 


.223 


.21537 


.279 


.26376 


.835 


.3079 


.391 ! 


.34632 


.447 


.37669 


.224 


.21626 


.28 


.26459 


.336 


.30864 


.392 


.34694 


.448 


.37714 


.22.1 


.21716 


.281 


.26511 


.337 ! 


.30938 


.393 


.34756 


.449 


.37758 


226 


.21805 


.282 


.26624 


.338 i 


.31012 


.394i 


.34818 


.45 


.37802 



AKEAS OF THE ZONES OF A CIRCLE. 



209 



Table— (Continued). 



H'ght. Area. Height.! Area. Height.) Area, 



.37845 
.37888 
.37931 
.37973 
.38014 
.38056 
.38096 
.38137 
.38177 
.38216 

This Table is 
eter. 



.461 
.462 
.463 
.464 
.465 
.466 
.467 
.468 
.469 
.47 



.38255 
.38294 
.38332 
.38369 
.38406 
.38443 
.38479 
.38514 
.38549 
.38583 



.471 
.472 
.473 

.474 
.475 
.476 
.477 
.478 
,479 
.48 



.38617 

.3865 

.38683 

.38715 

.38747 

.38778 



.494 
.495 
.496 
.497 
.498 
.499 
.5 

computed only for Zones, the longest chord of which 



.38867 
.38895 



Height. Area. (Height, 



.481 
.482 
.483 
.484 
.485 
.486 
.487 



.489 
.49 



.38923 

.3895 

.38976 

.39001 

.39026 

.3905 

.39073 

.39095 

.39117 

.39137 



.491 

.492 



Area. 



.39156 

.39175 

.39192 

.39208 

.39223 

.39236 

.39248 

.39258 

.39266 

.3927 

? diam- 



To Ascertain tlie Area of a Zone by th.e preceding 

Table. 

Kule 1. — When the Zone is Less than a Semicircle, Divide the height 
by the diameter, and find the quotient in the column of heights. Take 
out the area opposite to it in the next column on the right hand, and 
multiply it by the square of the longest chord ; the product will be the 
area of the zone. 

Example.— Required the area of a zone the diameter of which is 50, and its 
height 15. 

15 -h 50 = .3 ; and .3, as per table, = .28088. 
Hence .2S0S3X53 2 = 702.2 area. 

Eule 2. — When the Zone is Greater than a Semicircle, Take the height 
on each side of the diameter of the circle, and ascertain, by Rule 1, 
their respective areas ; add the areas of these two portions together, 
and the sum will be the area of the zone. 

Example. — Required the area of a zone, the diameter of the circle being 50, and 
the heights of the zone on each side of the diameter of the circle 20 and 15 respect- 
ively. 

20 -r- 50 = .4 ; .4, as per table, = .35182 ; and .35182x502 = 879.55. 
15 -h 50 = .3 ; .3, as per table, =? .2S0S8 ; and .2808SX502 =* 702.2. 
Hence 879.55 + 702.2 = 1531.75 area. 

Note. — When, in the division of a height by the chord, the quotient has a remain- 
der after the third place of decimals, and great accuracy is required, 

T;ike the area for the first three figures, subtract it from the next following area, 
multiply the remainder by the said fraction, and add the product to the first area ; 
the sum will be the area for the whole quotient. 

Example. — What is the area of a zone of a circle, the greater chord being 100 feet, 
and the breadth of it 14 feet 3 inches ? 

14 feet 3 inches = 14.25, and 14.25 -i- 100 = .1425; the tabular length for .142 = 
.14007, and for .143 = .14103, the difference between which is .00096. 
Then .5x. 00006 = .0004S. 
Hence .14* =.14007 
.0005 = .00048 

.14155, the sum by which the square of the greater chord is to be 
multiplied ; and .14055x1002 = 1405.5 feet. 

S* 



210 



SQUARES, CUBES, AND ROOTS. 



Table of Squares, Cubes, and. Square and. Cube 
Roots, of all Numbers from 1 to 1600. 

Square Root. Cube Root. 



Square. 



I 



1 


1 


1. 


1. 


4 


8 


1.4142 136 


1.2599 21 


9 


27 


1.7320 508 


1.4422 49G 


16 


64 


2. 


1.5874 011 


25 


125 


2.2360 68 


1.7099 759 


36 


216 


2.4494 897 


1.8171 206 


49 


343 


2.6457 513 


1.9129 312 


64 


512 


2.8284 271 


2. 


81 


729 


3. 


2.0800 837 


1 00 


1 000 


3.1622 777 


2.1544 347 


1 21 


1 331 


3.3166 248 


2.2239 801 


1 44 


1 728 


3.4641 016 


2.2894 286 


1 69 


2 197 


3.6055 513 


2.3513 347 


1 96 


2 744 


3.7416 574 


2.4101 422 


2 25 


3 375 


3.8729 833 


2.4662 121 


2 56 


4 096 


4. 


2.5198 421 


2 89 


4 913 


4.1231 056 


2.5712 816 


3 24 


5 832 


4.2426 407 


2.6207 414 


3 61 


6 859 


4.3585 989 


2.6684 016 


4 00 


8 000 


4.4721 36 


2.7144 177 


4 41 


9 261 


4.5825 757 


2.7589 243 


4 84 


10 648 


4.6904 158 


2.8020 393 


5 29 


12 167 


4.7958 315 


2.8438 67 


5 76 


13 824 


4.8989 795 


2.8844 991 


6 25 


15 625 


5. 


2.9240 177 


6 76 


17 576 


5.0990 195 


2.9224 96 


7 29 


19 683 


5.1961 524 


3. 


7 81 


21 952 


5.2915 026 


3.0365 889 


8 41 


24 389 


5.3851 648 


3.0723 168 


9 00 


27 000 


5.4772 256 


3.1072 325 


9 61 


29 791 


5.5677 644 


3.1413 806 


10 24 


32 768 


5.65C8 542 


3.1748 021 


10 89 


35 937 


5.7445 626 


3.2075 343 


11 56 


39 304 


5.8309 519 


3.2396 118 


12 25 


42 875 


5.9160 798 


3.2710 663 


12 96 


46 656 


6. 


3.3019 272 


13 69 


50 653 


6.0827 625 


3.3322 218 


14 44 


54 872 


6.1644 14 


3.3619 754 


15.21 


59 319 


6.2449 98 


3.3912 114 


16 00 


64 000 


6.3245 553 


. 3.4199 519 


16 81 


68 921 


6.4031 242 


3.4482 172 


17 64 


74 088 


6.4807 407 


3.4760 266 


18 49 


79 507 


6.5574 385 


3.5033 981 


19 36 


85 184 


6.6382 496 


3.5303 483 


20 25 


91 125 


6.7082 039 


3.5568 933 


21 16 


97 336 


6.7823 3 


3.5830 479 


22 09 


103 823 


6.8556 546 


3.6088 261 


23 04 


110 592 


6.9282 032 


3.6342 411 


24 01 


117 649 


7. 


3.6593 057 


25 00 


125 000 


7.0710 678 


3.6840 314 


26 01 


132 651 


7.1414 284 


3.7084 298 


27 04 


140 608 


7.2111 026 


3.7325 111 


28 09 


148 877 


7.2801 099 


3.7562 858 


29 16 


157 464 


7.3484 692 


3.7797 631 


30 25 


166 375 


7.4161 985 


3.8029 525 



SQUARES, CUBES, AND ROOTS. 



211 



Table— {Continued), 



Number. 


Square. 


Cube. 


| Square Root. 


Cube Root. 


56 


31 36 


175 616 


7.4833 148 


3.8258 624 


57 


32 49 


185 193 


7.5498 344 


3.8485 011 


58 


33 64 


195 112 


7.6157 731 


3.8708 766 


59 


34 81 


205 379 


7.6811 457 


3.8929 965 


60 


36 00 


216 000 


7.7459 667 


3.9148 676 


61 


37 21 


226 981 


7.8102 497 


3.9364 972 


62 


38 44 


233 328 


7.8740 079 


3.9578 915 


63 


39 69 


250 047 


7.9372 539 


3.9790 571 


64 


40 96 


262 144 


8. 


4. 


65 . 


42 25 


274 625 


8.0622 577 


4.0207 256 


66 


43 56 


287 496 


8.1240 384 


4.0412 401 


67 


44 89 


300 763 


8.1853 528 


4.0615 48 


68 


46 24 


314 432 


8.2462 113 


4.0816 551 


69 


47 61 


328 509 


8.3066 239 


4.1015 661 


70 


49 00 


343 000 


8.3666 003 


4.1212 853 


71 


50 41 


357 911 


8.4261 498 


4.1408 178 


72 


51 84 


373 248 


8.4852 814 


4.1601 676 


73 


53 29 


389 017 


8.5440 037 


4.1793 39 


74 


54 76 


405 224 


8.6023 253 


4.1983 364 


75 


56 25 


421 875 


8.6602 54 


4.2171 633 


76 


57 76 


438 976 


8.7177 979 


4.2358 236 


77 


59 29 


456 533 


8.7749 644 


4.2543 21 


78 


60 84 


474 552 


8.8317 609 


4.2726 586 


79 


62 41 


493 039 


8.8881 944 


4.2908 404 


80 


64 00 


512 000 


8.9442 719 


4.3088 695 


81 


65 61 


531 441 


9. 


4.3267 487 


82 


67 24 


551 368 


9.0553 851 


4.3444 815 


83 


68 89 


571 787 


9.1104 336 


4.3620 707 


84 


70 56 


592 704 


9.1651 514 


4.3795 191 


85 


72 25 


614 125 


9.2195 445 


4.3968 296 


86 


73 96 


636 056 


9.2736 185 


4.4140 049 


87 


75 69 


658 503 


9.3273 791 


4.4310 476 


88 


77 44 


681 472 


9.3808 315 


4.4479 602 


89 


79 21 


704 969 


9.4339 811 


4.4647 451 


90 


81 00 


729 000 


9.4868 33 


4.4814 047 


91 


82 81 


753 571 


9.5393 92 


4.4979 414 


92 


84 64 


778 688 


9.5916 63 


4.5143 574 


93 


86 49 


804 357 


9.6436 508 


4.5306 549 


94 


88 36 


830 584 


9.6953 597 


4.5468 359 


95 


90 25 


857 375 


9.7467 943 


4.5629 026 


96 


92 


16 


884 736 


9.7979 59 


4.5788 57 


97 


94 


09 


912 673 


9.8488 578 


4.5947 009 


98 


96 


04 


941 192 


9.8994 949 


4.6104 363 


99 


98 


01 


970 299 


9.9498 744 


4.6260 65 


100 


1 00 


00 


1 000 000 


10. 


4.6415 888 


101 


1 02 


01 


1 030 301 


10.0498 756 


4.6570 095 


102 


1 04 04 


1 061 208 


10.0995 049 


4.6723 287 


103 


1 06 09 


1 092 727 


10.1488 916 


4.6875 482 


104 


1 08 16 


1 124 864 


10.1980 39 


4.7026 694 


105 


1 10 25 


1 157 625 


10.2469 508 


4.7176 94 


106 


1 12 36 


1 191 016 


10.2956 301 


4.7326 235 


107 


1 14 49 


1 225 043 


10.3440 804 


4.7474 594 


108 


1 16 64 


1 259 712 


10.3923 048 


4.7622 032 


109 


1 18 81 


1 295 029 


10.4403 065 


4.7768 562 


110 


1 21 00 


1 331 000 


10.4880 885 


4.7911 199 


111 


1 23 


21 


1 367 631 


10.5356 538 


4.8058 995 



212 



SQUARES, CUBES, AXD ROOTS. 



Table— {Continued). 



Square Root, 



112 


1 25 44 


1 404 928 


10.5830 052 


4.8202 845 


113 


1 27 69 


1 442 897 


10 6301 458 


4.8345 881 


114 


1 29 96 


1 481 544 


10.6770 783 


4.8488 076 


115 


1 32 25 


1 520 875 


10.7238 053 


4.8629 442 


116 


1 34 56 


1 560 896 


10.7703 296 


4.8769 99 


117 


1 36 89 


1 601 613 


10.8166 538 


4.8909 732 


118 


1 39 24 


1 643 032 


10.8627 805 


4.9048 681 


119 


1 41 61 


1 685 159 


10.9087 121 • 


4.9186 847 


120 


1 44 00 


1 728 000 


10.9544 512 


4.9324 242 


121 


1 46 41 


1 771 561 


11. 


4.9460 874 


122 


1 48 84 


1 815 848 


11.0453 61 


4,9596 757 


1:3 


1 51 29 


1 860 867 


11.0905 365 


4.9731 898 


124 


1 53 76 


1 906 624 


11.1355 287 


4.9866 31 


125 


1 56 25 


1 953 125 


11.1803 399 


5. 


12; 


1 dS 76 


2 000 376 


11.2249 722 


5.0132 979 


127 


1 61 29 


2 048 383 


11 2694 277 


5.0265 257 


128 


1 63 84 


2 097 152 


11 3137 085 


5.0396 842 


129 


1 66 41 


2 146 689 


11.3578 167 


5.0527 743 


180 


1 69 00 


2 197 000 


11.4017 543 


5.0657 97 


131 


1 71 61 


2 248 091 


11.4455 231 


5.0787 531 


132 


1 74 24 


2 299 968 


11.4891 253 


5.0916 434 


J83 


1 76 89 


2 352 637 


11.5325 626 


5.1044 687 


134 


1 79 56 


2 406 104 


11.5758 369 


5.1172 299 


1$5 


1 82 25 


2 460 375 


11.6189 5 


5.1299 278 


136 


1 84 96 


2 515 456 


11.6619 038 


5.1425 632 


137 


1 87 69 


2 571 353 


11.7046 999 


5.1551 367 


138 


1 90 44 


2 628 072 


11.7473 401 


5.1676 493 


139 


1 93 21 


2 685 619 


11.7898 261 


5.1801 015 


140 


1 96 00 


2 744 000 


11.8321 596 


5.1924 941 


141 


1 98 81 


2 803 221 


11.8743 421 


5.2048 279 


142 


2 01 64 


2 863 288 


11.9163 753 


5.2171 034 


143 


2 04 49 


2 924 207 


41.9582 607 


5.2293 215 


144 


2 07 36 


2 985 984 


12. 


5.2414 828 


14") 


2 10 25 


3 048 625 


12.0415 946 


5.2535 879 


146 


2 13 16 


3 112 136 


12.0830 46 


5.2656 374 


147 


2 16 09 


3 176 523 


12.1243 557 


5.277G 321 


148 


2 19 04 


3 241 792 


12.1655 251 


5.2895 725 


149 


2 22 01 


3 307 949 


12.2065 556 


5.3014 592 


150 


2 25 00 


3 375 000 


12.2474 487 


5.3132 928 


151 


2 28 01 


3 442 951 


12.2*82 057 


5.3250 74 


152 


2 31 04 


3 511 008 


12.32*8 28 


5.3368 033 


153 


2 34 09 


3 581 577 


12.3693 169 


5.3484 812 


154 


2 37 16 


3 652 264 


12.409(5 736 


5.3601 084 


155 


2 40 25 


3 723 875 


12.4498 996 


5.3716 854 


156 


2 43 36 


3 796 416 


12.489!) 96 


5.3832 126 


157 


2 46 49 


3 869 893 


12.5299 641 


5.3946 907 


158 


2 49 64 


3 944 312 


12.5098 051 


5.4061 202 


159 


2 52 81 


4 019 679 


12.6095 202 


5.4175 015 


160 


2 56 00 


4 096 000 


12.6491 106 


5.428* 352 


161 


2 59 21 


4 173 281 


12.6*85 775 


5.4401 218 


162 


2 62 44 


4 251 528 


12.7279 221 


5.4513 618 


l<-3 


2 65 69 


4 330 747 


12.76)71 453 


5.4625 556 


164 


2 68 96 


4 410 944 


12.8062 485 


5.4737 037 


165 


2 72 25 


4 492 125 


12.8452 326 


5.4848 066 


166 


2 75 56 


4 574 296 


12.8840 987 


5.4958 647 


167 


2 78 89 


4 657 463 


12.9228 48 


5.5068 784 



SQUARES, CUBES, AND ROOTS. 



213 







Ta"ble— (Continued ) . 




Number. 


Square. 


. Cube. 


Square Root. 


Cnb« Root. 


168 


2 82 24 


4 741 632 


12.9614 814 


5.5178 484 


169 


2 85 61 


4 826 809 


13. 


5.5287 748 


170 


2 89 00 


4 913 000 


13.0384 048 


5.5396 583 


171 


2 92 41 


5 000 211 


13.0766 968 


5.5504 991 


172 


2 95 84 


5 088 448 


13.1148 77 


5.5612 978 


173 


2 99 29 


5 177 717 


13.1529 464 


5.5720 546 


174 


3 02 76 


5 268 024 


13.1909 06 


5.5827 702 


175 


3 06 25 


5 359 375 


13.2287 566 


5.5934 447 


176 


3 09 76 


5 451 776 


13.2664 992 


5.6040 787 


177 


3 13 29 


5 545 233 


13.3041 347 


5.6146 724 


178 


3 16 84 


5 639 752 


13.3416 641 


5.6252 263 


179 


3 20 41 


5 735 339 


13.3790 882 


5.6357 408 


180 


3 24 00 


5 832 000 


13.4164 079 


5.6462 162 


181 


3 27 61 


5 929 741 


13.4536 24 


5.6566 528 


182 


3 31 24 


6 028 568 


13.4907 376 


5.6670 511 


183 


3 34 89 


6 128 487 


13.5277 493 


5.6774 114 


184 


3 38 56 


6 229 504 


13.5646 6 


5.6877 34 


185 


3 42 25 


6 331 625 


13.6014 705 


5.6980 192 


186 


3 45 96 


6 434 856 


13.6381 817 


5.7082 675 


187 


3 49 69 


6 539 203 


13.6747 943 


5. 7284 791 


188 


3 53 44 


6 644 672 


13.7113 092 


5.7i86 543 


189 


3 57 21 


6 751 269 


13.7477 271 


5.7387 936 


190 


3 61 00 


6 859 000 


13.7840 488 


5.7488 971 


191 


3 64 81 


6 967 871 


13.8202 75 


5.7589 652 


192 


3 68 64 


7 077 888 


13.8564 065 


5.7689 982 


193 


3 72 49 


7 189 057 


13.8924 4 


5.7789 966 


194 


3 76 36 


7 301 384 


13.9283 883 


5.7889 604 


195 


3 80 25 


7 414 875 


13.9642 4 


5.7988 9 


196 


3 84 16 


7 529 536 


14. 


5.8087 857 


197 


3 88 09 


7 645 373 


14.0356 688 


5.8186 479 


198 


3 92 04 


7 762 392 


14.0712 473 


•5.8284 867 


199 


3 96 01 


7 880 599 


14.1067 36 


5.8382 725 


200 


4 00 00 


8 000 000 


14.1421 356 


5.8480 355 


201 


4 04 01 


8 120 601 


14.1774 469 


5.8577 66 


202 


4 08 04 


8 242 408 


14.2126 704 


5.8674 673 


203 


4 12 09 


8 365 427 


14.2478 068 


5.8771 307 


204 


4 16 16 


8 489 664 


14.2828 569 


5.8867 653 


205 


4 20 25 


8 615 125 


14.3178 211 


5.8963 685 


206 


4 24 36 


8 741 816 


14.3527 001 


5.9059 406 


207 


4 28 49 


8 869 743 


14.3874 946 


5.9154 817 


208 


4 32 64 


8 998 912 


14.4222 051 


5.9249 921 


209 


4 86 81 


9 129 329 


14.4568 323 


5.9344 721 


210 


4 41 00 


9 261 000 


14.4913 767 


5.9439 22 


211 


4 45 21 


9 393 931 


14.5258 39 


5.9533 418 


212 


4 49 44 


9 528 128 


14.5602 198 


5.9627 32 


213 


4 53 69 


9 663 597 


14.5945 195 


5.9720 926 


214 


4 57 96 


9 800 344 


14.6287 388 


5.9814 24 


215 


4 62 25 


9 938 375 


14.6628 783 


5.9907 264 


216 


4 66 56 


10 077 696 


14.6969 385 


6„ 


217 


4 70 89 


10 218 313 


14.7309 199 


6.0092 45 


218 


4 75 24 


10 360 232 


14.7648 231 


6.0184 617 


219 


4 79 61 


10 503 459 


14.7986 486 


6.0276 502 


220 


4 84 00 


10 648 000 


14.8323 97 


6.0368 107 


221 


4 88 41 


10 793 861 


14.8660 687 


6.0459 435 


222 


4 92 84 


10 941 048 


14.8996 644 


6.0550 489 


223 


4 97 29 


11 089 567 


14.9331 845 


6.0641 27 



214 



SQUARES, CUBES, AND ROOTS. 







Ta"ble— {Continued). 




'umber. 


Square. 


Cube. 


Square Root- 


Cube Root. 


"22~ 


5 01 76 


11 239 424 


14.9666 295 


6.0731 779 


225 


5 06 25 


11 390 625 


15. 


6.0822 02 


226 


5 10 76 


11 543 176 


15.0332 964 


6.0911 994 


227 


5 15 29 


11 697 083 


15.0665 192 


6.1001 702 


228 


5 19 84 


11 852 352 


15.0996 689 


6.1091 147 


229 


5 24 41 


12 008 989 


15.1327 46 


6.1180 332 


230 


5 29 00 


12 167 000 


15.1657 509 


6.1269 257 


231 


5 33 61 


12 326 391 


15.1986 842 


6.1357 924 


232 


5 38 24 


12 487 168 


15.2315 462 


6.1446 337 


233 


5 42 89 


12 649 337 


15.2643 375 


6.1534 495 


231 


5 47 56 


12 812 904 


15.2970 585 


6.1622 401 


235 


5 52 25 


12 977 875 


15.3297 097 


6.1710 058 


233 


5 56 £6 


13 144 256 


15.3622 915 


6.1797 466 


237 


5 61 69 


13 312 053 


15.3948 043 


6.1884 628 


238 


5 66 44 


13 481 272 


15.4272 486 


6.1971 544 


239 


5 71 21 


13 651 919 


15.4596 248 


6.2058 218 


210 


5 76 00 


13 824 000 


15.4919 334 


6.2144 65 


241 


5 SO 81 


13 997 521 


15.5241 747 


6.2230 843 


242 


85 u4 


14 172 488 


15.5563 49*2 


6.2316 797 


243 


5 90 49 


14 348 907 


15.5884 573 


6.2402 515 


244 


5 95 36 


14 526 784 


15.6204 994 


6.2487 998 


245 


6 00 25 


14 706 125 


15.6524 758 


6.2573 248 


246 


6 05 16 


14 886 936 


io.6843 871 


6.2658 266 


247 


6 10 09 


15 069 223 


15.7162 336 


6.2743 054 


248 


6 15 04 


15 252 992 


15.7480 157 


6.2827 613 


249 


6 20 01 


15 438 249 


15.7797 338 


6.2911 946 


250 


6 25 00 


15 625 000 


15.8113 883 


6,2996 053 


251 


6 30 01 


15 813 251 


15.8429 795 


6.3079 9?5 


252 


6 35 01 


16 003 008 


15.8745 079 


6 3163 596 


253 


6 40 09 


16 194 277 


15.9059 737 


6.3247 035 


254 


6 45 16 


16 387 064 


15.9373 775 


6.3330 256 


255 


6 50 25 


16 581 375 


15.9687 194 


6.3413 257 


256 


6 55 36 


16 777 216 


16. 


6.3496 042 


257 


6 60 49 


16-974 593 


16.0312 195 


6.3578 611 


258 


6 65 64 


17 173 512 


16.0623 784 


6.3660 968 


259 


6 70 81 


17 373 979 


16.0934 769 


6.3743 111 


260 


6 76 00 


17 576 000 


16.1245 155 


6.3825 043 


261 


6 81 21 


17 779 581 


16.1554 944 


6.3906 765 


262 


6 86 44 


17 984 728 


16.1864 141 


6.3988 279 


263 


6 91 69 


18 191 447 


16.2172 747 


6.4069 585 


264 


6 96 96 


18 399 744 


16.2480 768 


6.4150 687 


265 


7 02 25 


18 609 625 


16.2788 206 


6.4231 583 


266 


7 07 56 


18 821 096 


16.3095 064 


6.4312 276 


267 


7 12 89 


19 034 163 


16.3401 346 


6.4392 767 


268 


7 18 24 


19 248 832 


16.3707 055 


6.4473 057 


269 


7 23 61 


19 465 109 


' 16.4012 195 


6.4553 148 


270 


7 29 00 


19 683 000 


16.4316 767 


6.4633 041 


271 


7 34 41 


19 902 511 


16.4620 776 


6.4712 736 


272 


7 39 84 


20 123 648 


16.4924 225 


6.4792 236 


273 


7 45 29 


20 346 417 


16.5227 116 


6.4871 541 


274 


7 50 76 


20 570 824 


16.5529 454 


6.4950 653 


275 


7 56 25 


20 796 875 


16.5831 24 


6.5029 572 


276 


7 61 76 


21 024 576 


16.6132 477 


6.5108 3 


277 


7 67 29 


21 253 933 


1(5.6433 17 


6J)1^6 839 


278 


7 72 84 


21 484 952 


16.6783 32 


6.5265 189 


279 


7 78 41 


21 717 639 


16.7032 931 


6.5343 351 



SQUARES, CUBES, AND ROOTS. 



215 



Table— (Continued ). 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


280 


7 84 00 


21 952 000 


16.7332 005 


6.5421 326 


281 


7 89 61 


22 188 041 


16.7630 546 


6.5499 116 


282 


7 95 24 


22 425 768 


16.7928 556 


6.5576 722 


283 


8 00 89 


22 665 187 


16.8826 038 


6.5654 144 


284 


8 06 56 


22 906 304 


16.8522 995 


6.5731 385 


285 


8 12 25 


23 149 125 


16.8819 43 


6.5808 443 


286 


8 17 96 


23 393 656 


16.9115 345 


6.5885 323 


287 


8 23 69 


23 639 903 


16.9410 743 


6.5962 023 


988 


8 29 44 


23 887 872 


16.9705 627 


6.6038 545 


289 


8 35 21 


24 137 569 


17. 


6.6114 89 


290 


8 41 00 


24 389 000 


17.0293 864 


6.6191 06 


291 


8 46 81 


24 642 171 


17.0587 221 


6.6267 054 


292 


8 52 64 


24 897 088 


17.0880 075 


6.6342 874 


293 


8 58 49 


25 153 757 


17.1172 428 


6.6418 522 


294 


8 64 36 


25 412 184 


17.1464 282 


6.6493 998 


295 


8 70 25 


25 672 375 


17.1755 64 


6.6569 302 


296 


8 76 16 


25 934 336 


17.2046 505 


6.6644 437 


297 


8 82 09 


• 26 198 073 


17.2336 879 


6.6719 403 


298 


8 88 04 


26 463 592 


17.2626 765 


6.6794 2 


299 


8 94 01 


26 730 899 


17.2916 165 


6.6868 831 


300 


9 00 00 


27 000 000 


17.3205 081 


6.6943 295 


301 


9 06 01 


27 270 901 


17.3493 516 


6.7017 593 


302 


9 12 04 


27 543 608 


17.3781 472 


6.7091 729 


303 


9 18 09 


27 818 127 


17.4068 952 


6.7165 7 


304 


9 24 16 


28 094 464 


17.4355 958 


6.7239 508 


305 


9 30 25 


28 372 625 


17.4642 492 


6.7313 155 


306 


9 36 36 


28 652 616 


17.4928 557 


6.7386 641 


307 


9 42 49 


28 934 443 


17.5214 155 


6.7459 967 


308 


9 48 64 


29 218 112 


17.5499 288 


6.7533 134 


309 


9 54 


81 


29 503 609 


17.5783 958 


6.7606 143 


310 


9 61 


00 


29 791 000 


17.6068 169 


6.7678 995 


311 


9 67 


21 


30 080 231 


17.6151 921 


6.7751 69 


312 


9 73 44 


30 371 328 


17.6635 217 


6.7824 229 


313 


9 79 69 


30 664 297 


17.6918 06 


6.7896 613 


314 


9 85 96 


30 959 144 


17.7200 451 


6.7968 844 


315 


9 92 25 


31 255 875 


17.7482 393 


6.8040 921 


316 


9 98 56 


31 554 496 


17.7763 888 


6.8112 847 


317 


10 04 89 


31 855 013 


17.8044 938 


6.8184 62 


318 


10 11 24 


32 157 432 


17.8325 545 


6.8256 242 


319 


10 17 61 


32 461 759 


17.8605 711 


6.8327 714 


320 


10 24 


00 


32 768 000 


17.8885 438 


6.8399 037 


321 


10 30 


41 


33 076 161 


17.9164 729 


6.8470 213 


322 


10 36 


84 


33 386 248 


17.9443 584 


6.8541 24 


323 


10 43 


29 


33 698 267 


17.9722 008 


6.8612 12 


324 


10 49 


76 


34 012 224 


18. 


6.8682 855 


325 


10 56 


25 


34 328 125 


18.0277 564 


6.8753 433 


326 


10 62 


76 


34 645 976 


18.0554 701 


6.8823 888 


327 


10 69 


29 


34 965 783 


18.0831 413 


6.8894 188 


328 


10 75 


84 


35 287 552 


18.1107 703 


6.8964 345 


329 


10 82 


41 


35 611 289 


18.1383 571 


6.9034 359 


330 


10 89 


00 


35 937 000 


18.1659 021 


6.9104 232 


331 


10 95 61 


36 264 691 


18.1934 054 


6.9173 964 


332 


11 02 24 


36 594 368 


18.2208 672 


6.9243 556 


333 


11 08 89 


36 926 037 


18.2482 876 


6. 9313 088 


334 


11 15 56 


37 259 704 


18.2756 669 


6; 9382 821 


335 


11 22 


25 


37 595 375 


18.3030 052 


6.9151 496 



216 



SQUARES, CUBES, AND ROOTS. 



Table — (Continued). 



Number. | Square. 



11 28 96 
11 35 69 
11 42 44 
11 49 21 
11 56 00 
11 62 81 
11 69 64 
11 76 49 
11 83 36 
11 90 25 

11 97 16 

12 04 09 
12 11 04 
12 18 01 
12 25 00 
12 32 01 
12 39 04 
12 46 09 
12 53 16 
12 60 25 
12 67 36 
12 74 49 
12 81 64 
12 88 81 

12 96 00 

13 03 21 
13 10 44 
13 17 69 
13 24 96 
13 32 25 
13 39 06 
13 46 89 
13 54 24 
13 61 61 
13 69 00 
13 76 41 
13 83 84 
13 91 29 

13 98 76 

14 06 25 
14 13 76 
14 21 29 
14 28 84 
14 36 41 
14 44 00 
14 51 61 
14 59 24 
14 66 89 
14 74 56 
14 82 25 
14 89 96 

14 97 69 

15 05 44 
15 13 21 
15 21 00 
15 28 81 



Square Root. 



37 933 056 

38 272 753 
38 614 472 

38 958 219 

39 304 000 

39 651 821 

40 001 688 
40 353 607 

40 707 584 

41 063 625 
41 421 736 

41 781 923 

42 144 192 
42 508 549 

42 875 000 

43 243 551 
43 614 208 

43 986 977 

44 361 864 

44 738 875 

45 118 016 
45 499 293 

45 882 712 

46 268 279 

46 656 000 

47 045 831 
47 437 928 

47 832 147 

48 228 544 

48 627 125 

49 027 896 
49 430 863 

49 836 032 

50 243 409 

50 653 000 

51 064 811 
51 478 848 

51 895 117 

52 313 624 

52 734 375 

53 157 376 

53 582 633 

54 010 152 
54 439 939 

54 872 000 

55 306 341 

55 742 968 

56 181 887 
56 623 104 

56 066 625 

57 512 456 

57 960 603 

58 411 072 

58 863 869 

59 319 000 
59 776 471 



18.3303 028 
18.3575 598 
18.3847 763 
18.4119 526 
18.4390 889 
18.4661 853 
18.4932 42 
18.5202 592 
18.5472 37 
18.5741 756 
18.6010 752 
18.6279 36 
18.6547 581 
18.6815 417 
18.7082 869 
18.7349 94 
18.7616 63 
18.7882 942 
18.8148 877 
18.8414 437 
18.8679 623 
18.8944 436 
18.9208 879 
18.9472 953 
18.9736 66 
19. 

19.0262 976 
19.0525 589 
19.0787 84 
19.1049 732 
19.1311 265 
19.1572 441 
19.1833 261 
19.2093 727 
19.2353 841 
19.2613 603 
19.2873 015 
19.3132 079 
19.3390 796 
19.3649 167 
19.3907 194 
19.4164 878 
19.4422 221 
19.4679 223 
19.4935 887 
19.5192 213 
19.5448 203 
19.5703 858 
19.5959 179 
19.6214 169 
19.6468 827 
19.6723 156 
19.6977 156 
19.7230 829 
19.74*4 177 
19.7737 199 



Cvbe Root. 



6.9520 533 
6.9589 434 
6.9658 198 
6.9726 826 
6.9795 321 
6.9863 681 
6.9931 906 



7.0067 

7.0135 

7.0203 

7.0271 

7.0338 

7.0405 

7.0472 

7.0540 

7.0606 

7.0673 

7.0740 

7.0806 

7.0873 

7.0939 

7.1005 

7.1071 

7.1137 

7.1203 

7.1269 

7.1334 

7.1400 

7.1465 

7.1530 

7.1595 

7.1660 

7.1725 

7.1790 

7.1855 

7.1919 

7.1984 

7.2048 

7.2112 

7.2176 

7.2240 

7.2304 

7.2367 

7.2431 

7.2495 

7.2558 

7.2621 

7.2684 

7.2747 

7.2810 

7.5873 

7.2936 

7.2998 

7.3061 

7.3123 



962 

791 

49 

058 

497 

806 

r-87 

041 

967 

767 

44 

988 

411 

709 

885 

937 

£66 

674 

36 

925 

37 

695 

901 

i)88 

957 

809 

544 

162 

663 

05 

322 

479 

522 

45 

268 

972 

565 

045 

415 

6<5 

b24 

864 

794 

617 

33 

936 

436 

828 



SQUARES, CUBES, AND HOOTS. 



217 



Table— ( Continued ). 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


392 


15 36 64 


60 236 288 


19.7989 899 


7.3186 114 


393 


15 44 49 


60 698 457 


19.8242 276 


7.3248 295 


394 


15 52 36 


61 162 984 


19.8494 332 


7.3310 369 


395 


15 60 25 


61 629 875 


19.8746 069 


7.3372 339 


396 


15 68 16 


62 099 136 


19.8997 487 


7.3434 205 


397 


15 76 09 


62 570 773 


19.9248 588 


7.3495 966 


398 


15 84 04 


63 044 792 


19.9499 373 


7.3557 624 


399 


15 92 01 


63 521 199 


19.9749 844 


7.3619 178 


400 


16 00 00 


64 000 000 


20. 


7.3680 63 


401 


16 08 01 


64 481 201 


20.0249 844 


7.3741 979 


402 


16 16 04 


64 964 808 


20.0499 377 


7.3803 227 


403 


16 24 09 


65 450 827 


20.0748 599 


7.3864 373 


404 


16 32 16 


65 939 264 


20.0997 512 


7.3925 418 


405 


16 40 25 


66 430 125 


20.1246 118 


7.3986 363 


406 


16 48 36 


66 923 416 


20.1494 417 


7.4047 206 


407 


16 56 49 


67 419 143 


20.1742 41 


7.4107 95 


408 


16 64 64 


67 917 312 


20.1990 099 


7.4168 595 


409 


16 72 81 


68 417 929 


20.2237 484 


7.4229 142 


410 


16 81 00 


68 921 000 


20.2484 567 


7.4289 589 


411 


16 89 21 


69 426 531 


20.2731 349 


7.4349 938 


412 


16 97 44 


69 934 528 


20.2977 831 


7.4410 189 


413 


17 05 69 


70 444 997 


20.3224 014 


7.4470 342 


414 


17 13 96 


70 957 944 


20.3469 899 


7.4530 399 


415 


17 22 25 


71 473 375 


20.3715 488 


7.4590 359 


416 


17 30 56 


71 991 296 


20.3960 781 


7.4650 223 


417 


17 38 89 


72 511 713 


20.4205 779 


7.4709 991 


418 


17 47 24 


73 034 632 


20.4450 483 


7.4769 664 


419 


17 55 61 


73 560 059 


20.4694 895 


7.4829 242 


420 


17 64 00 


74 088 000 


20.4939 015 


7.4888 724 


421 


17 72 41 


74 618 461 


20.5182 845 


7.4948 113 


422 


17 80 84 


75 151 448 


20.5426 386 


7.5007 406 


423 


17 89 29 


75 686 967 


20.5669 638 


7.5066 607 


421 


17 97 76 


76 225 024 


20.5912 603 


7.5125 715 


425 


18 06 25 


76 765 625 


20.6155 281 


7.5184 73 


426 


18 14 76 


77 308 776 


20.6397 674 


7.5243 652 


427 


18 23 29 


77 854 483 


20.6639 783 


7.5302 482 


428 


18 31 84 


78 402 752 


20.6881 609 


7.5361 221 


429 


18 40 41 


78 953 589 


20.7123 152 


7.5419 867 


430 


18 49 00 


79 507 000 


20.7364 414 


7.5478 423 


431 


18 57 61 


80 062 991 


20.7605 395 


7.5536 888 


432 


18 66 24 


80 621 568 


20.7846 097 


7.5595 263 


433 


18 74 89 


81 182 737 


20.8086 52 


7.5653 548 


434 


18 83.56 


81 746 504 


20.8326 667 


7.5711 743 


435 


18 92 25 


82 312 875 


20.8566 536 


7.5769 849 


436 


19 00 96 


82 881 856 


20.8806 13 


7.5827 865 


437 


19 09 69 


83 453 453 


20.9045 45 


7.5885 793 


438 


19 18 44 


84 027 672 


20.9284 495 


7.5943 633 


439 


19 27 21 


84 604 519 


20.9523 268 


7.6001 385 


440 


19 36 00 


85 184 000 


20.9761 77 


7.6059 049 


441 


19 44 81 


85 766 121 


21. 


7.6116 626 


442 


19 53 64 


86 350 888 


21.0237 96 


7.6174 116 


443 


19 62 49 


86 938 307 


21.0475 652 


7.6231 519 


444 


19 71 36 


87 528 384 


21.0713 075 


7.6288 837 


445 


19 80 25 


88 121 125 


21.0950 231 


7.6346 067 


446 


19 89 16 


88 716 536 


21.1187 121 


7.6403 213 


447 


19 98 09 


89 314 623 


21.1423 745 


7.6460 272 



218 



SQUARES, CUBES, AND ROOTS.' 



Table— {Continued). 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


448 


20 07 04 


89 915 392 


21.1660 105 


7.6517 247 


449 


20 16 01 


90 518 849 


21.1896 201 


7.6574 138 


450 


20 25 00 


91 125 000 


21.2132 034 


7.6630 943 


451 


20 34 01 


91 733 851 


21.2367 606 


7.6687 665 


452 


20 43 04 


92 345 408 


21.2602 916 


7.6744 303 


453 


20 52 09 


92 959 677 


21.2837 967 


7.6800 857 


454 


20 61 16 


93 576 664 


21.3072 758 


7.6857 328 


455 


20 70 25 


94 196 375 


21.3307 29 


7.6913 717 


456 


20 79 36 


94 818 816 


21.3541 565 


7.6970 023 


457 


20 88 49 


95 443 993 


21.3775 583 


7.7026 246 


458 


20 97 64 


96 071 912 


21.4009 346 


7.7082 388 


459 


21 06 81 


96 702 579 


21.4242 853 


7.7138 448 


460 


21 16 00 


97 336 000 


21.4476 106 


7.7194 426 


461 


21 25 21 


97 972 181 


21.4709 106 


7.7250 325 


462 


21 34 44 


98 611 128 


21.4941 853 


7.7306 141 


463 


21 43 69 


99 252 847 


21.5174 348 


7.7361 877 


464 


21 52 96 


99 897 344 


21.5406 592 


7.7417 532 


465 


21 62 25 


100 544 625 


21.5638 587 


7.7473 109 


466 


21 71 56 


101 194 696 


21.5870 331 


7.7528 606 


467 


21 80 89 


101 847 563 


21.6101 828 


7.7584 023 


468 


21 90 24 


102 503 232 


21.6333 077 


7.7639 361 


469 


21 99 61 


103 161 709 


21.. 6564 078 


7.7694 62 


470 


22 09 00 


103 823 000 


21.6794 834 


7.7749 801 


471 


22 18 41 


104 487 111 


21.7025 344 


7.7804 904 


472 


22 27 84 


105 154 048 


21.7255 61 


7.7859 928 


473 


22 37 29 


105 823 817 


21.7485 632 


7.7914 875 


474 


22 46 76 


106 496 424 


21.7715 411 


7.7969 745 


475 


22 56 25 


107 171 875 


21.7944 947 


7.8024 538 


476 


22 65 76 


107 850 176 


21.8174 242 


7.8079 254 


477 


22 75 29 


108 531 333 


21.8403 297 


7.8133 892 


478 


22 84 84 


109 215 352 


21.8632 111 


7.8188 456 


479 


22 94 41 


109 902 239 


21.8860 686 


7.8242 942 


480 


23 04 00 


100 592 000 


21.9089 023 


7.8297 353 


481 


23 13 61 


111 284 641 


21.9317 122 


7.8351 688 


482 


23 23 24 


111 980 168 


21.9544 984 


7.8405 949 


483 


23 32 89 


112 678 587 


21.9772 61 


7.8460 134 


.484 


23 42 56 


113 379 904 


22. 


7.8514 244 


485 


23 52 25 


114 084 125 


22.0227 155 


7.8568 281 


486 


23 61 96 


114 791 256 


22.0454 077 


7.8622 242 


487 


23 71 69 


115 501 303 


22.0680 765 


7.8676 13 


488 


23 81 44 


116 214 272 


22.0907 22 


7.8729 944 


489 


23 91 21 


116 930 169 


22.1133 444 


7.8783 684 


490 


24 01 00 


117 649 000 


22.1359 486 


7.8837 352 


491 


24 10 81 


118 370 771 


22.1585 1?8 


7.8890 946 


492 


24 20 64 


119 095 488 


22.1810 73 


7.8944 468 


493 


24 30 49 


119 823 157 


22.2036 033 


7.8997 917 


494 


24 40 36 


120 553 784 


22.2261 108 


7.9051 294 


495 


24 50 25 


121 287 375 


22.2485 955 


7.9104 599 


496 


24 60 16 


122 023 936 


22.2710 575 


7.9157 832 


497 


24 70 09 


122 763 473 


22.2934 968 


7.9210 994 


498 


24 80 04 


123 505 992 


22.3159 136 


7.9264 085 


499 


24 90 01 


124 251 499 


22.3383 079 


7.9317 104 


500 


25 00 00 


125 000 000 


22.3606 798 


7.9370 053 


501 


25 10 01 


125 751 501 


22.3830 293 


7.9422 931 


502 


25 20 04 


126 506 008 


22.4053 565 


7.9475 739 


503 


25 30 09 I 


127 263 527 


22.4276 615 


7.9528 477 



SQUARES, CUBES, AND ROOTS. 



219 



Ta"ble— (Continued). 



Number. 


| Square. 


i Cube. 


j Square Root. 


Cube Root. 


504 


25 40 16 


128 024 064 


22.4499 443 


7.9581 144 


505 


25 50 25 


128 787 625 


22.4722 051 


7.9633 743 


506 


25 60 36 


129 554 246 


22.4944 438 


7.9686 271 


507 


25 70 49 


130 323 843 


22.5166 605 


7.9738 731 


508 


25 80 64 


131 096 512 


22.5388 553 


7.9791 122 


509 


25 90 81 


131 87? 229 


22.5610 283 


7.9843 444 


510 


26 01 00 


132 651 000 


22.5831 796 


7.9895 697 


511 


26 11 21 


133 432 831 


22.6053 091 


7.9947 883 


512 


26 21 44 


134 217 728 


22.6274 17 


8. 


513 


26 31 69 


135 005 697 


22.6495 033 


8.0052 049 


514 


26 41 96 


135 796 744 


22.6715 681 


8.0104 032 


515 


26 52 25 


136 590 875 


22.6936 114 


8.0155 946 


516 


26 62 56 


137 388 096 


22.7156 334 


8.0207 794 


517 


26 72 89 


138 188 413 


22.7376 340 


8.0259 574 


518 


26 83 24 


138 991 832 


22.7596 134 


8.0311 287 


519 


26 93 61 


139 798 359 


22.7815 715 


8.0362 935 


520 


27 04 00 


140 608 000 


22.8035 085 


8.0414 515 


521 


27 14 41 


141 420 761 


22.8254 244 


8.0466 03 


522 


27 24 84 


142 236 648 


22.8473 193 


8.0517 479 


523 


27 35 29 


143 055 667 


22.8691 933 


8.0568 862 


524 


27 45 76 


143 877 824 


22.8910 463 


8.0620 18 


525 


27 56 25 


144 703 125 


22.9128 785 


8.0671 432 


526 


27 66 76 


145 531 576 


22.9346 899 


8.0722 62 


527 


27 77 29 


146 363 183 


22.9564 806 


8.0773 743 


528 


27 87 84 


147 197 952 


22.9782 506 


8.0824 8 


529 


27 98 '41 


148 035 889 


23. 


8.0875 794 


530 


28 09 00 


148 877 000 


23.0217 289 


8.0926 723 


531 


28 19 61 


149 721 291 


23.0434 372 


8.0977 589 


532 


28 30 24 


150 568 768 


23.0651 252 


8.1028 39 


533 


28 40 89 


151 419 437 


23.0867 928 


8.1079 128 


534 


28 51 56 


152 273 304 


23.1084 4 


8.1129 803 


535 


28 62 25 


153 130 375 


23.1300 67 


8.1180 414 


536 


28 72 96 


153 990 656 


23.1516 738 


8.1230 962 


537 


28 83 69 


154 854 153 


23.1732 605 


8.1281 447 


538 


28 94 44 


155 720 872 


23.1948 27 


8.1331 87 


539 


29 05 21 


156 590 819 


23.2163 735 


8.1382 23 


540 


29 16 00 


157 464 000 


23.2379 001 


8.1432 529 


541 


29 26 81 


158 340 421 


23.2594 067 


8.1482 765 


542 


29 37 64 


159 220 088 


23.2808 935 


8.1532 939 


543 


29 48 49 


160 103 007 


23.3023 604 


8.1583 051 


544 


29 59 36 


160 989 184 


23.3238 076 


8.1633 102 


545 


29 70 25 


161 878 625 


23.3452 351 


8.1683 092 


546 


29 81 16 


162 771 336 


23.3666 429 


8.1733 02 


547 


29 92 09 


163 667 323 


23.3880 311 


8.1782 888 


548 


30 03 04 


164 566 592 


23.4093 998 


8.1832 395 


549 


30 U 01 


165 469 149 


23.4307 49 


8.1882 441 


550 


30 25 00 


166 375 000 


23.4520 788 


8.1932 127 


551 


30 36 01 


167 284 151 


23.4733 892 


8.1981 753 


552 


30 47 04 


168 196 608 


23.4946 802 


8.2031 319 


553 


30 58 09 


169 112 377 


23.5159 52 


8.2080 825 


554 


30 69 16 


170 031 464 


23.5372 046 


8.2130 271 


555 


30 80 25 


170 953 875 


23.5584 38 


8.2179 657 


556 


30 91 36 


171 879 616 : 


23.5796 522 


8.2228 986 


657 


31 02 49 


172 808 693 


23.6008 474 i 


8.2278 254 


558 


. 31 13 64 


173 741 112 


23.6220 236 


8.2327 463 


559 


31 24 81 


174 676 879 


23.6431 808 


8.2376 614 



220 



SQUARES. CUBES, AND BOOTS. 



Table— {Continued). 



Number. i 


Square. 


Cube. 


Square Root. 


Cube Root. 


560 


31 36 00 


175 616 000 


23.6643 191 


8.2425 706 


561 


31 47 21 


176 558 481 


23.6854 386 


8.2474 74 


562 


31 58 44 


177 504 328 


23.7065 392 


8.2523 715 


563 


31 69 69 


178 453 547 


23.7276 21 


8.2572 635 


564 


31 80 96 


179 406 144 


23.7486 842 


8.2621 492 


565 


31 92 25 


180 362 125 


23.7697 286 


8.2670 294 


566 


32 03 56 


181 321 496 


23.7907 545 


8.2719 039 


567 


32 14 89 


182 284 263 


23.8117 618 


8.2767 726 


568 


32 26 24 


183 250 432 


23.8327 506 


8.2816 255 


569 


32 37 61 


184 220 009 


23.8537 209 


8.2864 928 


570 


32 49 00 


185 193 000 


23.8746 728 


8.2913 444 


571 


32 60 41 


186 169 411 


23.8956 063 


8.2961 903 


572 


32 71 84 


187 149 248 


23.9165 215 


8.3010 304 


573 


32 83 29 


188 132 517 


23.9374 184 


8.3058 651 


574 


32 94 76 


189 119 224 


23.9582 971 


8.3106 941 


575 


33 06 25 


190 109 375 


23.9791 576 


8.3155 175 


576 


33 17 76 


191 102 976 


24. 


8.3203 353 


577 


33 29 29 


192 100 033 


24.0208 243 


8.3251 475 


578 


33 40 84 


193 100 552 


24.0416 306 


8.3299 542 


579 


33 52 41 


194 104 539 


24.0624 188 


8.3347 553 


580 


33 64 00 


195 112 000 


24.0831 891 


8.3395 509 


581 


33 75 61 


196 122 941 


24.1039 416 


8.3443 41 


582 


33 87 24 


197 137 368 


24.1246 762 


8.3491 256 


583 


33 98 89 


198 155 287 


24.1453 929 


8.3539 047 


584 


34 10 56 


199 176 704 


24.1660 919 


8.3586 784 


585 


34 22 25 


200 201 625 


24.1867 732* 


8.3634 466 


586 


34 33 96 


201 230 056 


24.2074 369 


8.3682 095 


587 


34 45 69 


202 262 003 


24.2280 829 


8.3729 668 


588 


34 57 44 


203 297 472 


24.2487 113 


8.3777 188 


589 


34 69 21 


204 336 469 


24.2693 222 


8.3824 653 


590 


34 81 00 


205 379 000 


24.2899 156 


8.3872 065 


591 


34 92 81 


206 425 071 


24.3104 916 


8.3919 423 


592 


35 04 64 


207 474 688 


24.3310 501 


8.3966 729 


593 


35 16 49 


208 527 857 


24.3515 913 


8.4013 981 


594 


35 28 36 


209 584 584 


24.3721 152 


8.4061 180 


595 


35 40 25 


210 644 875 


24.3926 218 


8.4108 326 


596 


35 52 16 


211 708 736 


24.4131 112 


8.4155 419 


597 


35 64 09 


212 776 173 


24.4335 834 


8.4202 46 


598 


35 76 04 


213 847 192 


24.4540 385 


8.4249 448 


599 


35 88 01 


214 921 799 


24.4744 765 


8.4296 383 


600 


36 00 00 


216 000 000 


24.4948 974 


8.4343 267 


601 


36 12 01 


217 081 801 


24.5153 013 


8.4390 098 


602 


36 24 04 


218 167 208 


24.5356 883 


8.4436 877 


603 


36 36 09 


219 256 227 


24.5560 583 


8.4483 605 


604 


36 48 16 


220 348 864 


24.5764 115 


8.4530 281 


605 


36 CO 25 


221 445 125 


24.5967 478 


8.4576 906 


606 


36 72 36 


222 545 016 


24.6170 673 


8.4623 479 


607 


36 84 49 


223 648 543 


24.6373 7 


8.467 


608 


36 96 64 


224 755 712 


24.6576 56 


8.4716 471 


609 


37 08 81 


225 866 529 


24.6779 254 


8.4762 892 


610 


37 21 00 


226 981 000 


24.6981 781 


8.4809 261 


611 


37 33 21 


228 099 131 


24.7184 142 


8.4855 579 


•612 


37 45 44 


229 220 928 


24.7386 338 


8.4901 848 


613 


37 57 69 


230 346 397 


24.7588 368 


8.4948 065 


614 


37 69 96 


231 475 544 


24.7790 234 


8.4994 233 


615 


37 82 25 


232 608 375 


24.7991 935 


8.5040 35 






SQUARES, CUBES, AND ROOTS. 



221 







Ta"ble— {Continued). 




Number. 


Square 


Cube 


l Square Root. 


1 Cube Root. 


616 


37 94 56 


233 744 896 


24.8193 473 


8.5086 417 


617 


38 06 89 


234 885 113 


24.8394 847 


8.5132 435 


618 


38 19 24 


236 029 032 


24.8596 058 


8.5178 403 


619 


38 31 61 


237 176 659 


24.8797 106 


8.5224 321 


620 


38 44 00 


238 328 000 


24.8997 992 


8.5270 189 


621 


38 56 41 


239 483 061 


24.9198 716 


8.5316 009 


622 


38 68 84 


240 641 848 


24.9399 278 


8.5361 78 


623 


38 81 29 


241 804 367 


24.9599 679 


8.5107 501 


624 


38 93 76 


242 970 624 


24.9799 92 


8.5453 173 


625 


39 06 25 


244 140 625 


25. 


8.5498 797 


620 


39 18 76 


245 134 376 


25.0199 92 


, 8.5544 372 


627 


39 31 29 


246 491 883 


25.0399 681 


8.5589 899 


628 


39 43 84 


247 673 152 


25.0599 282 


8.5635 377 


629 


39 56 41 


248 858 189 


25.0798 724 


8.5680 807 


630 


39 69 00 


250 047 000 


25.0998 008 


8.5726 189 


631 


39 81 61 


251 239 591 


25.1197 134 


8.5771 523 


632 


39 94 24 


252 435 968 


25.1396 102 


8.5816 809 


633 


40 06 89 


253 636 137 


25.1594 913 


8.5862 047 


634 


40 19 56 


254 840 104 


25.1793 566 


8.5907 238 


635 


40 32 25 


256 047 875 


25.1992 063 


8.5952 38 


636 


40 44 96 


257 259 456 


25.2190 404 


8.5997 476 


637 


40 57 69 


258 474 853 


25.2388 589 


8.6042 525 


638 


40 70 44 


259 694 072 


25.2586 619 


8.6087 526 


639 


40 83 21 


260 917 119 


25.2784 493 


8.6132 48 


640 


40 96 00 


262 144 000 


25.2982 213 


8.6177 388 


641 


41 08 81 


263 374 721 


25.3179 778 


8.6222 248 


642 


41 21 64 


264 609 288 


25.3377 189 


8.6267 063 


643 


41 34 49 


265 847 707 


25.3574 447 


8.6311 83 


644 


41 47 36 


267 089 984 


25.3771 551 


8.6356 551 


645 


41 60 25 


268 336 125 


25.3968 502 


8.6401 216 


646 


41 73 16 


269 585 136 


25.4165 301 


8.6445 855 


647 


41 86 09 


270 840 023 


25.4361 947 


8.6490 437 


618 


41 99 04 


272 097 792 


25.4558 441 


8.6534 974 


649 


42 12 01 


273 359 549 


25.4754 784 


8.6579 465 


650 


42 25 00 


274 625 000 


25.4950 976 


8.6623 911 


651 


42 38 01 


275 894 451 


25.5147 016 


8.6668 31 


652 


42 51 04 


277 167 808 


25.5342 907 


8.6712 665 


653 


42 64 09 


278 445 077 


25.5538 647 


8.6756 974 


654 


42 77 16 


279 726 264 


25.5734 237 


8.6801 237 


655 


42 90 25 


281 Oil 375 


25.5929 678 


8.6845 156 


656 


43 03 36 


282 300 416 


25.6124 969 


8.6889 63 


657 


43 16 49 


283 593 393 


25.6320 112 


8.6933 759 


658 


43 29 64 


284 890 312 


25.6515 107 


8.6977 843 


659 


43 42 81 


286 191 179 


25.6709 953 


8.7021 882 


660 


43 56 00 


287 496 000 


25.6904 652 


8.7065 877 


661 


43 69 21 


288 804 781 


25.7099 203 


8.7109 827 


662 


43 82 44 


290 117 528 


25.7293 607 


8.7153 734 


663 


43 95 69 


291 434 247 


25.7487 864 


8.7197 596 


664 


44 08 96 


292 754 944 


25.7681 975 


8.7241 414 


665 


44 22 25 


294 079 625 


25.7875 939 


8.7285 187 


666 


44 35 56 


295 408 296 


25.8069 758 


8.7328 918 


667 


44 48 89 


296 740 963 


25.8263 431 


8.7372 604 


668 


44 62 24 


298 077 632 


25.8456 96 


8.7416 246 


669 


44 75 61 


299 418 309 


25.8650 343 


8.7459 846 


670 


44 89 00 


300 763 000 


25.8843 582 


8.7503 401 


671 


45 02 41 


302 111 711 
T* 


25.9036 677 • 


8.7516 913 



222 



SQUARES, CUBES, AND ROOTS. 



Table — (Continued). 



Number. 


Square. 


Cube 


Square Root. 


l Cube Root. 


672 


45 15 84 


303 464 448 


25.9229 628 ■ 


8.7590 383 


673 


45 29 29 


304 821 217 


25.9422 435 


8.7633 809 


674 


45 42 76 


306 182 024 


25.9615 1 


8.7677 192 


675 


45 56 25 


307 546 875 


25.9807 621 


8.7720 532 


676 


45 69 76 


308 915 776 


26. 


8.7763 83 


677 


45 83 29 


310 288 733 


26.0192 237 


8.7807 084 


678 


45 96 84 


311 665 752 


26.0384 331 


8.7850 296 


679 


46 10 41 


313 046 839 


26.0576 284 


8.7893 46? 


680 


46 24 00 


314 432 000 


26.0768 096 


8.7936 593 


681 


46 37 61 


315 821 241 


26.0959 767 


8.7979 679 


682 


46 51 24 


317 214 568 


26.1151 297 


8.8022 721 


683 


46 64 89 


318 611 987 


26.1342 687 


8.8065 722 


684 


46 78 56 


320 013 504 


26.1533 937 


8.8108 681 


685 


46 92 25 


321 419 125 


26.1725 047 


8.8151 598 


686 


47 05 96 


322 828 856 


26.1916 017 


8.8194 474 


687 


47 19 69 


324 242 703 


26.2106 848 


8.8237 307 


688 


47 33 44 


325 660 672 


26.2297 541 


8.8280 099 


689 


47 47 21 


327 082 769 


26.2488 095 


8.8322 85 


690 


47 61 00 


328 509 000 


26.2678 511 


8.8365 559 


691 


47 74 81 


329 939 371 


26.2868 789 


8.8408 227 


692 


47 88 64 


331 373 888 


26.3058 929 


8.8450 854 


693 


48 02 49 


332 812 557 


26.3248 932 


8.8493 44 


694 


48 16 36 


334 255 384 


26.3438 797 


8.8535 985 


695 


48 30 25 


335 702 375 


26.3628 527 


8.8578 489 


696 


48 44 16 


337 153 536 


26.3818 119 


8.8620 952 


697 


48 58 09 


338 608 873 


26.4007 576 


8.8663 375 


698 


48 72 04 


340 068 392 


26.4196 896 


8c 8705 757 


699 


48 86 01 


341 532 099 


26.4386 081 ■ 


8.8748 099 


700 


49 00 00 


343 000 000 


26.4575 131 


8.8790 4 


701 


49 14 01 


344 472 101 


26.4764 046 


8.8832 661 


702 


49 28 04 


345 948 408 


26.4952 826 


8.8874 882 


703 


49 42 09 


347 428 927 


26.5141 472 


8.8917 063 


704 


49 56 16 


348 913 664 


26.5329 983 


8.8959 204 


705 


49 70 25 


350 402 625 


26.5518 361 


8.9001 304 


706 


49 84 36 


351 895 816 


26.5706 605 


8.9043 366 


707 


49 98 49 


353 393 243 


26.5894 716 


8.9085 387 


708 


50 12 64 


354 894 912 


26.6082 694 


8.9127 369 


709 


50 26 81 


356 400 829 


26.6270 539 


8.9169 311 


710 


50 41 00 


357 911 000 


26.6458 252 


8.9211 214 


711 


50 55 21 


359 425 431 


'26.6645 833 


8.9253 078 


712 


50 69 44 


360 944 128 


26.6833 281 


8.9294 902 


713 


50 83 69 


362 467 097 


26.7020 598 


8.9336 687 


714 


50 97 96 


363 994 344 


26.7207 784 


8.9378 433 


715 


51 12 25 


365 525 875 


26.7394 839 


8.9420 14 


716 


51 26 56 


367 061 696 


26.7581 763 


8.9461 809 


717 


51 40 89 


368 601 813 


26.7768 557 


8.9503 438 


718 


51 55 24 


370 146 232 


26.7955 22 


8.9545 029 


719 


51 69 61 


371 694 959 


26.8141 754 


8.9586 581 


720 


51 84 00 


373 248 000 


26.8328 157 


8.9628 095 


721 


51 98 41 


374 805 361 


26.8514 432 


8.9669 57 


722 


52 12 84 


376 367 048 


26.8700 577 


8.9711 007 


723 


52 27 29 


377 933 067 


26. m6 593 


8.9752 406 


724 


52 41 76 


379 503 434 


26.9072 481 


8.9793 766 


725 


52 56 25 


381 078 125 


26.9258 24 


8.9835 089 


72G 


52 70 76 


382 657 176 


26.9443 872 


8.9876 373 


727 


52 85 29 


384 240 583 


26.9629 375 


8.9917 62 



SQUARES, CUBES, AND ROOTS. 



223 



Table— {Continued ). 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


728 


52 99 84 


385 828 352 


26.9814 751 


8.9958 899 


729 


53 14 41 


387 420 489 


27. 


9. 


730 


53 29 00 


389 017 000 


27.0185 122 


9.0041 134 


731 


53 43 61 


390 617 891 


27.0370 117 


9.0082 229 


732 


53 58 24 


392 223 168 


27.0554 985 


9.0123 288 


733 


53 72 89 


393 832 837 


27.0739 727 


9.0164 309 


734 


53 87 56 


395 446 904 


27.0924 344 


9.0205 293 


735 


54 02 25 


397 065 375 


27.1108 834 


9.0246 239 


736 


54 16 96 


398 688 256 


27.1293 199 


9.0287 149 


737 


54 31 69 


400 315 553 


27.1477 439 


9.0328 021 


738 


54 46 44 


401 947 272 


27.1661 554 


9.0368 857 


739 


54 61 21 


403 583 419 


27.1845 544 


9.0409 655 


740 


54 76 00 


405 224 000 


27.2029 41 


9.0450 417 


741 


54 90 81 


406 869 021 


27.2213 152 


9.0491 142 


742 


55 05 64 


408 518 488 


27.2396 769 


9.0531 831 


743 


55 20 49 


410 172 407 


27.2580 263 


9.0572 482 


744 


55 35 36 


411 830 784 


27.2763 634 


9.0613 098 


745 


55 50 25 


413 493 625 


27.2946 881 


9.0653 677 


746 


55 65 16 


415 160 936 


27.3130 006 


9.0694 22 


747 


55 80 09 


416 832 723 


27.3313 007 


9.0734 726 


748 


55 95 04 


418 508 992 


27.3495 887 


9.0775 197 


749 


56 10 01 


420 189 749 


27.3678 644 


9.0815 631 


750 


56 25 00 ' 


421 875 000 


27.3861 279 


9.0856 03 


751 


56 40 01 


423 564 751 


27.4043 792 


9.0896 352 


752 


56 55 04 


425 259 008 


27.4226 184 


9.0936 719 


753 


56 70 09 


426 957 777 


27.4408 455 


9.0977 01 


754 


56 85 16 


428 661 064 


27.4590 604 


9.1017 265 


755 


57 00 25 


430 368 875 


27.4772 633 


9.1057 485 


756 


57 15 36 


432 081 216 


27.4954 542 


9.1097 669 


757 


57 30 49 


433 798 093 


27.5136 33 


9.1137 818 


758 


57 45 64 


435 519 512 


27.5317 998 


9.1177 931 


759 


57 60 81 


437 245 479 


27.5499 546 


9.1218 01 


760 


57 76 00 


438 976 000 


27.5680 975 


9.1258 053 


761 


57 91 21 


440 711 081 


27.5862 284 


9.1298 061 


762. 


58 06 44 


442 450 728 


27.6043 475 


9.1338 034 


763 


58 21 69 


444 194 947 


27.6224 546 


9.1377 971 


764 


58 36 96 


445 943 744 


27.6405 499 


9.1417 874 


765 


58 52 25 


447 697 125 


27.6586'334 


9.1457 742 


766 


58 67 56 


449 455 096 


27.6767 05 


9.1497 576 


767 


58 82 89 


451 217 663 


27.6947 648 


9.1537 375 


768 


58 98 24 


452 984 832 


27.7128 129 


9.1577 139 


769 


59 13 61 


454 756 609 


27.7308 492 


9.1616 869 


770 


59 29 00 


456 533 000 


27.7488 739 


9.1656 565 


771 


59 44 41 


458 314 Oil 


27.7668 868 


9.1696 225 


772 


59 59 84 


460 099 648 


27.7848 88 


9.1735 852 


773 


59 75 29 


461 889 917 


27.8028 775 


9.1775 445 


774 


59 90 76 


463 684 824 


27.8208 555 


9.1815 003 


775 


60 06 25 


465 484 375 


27.8388 218 


9.1854 527 


776 


60 21 76 


467 288 576 


27.8567 766 


9.1894 018 


777 


60 37 29 


469 097 433 


27.8747 197 


9.1933 474 


778 


60 52 84 


470 910 952 


27.8926 514 


9.1972 897 


779 


60 68 41 


472 729 139 


27.9105 715 


9.2012 286 


780 


60 84 00 


474 552 000 


27.9284 801 


9.2051 641 


781 


60 99 61 


476 379 541 


27.9463 772 


9.2090 962 


782 


61 15 24 


478 211 768 


27.9642 629 


9.2130 25 


783 


61 30 89 


480 048 687 


27.9821 372 I 


9.2169 505 



224 



SQUARES, CUBES, AND ROOTS. 



Table— (Continued). 



Number. I Square. 



Square Root. 



784 


61 46 56 


481 890 304 


28. 


9.2208 726 


785 


61 62 25 


483 736 625 


28.0178 515 


9.2247 914 


78G 


61 77 96 


485 587 656 


28.0356 915 


9.2287 068 


787 


61 93 69 


487 443 403 


28.0535 203 


9.2326 189 


788 


62 09 44 


489 303 872 


28.0713 377 


9.2365 277 


7S9 


62 25 21 


491 169 069 


28.0891 438 


9.2404 333 


790 


62 41 00 


493 039 000 


28.1069 386 


9.2443 355 


791 


62 56 81 


494 913 671 


28.1247 222 


9.2482 344 


792 


62 72 64 


496 793 088 


28.1424 946 


9.2521 3 


793 


62 88 49 


498 677 257 


28.1602 557 


9.2560 224 


794 


63 04 36 


500 566 184 


28.1780 056 


9.2599 114 


795 


63 20 25 


502 459 875 


28.1957 444 


9.2637 973 


796 


63 36 16 


504 358 336 


28.2134 72 


9.2676 798 


797 


63 52 09 


506 261 573 


28.2311 884 


9.2715 592 


798 


63 68 04 


508 169 592 


28.2488 938 


9.2754 352 


799 


63 84 01 


510 082 399 


28.2665 881 


9.2793 081 


800 


64 00 00 


512 000 000 


28.2842 712 


9.2831 777 


801 


64 16 01 


513 922 401 


28.3019 434 


9.2870 44 


802 


64 32 04 


515 849 608 


28.3196 045 


9.2909 072 


803 


64 48 09 


517 781 627 


28.3372 546 


9.2947 671 


804 


64 64 16 


519 718 464 


28.3548 938 


9.2986 239 


805 


64 80 25 


521 660 125 


28.3725 219 


9.3024 775 


806 


64 96 36 


523 606 616 


28.3901 391 


9.3063 278 


807 


65 12 49 


525 557 943 


28.4077 454 


9.3101 75 


808 


65 28 64 


527 514 112 


28.4253 408 


9.3140 19 


809 


65 44 81 


529 475 129 


28.4429 253 


9.3178 599 


810 


65 61 00 


531 441 000 


28.4604 989 


9.3216 975 


811 


65 77 21 


533 411 731 


28.4780 617 


9.3255 32 


812 


65 93 44 


535 387 328 


28.4956 137 


9.3293 634 


813 


66 09 69 


537 367 797 


28.5131 549 


9.3331 916 


814 


66 25 96 


539 353 144 


28.5306 852 


9.3370 1C7 


815 


66 42 25 


541 343 375 


28.5482 048 


9.3408 3S6 


816 


66 58 56 


543 338 496 


28.5657 137 


9.3446 575 


817 


66 74 89 


545 338 513 


28.5832 119 


9.3484 731 


818 


66 91 24 


547 343 432 


28.6006 993 


9.3522 857 


819 


67 07 61 


549 353 239 


28.6181 76 


9.3560 952 


820 


67 24 00 


551 368 000 


28.6356 421 


9.3599 016 


821 


67 40 41 ' 


553 387 661 


28.6530 976 


9.3637 049 


822 


67 56 84 


555 412 248 


28.6705 424 


9.3675 051 


823 


67 73 29 


557 441 767 


28.6879 766 


9.3713 022 


824 


67 89 76 


559 476 224 


28.7054 002 


9.3750 963 


825 


68 06 25 


561 515 625 


28.7228 132 


9.3788 873 


826 


68 22 76 


563 559 976 


28.7402 157 


9.3826 752 


827 


68 39 29 


565 609 283 


28.7576 077 


9.3864 6 


828 


68 55 84 


567 663 552 


28.7749 891 


9.3902 419 


829 


68 72 41 


569 722 789 


28.7923 601 


9.3940 206 


830 


68 89 00 


571 787 000 


28.8097 206 


9.3977 964 


831 


69 05 61 


573 856 191 


28.8270 706 


9.4015 691 


832 


69 22 24 


575 930 368 


28.8444 102 


9.4053 387 


833 


69 38 89 


578 009 537 


28.8617 394 


9.4091 054 


834 


69 55 56 


580 093 704 


28.8790 582 


9.4128 69 


835 


69 72 25 


582 182 875 


28.8963 666 


9.4166 297 


836 


69 88 96 


5*4 277 056 


28.9136 646 


9.4203 873 


837 


70 05 69 


586 376 253 


28.9309 523 


9.4241 42 


838 


70 22 44 


588 4,90 472 


28.9482 297 


9.4278 936 


839 


70 39 21 


590 589 719 1 


28.9654 967 


9.4316 423 



SQUARES, CUBES, AND ROOTS. 



225 



Table— (Continued). 



Number. 


Square. 


Cube. 


1 Square Root. 


j Cube Root. 


840 


70 56 00 


592 704 000 


28.9827 535 


9.4353 8 


841 


70 72 81 


594 823 321 


29. 


9.4391 307 


842 


70 89 64 


596 947 688 


29.0172 363 


9.4428 704 


843 


71 06 49 


599 077 107 


29.0344 623 


9.4466 072 


844 


71 23 36 


601 211 584 


29.0516 781 


9.4503 41 


845 


71 40 25 


603 351 125 


29.0688 837 


9.4540 719 


846 


71 57 16 


605 495 736 


29.0860 791 


9.4577 999 


847 


71 74 09 


607 645 423 


29.1032 644 


9.4615 249 


848 


71 91 04 


609 800 192 


29.1204 396 


9.4652 47 


849 


72 08 01 


611 960 049 


29.1376 046 


9.4689 661 


850 


72 25 00 


614 125 000 


29.1547 595 


9.4726 824 


851 


72 42 01 


616 295 051 


29.1719 043 


9.4763 957 


852 


72 59 04 


618 470 208 


29.1890 39 


9.4801 061 


853 


72 76 09 


620 650 477 


29.2061 637 


9.4838 136 


854 


72 93 16 


622 835 864 


29.2232 784 


9.4875 182 


855 


73 10 25 


625 026 375 


29.2403 83 


9.4912 2 


856 


73 27 36 


627 222 016 


29.2574 777 


9.4949 188 


857 


73 44 49 


629 422 793 


29.2745 623 


9.4986 147 


858 


73 61 64 


631 628 712 


29.2916 37 


9.5023 078 


859 


73 78 81 


633 839 779 


29.3087 018 


9.5059 98 


860 


73 96 00 


636 056 000 


29.3257 566 


9.5096 854 


861 


74 13 21 


638 277 381 


29.3428 015 


9.5133 699 


862 


74 30 44 


640 503 928 


29.3598 365 


9.5170 515 


863 


74 47 69 


642 735 647 


29.3768 "616 


9.5207 303 


864 


74 64 96 


644 972 544 


29.3938 769 


9.5244 063 


865 


74 82 25 


647 214 625 


29.4108 823 


9.5280 794 


866 


74 99 56 


649 461 896 


29.4278 779 


9.5317 497 


867 


75 16 89 


651 714 363 


29.4448 637 


9.5354 172 


868 


75 34 24 


653 972 032 


29.4618 397 


9.5390 818 


869 


75 51 61 


656 234 909 


29.4788 059 


9.5427 437 


870 


75 69 00 


658 503 000 


29.4957 624 


9.5464 027 


871 


75 86 41 


660 776 311 


29.5127 091 


9.5500 589 


872 


76 03 84 


663 054 848 


29.5296 461 


9.5537 123 


873 


76 21 29 


665 338 617 


29.5465 734 


9.5573 63 


874 


76 38 76 


667 627 624 


29.5634 91 


9.5610 108 


875 


76 56 25 


669 921 875 


29.5803 989 


9.5646 559 


876 


76 73 76 


672 221 376 


29.5972 972 


9.5682 982 


877 


76 91 29 


674 526 133 


29.6141 858 


9.5719 377 


878 


77-08 84 


676 836 152 


29.6310 648 


9.5755 745 


879 


77 26 41 


679 151 439 


29.6479 342 


9.5792 085 


880 


77 44 00 


681 472 000 


29.6647 939 


9.5828 397 


881 


77 61 61 


683 797 841 


29.6816 442 


9.5864 682 


882 


77 79 24 


686 128 968 


29.6984 848 


9.5900 937 


883 


77 96 89 


688 465 387 


29.7153 159 


9.5937 169 


884 


78 14 56 


690 807 104 


29.7321 375 


9.5973 373 


885 


78 32 25 


693 154 125 


29.7489 496 


9.6009 548 


886 


78 49 96 


695 506 456 


29.7657 521 


9.6045 696 


887 


78 67 69 


697 864 103 


29.7825 452 


9.6081 817 


888 


78 85 44 


700 227 072 


29.7993 289 


9.6117 911 


889 


79 03 21 


702 595 3G9 


29.8161 03 


9.6153 977 


890 


79 21 00 


704 969 000 


29.8328 678 


9.6190 017 


891 


79 38 81 


707 347 971 


29.8496 231 


9.6226 03 


892 


79 56 64 


707 932 288 


29.8663 69 


9.6262 016 


893 


79 74 49 


712 121 957 


29.8831 056 


9.6297 975 


894 


79 92 36 


714 516 984 


29.8998 328 


9.6333 907 


895 


80 10 25 


716 917 375 


29.9165 506 


9.6369 812 



226 



SQUARES, CUBES, AND ROOTS. 







Tatole— (Continual). 




Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


896 


80 28 16 


719 323 136 


29.9332 591 


9.6405 69 


897 


80 46 09 


721 734 273 


29.9499 583 


9.6441 542 


898 


80 64 04 


724 150 792 


29.9666 481 


9.6477 367 


899 


80 82 01 


726 572 699 


29.9833 287 


9.6513 166 


900 


81 00 00 


729 000 000 


30. 


9.6548 938 


901 


81 18 01 


731 432 701 


30.0166 62 


9.6584 684 


902 


81 36 04 


733 870 808 


30.0333 148 


9.6620 403 


903 


81 54 09 


736 314 327 


30.0499 584 


9.6656 096 


904 


81 72 16 


738 763 264 


30.0665 928 


9.6691 762 


905 


81 90 25 


741 217 625 


30.0832 179 


9.6727 403 


90G 


82 08 36 


743 677 416 


30.0998 339 


9-6763 017 


907 


82 26 49 


746 142 643 


30.1164 407 


9-6798 604 


908 


82 44 64 


748 613 312 


30.1330 383 


9.6834 166 


909 


82 62 81 


751 089 429 


30.1496 269 


9.6869 701 


910 


82 81 00 


753 571 000 


30.1662 063 


9.6905 211 


911 


82 99 21 


756 058 031 


30.1827 765 


9.6940 694 


912 


83 17 44 


758 550 825 


30.1993 377 


9.6976 151 


913 


83 35 69 


761 048 497 


30.2158 899 


9.7011 583 


9U 


83 53 96 


763 551 944 


30.2324 329 


9.7046 989 


915 


83 72 25 


766 060 875 


30.2489 669 


9.7082 369 


916 


83 90 56 


768 575 296 


30.2654 919 


9.7117 723 


917 


84 08 89 


771 095 213 


30.2820 079 


9.7153 051 


918 


84 27 24 


773 620 632 


30.2985 148 


9.7188 354 


919 


84 45 61 


776 151 559 


30.3150 128 


9.7223 631 


920 


84 64 00 


778 688 000 


30.3315 018 


9.7258 8S3 


921 


84 82 41 


781 229 961 


30.3479 818 


9.7294 109 


922 


85 00 84 


783 777 448 


30.3644 529 


9.7329 309 


923 


85 19 29 


786 330 467 


30.3809 151 


9.7364 484 


924 


85 37 76 


788 889 024 


30.3973 683 


9.7399 634 


925 


85 56 25 


791 453 125 


30.4138 127 


9.7434 758 


926 


85 74 76 


794 022 776 


30.4302 481 


9.7469 857 


927 


85 93 29 


796 597 983 


30.4466 747 


9.7504 93 


928 


86 11 84 


799 178 752 


30.4630 924 


9.7539 979 


929 


86 30 41 


801 765 089 


30.4795 013 


9.7575 002 


930 


86 49 00 


804 357 000 


30.4959 014 


9.7610 001 


931 


86 67 61 


806 954 491 


30.5122 926 


9.7644 974 


932 


86 86 24 


809 557 568 


30.5286 75 


9.7679 922 


933 


87 04 89 


812 166 237 


30.5450 487 


9.7714 845 


934 


87 23 56 


814 780 504 


30.5614 136 


9.7749 743 


935 


87 42 25 


817 400 375 


30.5777 697 


9.7784 616 


936 


87 60 96 


820 025 856 


30.5941 171 


9.7829 466 


937 


87 79 69 


822 656 953 


30.6104 557 


9.7854 288 


938 


87 98 44 


825 293 672 


30.6267 857 


9.7889 087 


939 


83 17 21 


827 936 019 


30.6431 069 


9.7923 861 


910 


88 36 00 


830 584 000 


30.6594 194 


9.7958 611 


941 


88 54 81 


833 237 621 


30.6757 233 


9.7993 336 


942 


88 73 64 


835 806 888 


30.6920 185 


9.8028 036 


943 


88 92 49 


838 561 807 


30.7083 051 


9.8062 711 


944 


89 11 36 


841 232 384 


30.7245 83 


9.8097 362 


945 


89 30 25 


843 908 625 


30.7408 523 


9.8131 989 


946 


89 49 16 


846 590 536 


30.7571 13 


9.8166 591 


947 


89 68 09 


849 278 123 


30.7733 651 


9.8201 169 


918 


89 87 04 


851 971 392 


30.7896 086 


9.8235 723 


919 


90 06 01 


854 670 349 


30.8058 436 


9.8270 252 


950 


90 25 00 


857 375 000 


30.8220 7 


9.8304 757 


951 


90 44 01 


860 085 351 


30.8382 879 


9.8339 238 



SQUARES, CUBES, AND ROOTS. 



227 



Table— (Continued ). 



Number. 


Square. 


| Cube. 


Square Root. 


| Cube Root. 


952 


90 63 04 


862 801 408 


30.8544 972 


9.8373 695 


953 


90 82 09 


865 523 177 


30,8706 981 


9.8408 127 


954 


91 01 16 


868 250 664 


30.8868 904 


9.8442 536 


955 


91 20 25 


870 983 875 


30.9030 743 


9.8476 92 


956 


91 39 36 


873 722 816 


30.9192 477 


9.8511 28 


957 


91 58 49 


876 467 493 


30.9354 166 


9.8545 617 


958 


91 77 64 


879 217 912 


30.9515 751 


9.8579 929 


959 


91 96 81 


881 974 079 


30.9677 251 


9.8614 218 


960 


92 16 00 


884 736 000 


30.9838 668 


9.8648 483 


961 


92 35 21 


887 503 681 


31. 


9.8682 724 


962 


92 54 44 


890 277 128 


31.0161 248 


9.8716 941 


963 


92 73 69 


893 056 347 


31.0322 413 


9.8751 135 


964 


92 92 96 


895 841 344 


31.0483 494 


9.8785 305 


965 


93 12 25 


898 632 125 


31.0644 491 


9.8819 451 


966 


93 31 56 


901 428 696 


31.0805 405 


9.8853 574 


967 


93 50 89 


904 231 063 


31.0966 236 


9.8887 673 


968 


93 70 24 


907 039 232 


31.1126 984 


9.8921 749 


969 


93 89 61 


909 853 209 


31.1287 648 


9.8955 801 


970 


94 09 00 


912 673 000 


31.1448 23 


9.8989 83 


971 


94 28 41 


915 498 611 


31.1608 729 


9.9023 835 


972 


94 47 84 


918 330 048 


31.1769 145 


9.9057 817 


973 


94 67 29 


921 167 317 


31.1929 479 


9.9091 776 


974 


94 86 76 


924 010 424 


31.2089 731 


9.9125 712 


975 


95 06 25 


926 859 375 


31.2249 9 


9.9159 624 


976 


95 25 76 


929 714 176 


31.2409 987 


9.9193 513 


977 


95 45 29 


932 574 833 


31.2569 992 


9.9227 379 


978 


95 64 84 


935 441 352 


31. 2729' 915 


9.9261 222 


979 


95 84 41 


938 313 739 


31.2889 757 


9.9295 042 


980 


96 04 00 


941 192 000 


31.3049 517 


9.9328 839 


981 


96 23 61 


944 076 141 


31.3209 195 


9.9362 613 


982 


96 43 24 


946 966 168 


31.3368 792 


9.9396 363 


983 


96 62 89 


949 862 087 


31.3528 308 


9.9430 092 


984 


96 82 56 


952 763 904 


31.3687 743 


9.9463 797 


985 


97 02 25 


955 671 625 


31.3847 097 


9.9497 479 


986 


97 21 96 


958 585 256 


31.4006 369 


9.9531 138 


987 


97 41 69 


961 504 803 


31.4165 561 


9.9564 775 


988 


97 61 44 


964 430 272 


31.4324 673 


9.9598 389 


989 


97 81 21 


967 361 669 . 


31.4483 704 


9.9631 981 


900 


98 01 00 


970 299 000 


31.4642 654 


9.00G5 549 


991 


98 20 81 


973 242 271 


31.4801 525 


9.9699 095 


992 


98 40 64 


976 191 488 


31.4960 315 


9.9732 619 


993 


98 60 49 


979 146 657 


31.5119 025 


9.9766 12 


994 


98 80 36 


982 107 784 


31.5277 655 


9.9799 599 


995 


99 00 25 


985 074 875 


31.5436 206 


9.9833 055 


996 


99 20 16 


988 047 936 


31.5594 677 


9.9866 488 


997 


99 40 09 


991 026 973 


31.5753 068 


9.9899 9 


998 


99 60 04 


994 Oil 992 


31.5911 38 


9.9933 289 


999 


99 80 01 


997 002 999 


31.6069 613 


9.9966 656 


1000 


1 00 00 oo 


1 000 000 000 


31.6227 766 


10. 


1001 


1 00 02 01 


1 003 003 001 


31.6385 84 


10.0033 222 


1002 


1 00 40 04 


1006 012 008 


31.6543 836 


10.0066 622 


1003 


1 00 60 09 


1 009 027 027 


31.6701 752 


10.0099 899 


1004 


1 00 80 16 


1 012 048 064 


31.6859 59 


10.0183 155 


1005 


1 01 00 25 


1 015 075 125 


31.7017 349 


10.0166 389 


1006 


1 01 00 36 


1018 108 216 


31 .7175 03 


10.0199 601 


1007 


1 01 40 49 


1 021 147 343 


31.7332 633 


10.0232 791 



228 



SQUARES, CUBES, AND HOOTS. 



Table — {Continued). 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


1008 


1 01 60 64 


1 024 192 512 


31.7490 157 


10.0265 958 


1009 


1 01 80 81 


1 027 243 729 


31.7647 G03 


10.0299 104 


1010 


1 02 01 00 


1 030 301 000 


31.7804 972 


10.0332 228 


1011 


1 02 01 21 


1 033 364 331 


31.7962 262 


10.0365 33 


1012 


1 02 41 44 


1 036 433 728 


31.8119 474 


10.0398 41 


1013 


1 02 61 69 


1 039 509 197 


31. £276 609 


10.0431 469 


1014 


102 81 96 


1 042 590 744 


31.8433 666 


10.0464 506 


1015 


1 03 02 25 


1 045 678 375 


31.8590 646 


10.0497 521 


1016 


1 03 22 56 


1 048 772 096 


31.8747 549 


10.0530 514 


1017 


1 03 42 89 


1 051 871 913 


31.8904 374 


10.0563 485 


1018 


1 03 63 24 


1 054 977 832 


31.9061 123 


10.0596 435 


1019 


1 03 83 61 


1 058 089 859 


31.9217 794 


10.0629 364 


1020 


1 04 04 00 


1 061 208 000 


31.9374 388 


10.0662 271 


1021 


1 04 24 41 


1 064 332 261 


31.9530 906 


10.0695 156 


1022 


1 04 44 84 


1067 462 648 


31.9687 347 


10.0728 02 


1023 


1 04 65 29 


1 070 599 167 


31.9843 712 


10.0760 863 


102. i 


1 04 85 76.. 


1 073 741 824 


32. 


10.0793 684 


Ufe5 


1 05 06 25 


1 076 890 625 


32.0156 212 


10.0826 484 


1026 


1 05 26 76 


1 080 045 576 


32.0312 348 


10.0859 262 


1027 


1 05 47 29 


1 083 206 683 


32.0468 407 


10.0892 019 


1028 


1 05 67 84 


1 086 373 952 


32.0624 391 


10.0924 755 


1029 


1 05 88 41 


1 089 547 389 


32.0780 298 


10.0957 469 


1030 


1 06 09 00 


1 092 727 000 


32.0936 131 


10.0990 163 


1031 


1 06 29 61 


1 095 912 791 


32.1091 887 


10.1022 835 


1032 


1 06 50 24 


1 099 104 768 


32.1247 568 


10.1055 487 


1033 


1 06 70 89 


1 102 302 937 


32.1403 173 


10.1088 117 


1034 


1 06 91 56 * 


1 105 507 304 


32.1558 704 


10.1120 726 


1035 


1 07 12 25 


1 108 717 875 


32.1714 159 


10.1155 314 


1036 


1 07 32 96 


1 111 934 656 


32.1869 539 


10.1185 882 


1037 


1 07 53 69 


1 115 157 653 


32.2024 844 


40.1218 428 


1038 


1 07 74 44 


1 118 386 872 


32.2180 074 


10.1260 953 


1039 


1 07 95 21 


1 121 622 319 


32.2335 229 


10.1283 457 


1040 


1 08 16 00 


1 124 864 000 


32.2490 31 


10.1315 941 


1041 


1 08 36 81 


1 128 111 921 


32.2645 316 


10.1348 403 


1042 


1 08 57 64 


1 131 366 088 


32.2800 248 


10.1380 845 


1043 


1 08 78 49 


1 134 626 507 


32.2955 105 


10.1413 266 


1044 


1 08 99 36 


1 137 893 184 


32.3109 888 


10.1445 667 


1045 


1 09 20 25 


1 141 166 125 


32.3264 598 


10.1478 047 


1046 


1 09 41 16 


1 144 445 336 


32.3419 233 


10.1510 406 


1047 


1 09 62 09 


1 147 730 823 


32.3573 794 


10.1542 744 


1048 


1 09 83 04 


1 151 022 592 


32.3728 281 


10.1575 062 


1049 


1 10 04 01 


1 154 320 649 


32.3882 695 


10.1607 359 


1050 


1 10 25 00 


1 157 625 000 


32.4037 035 


10.1639 636 


1051 


1 10 46 01 


1 160 935 651 


32.4191 301 


10.1671 893 


1052 


1 10 67 04 


1 164 252 608 


32.4345 495 


10.1704 129 


1053 


1 10 88 09 


1 167 575 877 


32.4499 615 


10.1736 344 


1054 


1 11 09 16 


1 170 905 464 


32.4653 662 


10.1768 5S9 


1055 


1 11 30 25 


1 174 241 375 


32.4807 635 


10.1800 714 


1056 


1 11 51 36 


1 177 583 616 


32.4961 536 


10.1832 868 


1057 


1 11 72 49 


1 180 932 193 


32.5115 364 


10.1865 002 


1058 


1 11 93 64 


1 184 287 112 


32.5269 119 


10.1897 116 


1059 


1 12 14 81 


1 187 648 379 


32.5422 802 


10.1929 209 


1060 


112 36 00 


■ 1 191 016 000 


32.5576 412 


10.1961 283 


1061 


1 12 57 21 


1 194 389 981 


32.5729 949 


10.1993 3S6 


1062 


1 12 78 44 


1 197 770 328 


32.5883 415 


10.2025 369 


1063 


1 12 99 69 


1 201 157 047 


32.6036 807 


10.2057 382 









SQUARES, CUBES, AND ROOTS. 



229 



Ta"ble— (Continued). 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


1064 


1 13 20 96 


1 204 550 144 


32.6190 129 


10.2089 375 


1065 


1 13 42 25 


1 207 949 625 


32.6343 377 


10.2121347 


1066 


1 13 63 b6 


1 211 355 496 


32.6496 554 


10.2153 3 


1067 


1 13 84 89 


1 214 767 763 


32.6649 659 


10.2185 233 


1068 


1 14 06 24 


1 218 186 432 


32.6802 693 


10.2217 146 


1069 


1 14 27 61 


1 221 611 509 


32.6955 654 


10.2249 039 


1070 


1 14 49 00 


1 225 043 000 


32.7108 544 


10.2280 912 


1071 


1 14 70 41 


1 228 480 911 


32.7261363 


10.2312 766 


1072 


1 14 91 84 


1 231 925 248 


32.7414111 


10.2344 599 


1073 


1 15 13 29 


1 235 376 017 


32.7566 787 


10.2376 413 


1074 


1 15 34 76 


1 238 833 224 


32.7719 392 


10.2408 207 


1075 


1 15 5.6 25 


1 242 296 875 


32.7871926 


10.2439 981 


1076 


1 15 77 76 


1 245 766 976 


32.8024 389 


10.2471735 


1077 


1 15 99 29 


1 249 243 533 


32.8176 782 


10.2503 47 


1078 


1 16 20 84 


1 252 726 552 


32.8329103 


10.2535 186 


1079 


J 1 16 42 41 


1 256 216 039 


32.8481354 


10.2566 881 


1080 


1 16 64 00 


1 259 712 000 


32.8633 535 


10.2598 55' 


1081 


1 16 85 61 


1 263 214 441 


32.8785 644 


10 2630 213 


1082 


1 17 07 24 


1 266 723 368 


32.8937 684 


iO.Sfe6185 


1083 


1 17 28 89 


1 270 238 787 


32.9089 653 


10.2fe-67 


1084 


1 17 50 56 


1 273 760 704 


32.9241553 


10.2725 065 


1085 


1 17 72 25 


1 277 289 125 


32.9393 382 


10.2756 644 


1086 


1 17 93 96 


1 280 824 056 


32.9545141 


10.2788 203 


1087 


1 18 15 69 


1 284 365 503 


32.9696 83 


10.2819 743 


1088 


1 18 37 44 


1 287 913 472 


32.9848 45 


10.2851264 


1089 


1 18 59 21 


1 291 467 969 


33. 


10.2882 765 


1090 


1 18 81 00 


1 295 029 000 


33.015148 


10.2914 247 


1091 


1 19 02 81 


1 298 596 571 


' 33.0302 891 


10.2945 709 


1092 


1 19 24 64 


1 302 170 688 


33.0454 233 


10.2977 153 


1093 


1 19 46 49 


1 £05 V I 357 


33.0605 505 


10.3008 577 


1094 


1 19 68 36 


1 30f bJS 584 


33.0756 708 


10.3039 982 


1095 


1 19 90 25 


1 31'. 932 375 


33.0907 842 


10.3071368 


1096 


1 20 12 16 


1 316 532 736 


33.1058 907 


10.3102 735 


1097 


1 20 34 09 


1 320 139 673 


33.1209 903 


10.3134 083 


1098 


1 20 56 04 


1 823 753 192 


33.1360 83 


10.3165 411 


1099 ' 


1 20 78 01 


1 i >7 373 299 


33.1511689 


10.3196 721 


1100 


1 21 00 00 


1 331 000 000 


33.1662 479 


10.3228 012 


1101 


1 21 22 01 


1 334 633 301 


33.1813 2 


10.3259 284 


1102 


1 21 44 04 


1 338 273 208 


33.1963 853 


10.3290 537 


1103 


1 21 m 09 


1 341 919 727 


33.2114 438 


10.3321 77 


1104 


1 21 88 16 


1 45 572 864 


33.2266 955 


10.3352 985 


1105 


1 22 10 25 


1 U9 232 625 


33.2415 403 


10.3384181 


1106 


1 22 32 36 


i 352 899 016 


33.2565 783 


10.3415 358 


1107 


1 22 54 49 


1 356 572 043 


33.2716 095 


10.3446 517 


1108 


1 22 76 64 


1 360 251 712 


33.2866 339 


10.3477 657 


1109 


1 22 98 81 


1 363 938 029 


33.3016 516 


10.3508 778 


1110 


1 23 21 00 


1 367 631 000 


33.3166 625 


10.3539 88 


1111 


1 23 43 21 


1 371 330 631 


33.3316 666 


10.3570 964 


1112 


1 23 65 44 


1 375 036 928 


33.3466 64 


10.3602 029 


1113 


1 23 87 69 


1 378 749 897 


33.3616 546 


10.3633 076 


1114 


1 24 09 96 


1 382 469 544 


33.3766 385 


10.3664 103 


1115 


1 24 32 25 


1 386 195 875 


33.3916 157 


10.3695 113 


1116 


1 24 54 56 


1 389 928 896 


33.4065 862 


10.3726 103 


1117 


1 24 76 89 


1 393 668 613 


33.4215 499 


10.3757 076 


1118 


1 24 99 24 


1397 415 032 


33.4365 07 


10.3788 03 


1119 


125 2161 


1 401 168 159 


33.4514 573 


10.3818 965 



u 



230 



SQUARES, CUBES, AND EOOTS. 



Table— (.Continued). 



Square. 



_L 



Square Root. 



Cube Root. 



1 25 44 00 
1 25 66 41 
1 25 88 84 
1 26 11 29 
1 26 33 76 
1 26 56 25 
1 26 78 76 
1 27 01 29 
1 27 23 84 
1 27 46 41 
1 27 69 00 
1 27 91 61 
1 28 14 24 
1 28 36 89 
1 28 59 56 
1 28 82 25 
1 29 04 96 
1 29 27 69 
1 29 50 44 
1 29 73 21 
1 29 96 00 
1 30 18 81 
1 30 41 64 
1 30 64 49 
1 30 87 36 
1 31 10 25 
1 31 33 16 
1 31 56 09 
1 31 79 04 
1 32 02 01 
1 32 25 00 
1 32 48 01 
1 32 71 04 
1 32 94 09 
1 33 17 16 
1 33 40 25 
1 33 63 36 
1 33 86 49 
1 34 09 64 
1 34 32 81 
1 34 56 00 
1 31 79 21 
1 35 02 44 
1 35 25 69 
1 35 48 96 
1 35 72 25 
1 35 95 56 
1 36 18 89 
1 36 42 24 
1 36 65 61 
1 36 89 00 
1 37 12 41 
1 37 35 84 
1 37 59 29 
1 37 82 76 
1 38 06 25 



1 404 928 000 
1 408 694 561 
1 412 467 848 
1 416 247 867 
1 420 034 624 
1 423 828 125 
1 427 628 376 
1 431 435 383 
1 435 249 152 
1 439 069 689 
1 442 897 000 
1 446 731 091 
1 450 571 968 
1454 419 637 
1 458 274 104 
1 462 135 375 
1466 003 456 
1 469 878 353 
1 473 760 072 
1 477 648 619 
1 481 544 000 
1 485 446 221 
1 489 355 288 
1 493 271 207 
1 497 193 984 
1 501 123 625 
1 505 060 136 
1 509 603 523 
1 512 953 792 
1 516 910 949 
1 520 875 000 
1 524 845 951 
1 528 823 808 
1 532 808 577 
1 536 800 264 
1 540 798 875 
1 544 804 416 
1 548 816 893 
1 552 836 312 
1 556 862 679 
1 560 896 000 
1 564 936 281 
1 568 983 528 
1 573 037 747 
1 577 098 944 
1581167 125 
1 585 242 296 
1 589 324 463 
1 593 413 632 
1 597 509 809 
1601613 000 
1605 723 211 
1 609 840 448 
1613 964 717 
1618 096 024 
1 622 234 375 



33.4664 011 
33.4813 381 
33.4962 684 
33.5111921 
33.5261092 
33.5410 196 
33.5559 234 
33.5708 206 
33.5857 112 
33.6005 952 
33.6154 726 
33.6303 434 
33.6452 077 ' 
33.6600 653 
33.6749 165 
33.6897 61 
33.7045 991 
33.7194 306 
33.7340 556 
33.7490 741 
33.7638 860 
33.7786 915 
33.7934 905 
33.8082 83 
33.82.30 691 
33.8378 486 
33.8526 218 
33.8673 884 
33.8821487 
33.8969 025 
33.9116 499 
33.9263 909 
33.9411255 
33.9558 537 
33.9705 755 
33.9852 91 
34. 

34.0147 027 
34.0293 99 
34.0440 89 
34.0587 727 
84.0734 501 
34.0881211 
34.1027 858 
34.1174 442 
34.1320 963 
34.1467 422 
34.1613 817 
34.1760 15 
34.1906 42 
34.2052 627 
34.2198 773 
34.2344 855 
34.2490 875 
34.2636 834 
34.2782 73 . 



10.3849 882 
10.3880 781 
10.3911661 
10.3942 523 
10.3973 366 
10.4004 192 
10.4034 999 
10.4065 787 
10.4096 557 
10.4127 31 
10.4158 044 
10.4188 76 
10.4219 458 
10.4250 138 
10.4280 8 
10.4311443 
10.4342 069 
10.4372 677 
10.4403 267 
10.4433 839 
10.4464 393 
10.4494 929 
10.4525 448 
10.4555 948 
10.4586 431 
10.4616 896 
10.4647 343 
10.4677 773 
10.4708 185 
10.4738 579 
10. '768 955 
1C ,99 314 
10.4829 656 
10.4859 98 
10.4890 286 
10.4920 575 
10.4950 847 
10.4981101 
10.5011 337 
10.5041556 
10.5071 757 
10.5101 942 
10.5132 109 
10.5162 259 
10.5192 391 
10.5222 506 
10.5252 604 
10.5282 685 
10.5312 749 
10.5342 795 
10.5372 825 
10.5402 837 
10.5432 832 
10.5462 81 
10.5492 771 
10.5522 715 






SQUARES, CUBES, AND ROOTS. 



231 



Ta~bl e — {Continued). 



Square 



1 38 29 76 
1 38 53 29 
1 38 76 84 ■ 
1 39 00 41 
1 39 24 00 
1 39 47 61 
1 39 71 24 
1 39 94 89 
1 40 18 b6 
1 40 42 25 
1 40 65 96 
1 40 89 69 
1 41 13 44 
1 41 37 21 
1416100 
1 41 84 81 
1 42 08 64 
1 42 32 49 
1 42 56 36 
1 42 80 25 
143 04 16 
1 43 28 09 

1 43 52 04 
1 43 76 01 
1 44 00 00 
1 44 24 01 
1 44 48 04 
1 44 72 09 

144 96 16 

145 20 25 
1 45 44 36 

.145 68 49 
1 45 92 '64 

146 16 81 
1 46 41 00 
1 46 65 21 
1 46 89 44 
1 47 13 69 
1 47 37 96 
1 47 62 25 
1 47 86 56 
1 48 10 89 
1 48 35 24 

148 59 61 
1 4'8 84 00 

149 08 41 
1 49 32 84 
1 49 57 29 
1 49 81 76 
1 50 06 25 
1 50 30 76 
1 50 55 29 
1 50 79 84 
1 51 04 41 
15129 00 
1 51 53 61 



1 626 379 776 
1 630 532 233 
1 634 691 752 
1 638 858 339 
1 643 032 000 
1 647 212 741 
1 651 400 568 
1 655 595 487 
1 659 797 504 
1 664 006 625 
1 668 222 856 
1 672 446 203 
1 676 676 672 
1 680 914 629 
1 685 159 000 
1 689 410 871 
1 693 669 888 
1 697 936 057 
1 702 209 384 
1 706 489 875 
1 710 777 536 
1 715 072 373 
1 719 374 392 
1 723 683 599 
1 728 000 000 
1 732 323 601 
1 736 654 408 
1 740 992 427 
1 745 337 664 
1 749 690 125 
1 754 049 816 
1758 416 743 
1 762 790 912 
1 767 172 329 
1 771 561 000 
1 775 956 931 
1780 360 128 
1 784 770 597 
1 789 188 344 
1 793 613 375 
1 798 045 696 
1802 485-313 
1 806 932 232 
1 811 386 459 
1 815 848 000 
1 820 316 861 
1 824 793 048 
1 829 276 567 
1 833 767 244 
1 838 265 625 
1 842 771 176 
1 847 284 083 
1 851 804 352 
1 856 331 989 
1 860 867 000 
1 865 409 391 



Square Root. 



34.2928 564 
34.3074 336 
34.3220 046 
34.3365 694 
34.3511 281 
34.3656 805 
34.3802 268 
34.3947 67 
34.4093 Oil 
34.4238 289 
34.4383 507 
34.4528 663 
34.4673 759 
34.4818 793 
34.4963 766 
34.5108 678 
34.5253 53' 
34.5398 321 
34.5543 051 
34.5687 72 
34.5832 329 
34.5976 879 
34.6121 366 
34.6265 794 
34.6410 162 
34.6554 469 
34.6698 716 
34.6842 904 
34.6987 031 
34.7131 099 
34.7275 107 
34.7419 055 
34.7562 944 
34.7706 773 
34.7850 543 
34.7994 253 
34.8137 904 
34.8281 495 
34.8425 028 
34.8568 501 
34.8711 915 
34.8855 271 
34.8998 567 
34.9141 805 
34.9284 984 
34.9428 104 
34.9571 166 
34.9714 169 
34.9857 114 
35. 

35.0142 828 
35.0285 598 
35.0428 309 
35.0570 963 
35.0713 558 
35.0856 096 



Cube Root. 



10.5552 642 
10.5582 552 
10.5612 445 
10.5642 322 
10.5672 181 
10.5702 024 
10.5731849 
10.5761658 
10.5791449 
10.5821225 
10.5850 983 
10.5880 725 
10.5910 45 
10.5940158 
10.5969 85 
10.5999 525 
10.6029184 
10.6058 826 
10.6088 451 
10.6118 06 
10.6147 652 
10.6177 228 
10.6206 788 
10.6236 331 
10.6265 857 
10.6295 367 
10.6324 86 
10.6354 338 
10.6383 799 
10.6413 244 
10.6442 672 
10.6472 C85 
10.650148 
10.6530 86 
10.6560 223 
10.6589 57 
10.6618 902 
10.6648 217 
10.6677 516 
10.6706 799 
10.6736 066 
10.6765 317 
10.6794 552 
10.6823 771 
10.6852 973 
10.6882 16 
10.6911331 
10.6940 486 
10.6969 625 
10.6998 748 
10.7027 855 
10.7056 947 
10.7086 023 
10.7115 083 
10.7144 127 , 
10.7173 155 



232 



SQUARES, CUBES, AND HOOTS. 



Ta"ble— (Continued). 



Square. 



Cube. 



1 51 78 24 
1 52 02 89 
1 52 27 56 
1 52 52 25 
1 52 76 96 
1 53 01 69 
1 53 26 44 
1 53 51 21 
1 53 76 00 
1 54 00 81 
1 54 25 64 
1 54 50 49 
1 54 75 36 
1 55 00 25 
1 55 25 16 
1 55 50 09 
1 55 75 04 
1 56 00 01 
1 56 25 00 
1 56 50 01 
1 56 75 04 
1 57 00 09 
157 25 16 
1 57 50 25 
1 57 75 36 
1 58 00 49 
1 58 25 64 
1 58 50 81 
1 58 76 00 
1 59 01 21 
1 59 26 44 
1 59 51 69 
1 59 76 96 
1 60 02 25 
1 60 27 56 
1 60 52 89 
1 60 78 24 
1 61 03 61 
1 61 29 00 
1 61 54 41 
1 61 79 84 
1 62 05 29 
1 62 30 76 
1 62 bQ 25 
1 62 81 76 
1 63 07 29 
1 63 32 84 
1 63 58 41 
1 63 84 00 
1 64 09 61 
1 64 35 24 
1 64 60 89 
1 64 86 56 
1 65 12 25 
1 65 37 96 
1 65 63 69 



1 869 959 168 
1 874 516 337 
1 879 080 904 
1 883 652 875 
1 888 232 256 
1892 819 053 
1 897 413 272 
1 902 014 919 
1 906 624 000 
1 911 240 521 
1 915 864 488 
1 920 495 907 
1 925 134 784 
1 929 781 125 
1 934 434 936 
1 939 096 223 
1 943 764 992 
1 948 441 249 
1 953 125 000 
1957 816 251 
1 962 515 008 
1 967 221 277 
1 971 945 084 
1 976 656 375 
1 981 385 216 
1 986 121 593 
1 990 865 512 

1 995 616 979 

2 000 376 000 
2 005 142 581 
2 009 916 728 
2 014 698 447 
2 019 487 744 
2 024 284 625 
2 029 089 096 
2 033 901 163 
2 038 720 832 
2 043 548 109 
2 048 383 000 
2 053 225 511 
2 058 075 648 
2 062 933 417 
2 067 798 824 
2 072 671 875 
2 077 552 576 
2 082 440 933 
2 087 336 952 
2 092 240 639 
2 097 152 000 
2 102 071841 
2 106 997 768 
2111932187 
2116 874 304 
21218-24 125 
2 126 781 656 
2 131 746 903 



Square Root. 



35.0998 575 
35.1140 997 
35.1283 361 
35.1425 568 
35.1567 917 
35.1710 108 
35.1852 242 
35.1994 318 
35.2136 337 
35.2278 299 
35.2420 204 
35.2562 051 
35.2703 842 
35.2845 575 
35.2987 252 
35.3128 872 
35.3270 435 
35.3411941 
35.3553 391 
35.3694 784 
35.383612 
35.3977 4 
35.4118 624 
35.4259 792 
35.4400 903 
35.4541958 
35.4682 957 
35.4823 9 
35.4964 787 
35.5105 618 
35.5246 393 
35.5387 113 
35.5527 777 ' 
35.5668 385 
35.5808 937 
35.5949 434 
35.6089 876 
35.6230 262 
35.6370 593 
35.6510 869 
35.665109 
35.6791255 
35.6931366 
35.7071421 
35.7211422 
35.7351367 
35.7491258 
35.7631095 
35.7770 876 
35.7910 603 
35.8050 276 
35.8189 894 
85.83(29 457 
35.8468 966 
35.8608 421 
35.8747 822 



Cube Root. 



10.7202 168. 
10.7231165 
10.7260 146 
10.7289 112 
10.7318 062 
10.7346 997 
10.7375 916 
10.7404 819 
10.7433 707 
10.7462 579 
10.7491436 
10.7520 277 
10.7549103 
10.7577 913 
10.7606 708 
10.7635 488 
10.7664 252 
10.7693 001 
10.7721735 
10.7750 453 
10.7779156 
10.7807 843 
10.7836 516 
10.7865173 
10.7893 815 
10.7922 441 
10.7951053 
10.7979 649 
10.8008 23 
10.8036 797 
10.8065 348 
10.8093 884 
10.8122 404 
10.8150 909 
10.81794 
10.8207 876 
10.8236 336 
10.8264 782 
10.8293 213 
10.8321029 
10.8350 03 
10.8378 416 
10.8400 788 
10.8435144 
10.8463 485 
10.8491812 
10.8520125 
10.8548 422 
10.8576 704 
10.8604 972 
10.8633 225 
10.8661464 
10.8689 687 
10.8717 897 
10.8746 091 
10.8774 271 



SQUARES, CUBES, AND ROOTS. 



233 



Tatole— (Continued). 



Square. 



Cube. 



I 



Square Root. 



Cube Root. 



1 65 89 44 
1 66 15 21 

1 66 41 00 
1 66 66 81 
1 66 92 64 
1 67 18 49 
1 67 44 36 
1 67 70 25 
1 67 96 16 
1 68 22 09 
1 68 48 04 
1 68 74 01 
1 69 00 00 
1 69 26 01 
1 69 52 04 
1 69 78 09 
170 04 16 
1 70 SO 25 
1 70 50 36 
1 70 82 49 
1 71 08 64 
17134 81 
1716100 
17187 21 
172 13 44 
1 72 39 69 
1 72 65 96 
172 92 25 
17318 56 
1 73 44 89 
1 73 71 24 
1 73 97 61 
1 74 24 00 

174 50 41 
1 74 76 84 

175 93 29 
1 75 29 76 
1 75 56 25 
1 75 82 76 
1 76 09 29 
1 76 35 84 
1 76 62 41 
1 76 89 00 
177 15 61 
1 77 42 24 
1 77 68 89 
1 77 95 56 
1 78 22 25 
1 78 48 96 
1 78 75 69 
179 02 44 
1 79 29 21 
1 79 56 00 
179 82 81 
1 80 09 64 
1 80 36 49 



2 136 719 872 
2 141 700 569 
2 146 689 000 
2 151 685 171 
2 156 689 088 
2 161 700 757 
2 166 720 184 
2 171 747 375 
2 176 782 336 
2 181 825 073 
2 186 875 592 
2 191933 899 
2 197 000 000 
2 202 073 901 
2 207 155 608 
2 212 2,15 127 
2 217 342 464 
2 222 447 625 
2 227 560 616 
2 232 681 443 
2 237 810 112 
2 242 946 629 
2 248 091 000 
2 253 243 231 
2 258 403 328 
2 263 571 297 
2 268 747 144 
2 273 930 875 
2 279 122 496 
2 284 322 013 
2 289 529 432 
2 294 744 759 
2 299 968 000 
2 305 199161 
2 310 438 248 
2 315 685 267 
2 320 940 224 
2 326 203 125 
2 331 473 976 
2 336 752 783 
2 342 039 552 
2 347 334 289 
2 352 637 000 
2 357 947 691 
2 363 266 368 
2 368 593 037 
2 373 927 704 
2 379 270 375 
2 384 621 056 
2 889 979 753 
2 395 346 472 
2 400 721219 
2 406 104 000 
2 411494 821 
2 416 893 6X8 
2 422 300 607 
U* 



35.8887 169 
35.9026 461 
35.9165 699 
35.9304 884 
35.9444 015 
35.9583 092 
35.9722 115 
35.9861084 
36. 

36.0138 862 
36.0277 671 
36.0416 426 
36.0555128 
36.0693 776 
36.0832 371 
36.0970 913 
36.1109 402 
36.1247 837 
36.1386 22 
36.1524 55 
36.1662 826 
36.180105 
36.1939 221 
36.2077 34 
36.2215 406 
36.2353 419 
36.2491379 
36.2626 287 
36.2767143 
36.2904 946 
36.3042 697 
36.3180 396 
36.3318 042 
36.3455 637 
36.3593179 
36.3730 67 
36.3868 108 
36.4005 494 
36.4142 829 
36.4280 112 
36.4417 343 
36.4554 523 
36.469165 
36.4828 727 
36.4965 752 
36.5102 725 
36.5239 647 
36.5376 518 
36.5513 388 
36.5650 106 
36.5786 823 
36.5923 489 
36.6060104 
36.6196 668 
36.6333 181 
36.6469 644 



10.8802 436 
10.8830 587 
10.8858 723 
10.8886 845 
10.8914 952 
10.8943 044 
10.8971123 
10.8999186 
10.9027 235 
10.9055 269 
10.9083 29 
10.9111296 
10.9139 287 
10.9167 265 
10.9195 228 
10.9223177 
10.9251111 
10.9279 031 
10.9306 937 
10.9334 829 
10.9362 706 
10.9390 569 
10.9418 418 
10.9446 253 
10.9475 074 
10.950188 
10.9529 673 
10.9557 451 
10.9585 215 
10.9612 965 
10.9640 701 
10.9668 423 
10.9696 131 
10.9723 825 
10.9751505 
10.9779171 
10.9806 823 
10.9834 462 
10.9862 086 
10.9889 696 
10.9917 293 
10.9944 876 
10.9972 445 
11. 

11.0027 541 
11.0055 069 
11.0082 583 
11.0110082 
11.0137 569 
11.0165 041 
11.0192 5 
11.0219 9-15 
11.0217 377 
11.0274 795 
11.0302199 
11.0329 59 



234 



SQUARES, CUBES, AND ROOTS. 



Table— {Continued). 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


1344 


1 80 63 36 


2 427 715 584 


36.6606 056 


11.0356 967 


1345 


1 80 90 25 


2 433 138 625 


36.6742 416 


11.0384 33 


1346 


1 81 17 16 


2 438 569 736 


36.6878 726 


11.0411 68 


1347 


1 81 44 09 


2 444 008 923 


36.7014 986 


11.0439 017 


1348 


1 81 71 04 


2 449 456 192 


36.7151195 


11.0466 339 


1349 


1 81 98 01 


2 454 911 549 


36.7287 353 


11.0493 649 


1350 


1 82 25 00 


2 460 375 000 


36.7423 461 


11.0520 945 


1351 


1 82 52 01 


2 465 846 551 


36.7559 519 


11.0548 227 


1352 


1 82 79 04 


2 471 326 208 


36.7695 526 


11.0575 497 


1353 


1 83 06 09 


2 476 813 977 


36.7831483 


11.0602 752 


1354 


1 83 33 16 


2 482 309 864 


36.7967 39 


11.0629 994 


1355 


1 83 60 25 


2 487 813 875 


36.8103 246 


11.0657 222 


1356 


1 83 87 36 


2 493 326 016 


36.8239 053 


11.0684 437 


1357 


1 84 14 49 


2 498 846 293 


36.8374 809 


11.0711 639 


1358 


184 4164 


2 504 374 712 


36.8510 515 


11.0738 828 


1359 


1 84 68 81 


2 509 911 279 


36.8646 172 


11.0766 003 


1360 


1 84 96 00 


2 515 456 000 


36.8781778 


11.0793165 


1361 


1 85 23 21 


2 521 008 881 


36.8917 335 


11.0820 314 


1362 


1 85 50 44 


2 526 569 928 


36.9052 842 


11.0847 449 


1363 


1 85 77 69 


2 532 139 147 


36.9188 299 


11.0874 571 


1364 


1 86 04 96 


2 537 716 544 


36.9323 706 


11.0901679 


1365 


1 86 32 25 


2 543 302 125 


36.9459 064 


11.0928 775 


1366 


1 86 59 56 


2 548 895 896 


36.9594 372 


11.0955 857 


1367 


186 86 89 


2 554 497 863 


36.9729 631 


11.0982 926 


1368 


1 87 14 24 


2 560 108 032 


36.9864 84 


11.1009 982 


1369 


1 87 41 61 


2 565 726 409 


37. 


11.1037 025 


1370 


1 87 69 00 


2 571 353 000 


37.0135 11 


11.1064 054 


1371 


1 87.96 41 


2 576 987 811 


37.0270172 


11.109107 


1372 


1 88 23 84 


2 582 630 848 


37.0405 184 


11.1118 073 


1373 


1 88 51 29 


2 588 282 117 


37.0540146 


11.1145 064 


1374 


1 88 78 76 


2 593 941.624 


37.0675 06 


11.1172 041 


1375 


1 89 06 25 


2 599 609 375 


37.0899 924 


11.1199 004 


1376 


1 89 33 76 


2 605 285 376 


37.0944 74 


11.1225 955 


1377 


1 89 61 29 


2 610 969 633 


37.1079 506 


11.1252 893 


1378 


1 89 88 84 


2 616 662 152 


37.1214 224 


11.1279 817 


1379 


19016 41 


2 622 362 939 


37.1348 893 


11.1306 729 


1380 


1 90 44 00 


2 628 072 000 


37.1483 512 


11.1333 628 


1381 


1 90 71 61 


2 633 789 341 


37.1618 084 


11.1360 514 


1382 


1 90 99 24 


2 639 514 968 


37.1752 606 


11.1387 386 


1383 


1 91 26 89 


2 645 248 887 


37.1887 079 


11.1414 246 


1384 


1 91 54 56 


2 650 991 104 


37.2021505 


11.1441093 


1385 


1 91 82 25 


2 656 741 625 


37.2155 881 


11.1467 926 


1386 


1 92 09 96 


2 662 500 456 


37.2290 209 


11.1494 747 


1387 


1 92 37 69 


2 668 267 603 


37.2424 489 


11.1521555 


1388 


1 92 65 44 


2 674 043 072 


37.2558 72 


11.1548 35 


1389 


1 92 93 21 


2 679 826 869 


37.2692 903 


11.1575133 


1390 


1 93 21 00 


2 685 619 000 


37.2827 037 


11.1601903 


1391 


1 93 48 81 


2 691419 471 


37.2961124 


11.1628 659 


1392 


1 93 76 64 


2 697 228 288 


37.3095162 


11.1655 403 


1393 


1 94 04 49 


2 703 045 457 


37.3229 152 


11.1682134 


1394 


1 94 32 36 


2 708 870 984 


37.3363 094 


11.1708 852 


1395 


1 94 60 25 


2 714 704 875 


37.3496 988 


11.1735 558 


1396 


1 94 88 16 


2 720 547 136 


37.3630 834 


11.1762 25 


1397 


1 95 16 09 


2 726 397 773 


37.37C4 632 


11.1788 93 


1398 


1 95 44 04 


2 732 256 792 


37.3898 382 


11.1815 598 


1399 


1 95 72 01 


2 738 124 199 


37.4032 084: 


11.1842 252 



SQUARES, CUBES, AND ROOTS. 



235 







Table 


— (Conti7iued). 




Number. 


Square. 


Cube. 


Square Root. 


! Cube Root. 


1400 


1 96 00 00 


2 744 000 000 


37.41L5 738 


11.1868894 


1401 


1 96 28 01 


2 749 884 201 


37.4299 345 


11.1895 523 


1402 


1 96 56 04 


2 755 776 808 


37.4432 904 


11.1922 139 


1403 


1 96 84 09 


2 761 677 827 


37.4566 416 


11.1948 743 


1404 


1 97 12 16 


2 767 587 


264 


37.4699 88 


11.1975 334 


1405 


1 97 40 25 


2 773 505 


125 


37.4833 296 


11.2001913 


1405 


1 97 68 36 


2 779 431 


416 


37.4966 665 


11.2028 479 


1407 


1 97 96 49 


2 785 366 


143 


37.5099 987 


11.2055 032 


1408 


1 98 24 64 


2 791 309 


312 


37.5233 261 


11.2081573 


1409 


1 98 52 81 


2 797 260 


929 


37.5366 487 


11.2108 101 


1410 


1 98 81 00 


2 803 221 


000 


37.5499 667 


11.2134 617 


1411 


1 99 09 21 


2 809 189 


531 


37.5632 799 


11.2161 12 


1412 


1 99 37 44 


2 815 166 


528 


37.5765 885 


11.2187 611 


1413 


1 99 65 69 


2 821 151 


997 


37.5898 922 


11.2214 089 


1414 


I 99 93 96 


2 827 145 944 


37.6031 913 


11.2240 054 


1415 


2 00 22 25 


2 833 148 375 


37.6164 857 


11.2267 007 


1416 


2 00 50 56 


2 839 159 296 


37.6297 754 


11.2293 448 


1417 


2 00 78 89 


2 845 178 713 


37.6430 604 


11.2319 876 


1418 


2 01 07 24 


2 851 206 632 


37.6563 407 


11.2346 292 


1419 


2 01 35 61 


2 857 243 059 


37.6696 164 


11.2372 696 


1420 


2 01 64 00 


2 863 288 000 


37.6828 874 


11.2399 087 


1421 


2 01 92 41 


2 869 341461 


37.6961 536 


11.2425 465 


1422 


2 02 20 84 


2 875 403 448 


37.7094153 


11.2451 831 


1423 


2 02 49 29 


2 881 473 967 


37.7226 722 


11.2478 185 


1424 


2 02 77 76 


2 887 553 024 


37.7359 245 


11.2504 527 


1425 


2 03 06 25 


2 893 640 625 


37.7491722 


11.2530 856 


1426 


2 03 34 76 


2 899 736 776 


37.7624152 


11.2557 173 


1427 


2 03 63 29 


2 905 841 483 


37.7756 535 


11.2583 478 


1428 


2 03 91 84 


2 911 954 752 


37.7888 873 


11.2609 77 


1429 


2 04 20 41 


2 918 070 589 


37.8021163 


11.2636 05 


1430 


2 04 49 00 


2 924 207 000 


37.8153 408 


11.2662 318 


1431 


2 04 77 61 


2 930 345 991 


37.8285 606 


11.2683 573 


1432 


2 05 06 24 • 


2 936 493 568 


37.8417 759 


11.2714 816 


1433 


2 05 34 89 


2 942 649 737 


37.8549 864 


11.2741047 


1434 


2 05 63 56 


2 948 814 504 


37.8681924 


11.2767 266 


1435 


2 05 92 25 


2 954 987 875 


37.8813 938 


11.2793 472 


1436 


2 06 20 96 


2 961 169 856 


37.8945 906 


11.2819 666 


1437 


2 06 49 69 


2 967 360 453 


37.9077 828 


11.2845 849 


1438 


2 06 78 44 


2 973 559 672 


37.9209 704 


11.2872 019 


1439 


2 07 07 21 


2 979 767 519 


37.9341535 


11.2898 177 


1440 


2 07 36 00 


2 985 984 000 


37.9473 319 


11.2924 323 


1441 


2 07 64 81 


2 992 209 121 


37.9605 058 


11.2950 457 


1442 


2 07 93 64 


2 938 442 888 


37.9736 751 


11.2976 579 


1443 


2 08 22 49 


3 004 685 307 


37.9868 398 


11.3002 688 


1444 


2 08 51 36 


3 010 936 384 


38. 


11.3028 786 


1445 


2 08 80 25 


3 017 196 125 


38.0131556 


11.3054 871 


144(5 


2 09 09 16 


3 023 464 536 


38.0263 067 


11.3080 945 


1447 


2 09 38 09 


3 029 741 623 


38.0394 532 


11.3107 006 


1448 


2 09 67 04 


3 036 027 392 


38.0525 952 


11.3133 056 


14-19 


2 09 96 01 


3 042 321 849 


38.0657 326 


11.3159 094 


1450 


210 25 00 


3 048 625 000 


88.0788 655 


11.3185 119 


1451 


2 10 54 01 


3 054 936 851 


38.0919 939 


11.3211132 


1452 


2 10 83 04 


3 061 257 408 


38.1051 178 


11.3237 134 


1453 


2 11 12 09 


3 067 586 677 


3S. 1182 371 


11.326:5 124 


1454 


2 114116 


3 073 924 664 


38. 1313 519 


11.3289 102 


1455 


2 11 70 25 


3 080 271 


375 


38.1444 622 


11.3315 067 



236 



SQUARES, CUBES, AND ROOTS. 



TaTole— {Continued). 



Square. 



Square Root. 



Cube Root. 



2 11 99 36 

2 12 28 49 
2 12 57 64 
2 12 86 81 
213 16 00 
2 13 45 21 
2 13 74 44 
2 14 03 69 
2 14 32 96 
2 14 62 25 
2 14 91 56 
2 15 20 89 
2 15 50 24 
2 15 79 61 
2 16 09 00 
2 16 38 41 
2 16 67 84 
2 16 97 29 
2 17 26 76 
2 17 56 25 
2 17 85 76 
2 18 15 29 
2 18 44 84 
2 18 74 41 
2 19 04 00 
219 33 61 
2 19 63 24 
2 19 92 89 
2 20 22 hQ 
2 20 52 25 
2 20 81 96 
2 21 11 69 
2 21 41 44 
2 21 71 21 
2 22 01 00 
2 22 30 81 
2 22 GO 64 
2 22 90 49 
2 23 20 36 
2 23 50 25 
2 23 80 16 
2 24 10 09 
2 24 40 04 
2 24 70 01 
2 25 00 00 
2 25 30 01 
2 25 GO 04 
2 25 90 09 
2 26 2016 
2 26 50 25 
2 26 80 36 
2 27 10 49 
2 27 40 64 
2 27 70 81 
2 28 01 00 
2 28 31 21 



3 086 626 816 
3 092 990 993 
3 099 363 912 
3 105 745 579 
3 112 136 000 
3 118 535 181 
3 124 943128 
3 131 359 847 
3 137 785 344 
3 144 219 625 
3 150 662 696 
3 157 114 563 
3 163 575 232 
3 170 044 709 
3 176 523 000 
3183 010 111 
3 189 506 048 
3 196 010 817 
3 202 524 424 
3 209 046 875 
3 215 578 176 
3 222118 333 
3 228 667 352 
3 235 225 239 
3 241 792 000 
3 248 367 641 
3 254 952 168 
3 261 545 587 
3 268 147 904 
3 274 759 125 
3 281 379 256 
3 288 008 303 
3 294 646 272 
3 S01 293 169 
3 307 949 000 
3 314 613 771 
3 321 287 488 
3 327 970157 
3 334 661 784 
3 341 362 375 
3 348 071 936 
3 354 790 473 
3 361517 992 
3 368 254 499 
3 375 000 000 
3 381 754 501 
3 388 518 008 
3 395 290 527 
3 402 072 064 
3 408 862 625 
3 415 662 216 
3 422 470 843 
3 429 288 512 
3 436 115 229 
3 442 951 000 
3 449 795 831 



38.1575 681 
38.1706 693 
38.1837 662 
38.1968 5S5 
38.2099 463 
38.2230 297 
38.2361085 
38.2491829 
38.2622 529 
38.2753 184 
38.2883 794 
38.3014 36 
38.3144 881 
38.3275 358 
38.3405 79 
38.3536 178 
38.3666 522 
38.3796 821 
38.3927 076 
38.4057 287 
38.4187 454 
38.4317 577 
38.4447 656 
38.4577 691 
38.4707 681 
38.4837 627 
38.4967 53 
38.5097 39 
38.5227 206 
38.5356 977 
38.5486 705 
38.5616 389 
38.5746 03 
38.5875 627 
38.6005 181 
38.6134 691 
88.6264 158 
38.6393 582 
38.6522 962 
38.6652 299 
38.6781593 
38.6910 843 
38.7040 05 
38.7169 214 
38.7298 335 
38.7427 412 
38.7556 447 
38*7685 439 
38.7814 389 
38.7943 294 
38.8072 158 
38.8200 978 
38.8329 757 
38.8458 491 
38.8587 184 
38.8715 834 



11.3341022 
11.3366 964 
11.3392 894 
~ 11.3418 813 
11.3444 719 
11.3470 614 
11.3496 497 
11.3522 368 
11.3548 227 
11.3574 075 
11.3599 911 
11.3625 735 
11.3651547 
11.3677 347 
11.3703 136 
11.3728 914 
11.3754 679 
11.3780 433 
11.3806 175 
11.3831906 
11.3857 625 
11.3883 332 
11.3909 028 
11.3934 712 
11.3960 384 
11.3986 045 
11.4011695 
11.4037 332 
11.4062 959 
11.4088 574 
11.4114 177 
11.4139 769 
11.4165 349 
11.4190 918 
11.4206 476 
11.4242 022 
11.4267 556 
11.4293 079 
11.4318 591 
11.4344 092 
11.4369 581' 
11.4395 059 
11.4420 525 
11.4445 98 
11.4471424 
11.4496 857 
11.4522 278 
11.4547 688 
11.4573 087 
11.4598 474 
11.4623 85 
11.4649 215 
11.4674 568 
11.4699 911 
11.4725 242 
11.4750 562 



SQUARES, CUBES, AND ROOT& 



237 



Square. 



Ta"bl e— (Continued ) . 

Cube. Square Root. 



Cube Root. 



2 28 61 44 
2 28 91 69 
2 29 91 96 
2 29 52 25 
2 29 82 56 
2 30 12 89 
2 80 43 24 
2 30 73 61 
2 31 04 00 
2 31 34 41 
2 31 64 84 
2 31 95 29 
2 32 25 76 
2 32 56 25 
2 32 86 76 
2 33 17 29 
2 33 47 84 
2 33 78 41 
2 34 09 00 
2 34 39 61 
2 34 70 24 
2 35 00 89 
2 35 31 56 
2 35 62 25 
2 35 92 96 
2 36 23 69 
2 36 54 44 
2 36 85 21 
2 37 16 00 
2 37 46 81 
2 37 77 64 
2 38 08 49 
2 38 39 36 
2 38 70 25 
2 39 01 16 
2 39 32 09 
2 39 63 04 
2 39 94 01 
2 40 25 00 
2 40 56 01 
2 40 87 04 
2 41 18 09 
2 41 49 16 
2 41 80 25 
2 42 11 36 
2 42 42 49 
2 42 73 64 
2 43 04 81 
2 43 36 00 
2 43 67 21 
2 43 98 44 
2 44 29 69 
2 44 60 96 
2 44 92 25 
2 45 23 56 
2 45 54 89 



3 456 649 728 
3 463 512 697 
3 470 384 744 
3 477 265 875 
3 484 156 096 
3 491 055 413 
3 497 963 832 
3 504 881 359 
3 511808 000 
3 518 743 761 
3 525 688 648 
3 532 642 667 
3 539 605 824 
3 546 578 125 
3 553 559 576 
3 560 558 183 
3 567 549 552 
3 574 558 889 
3 581 577 000 
3 588 604 291 
3 595 640 768 
3 602 686 437 
3 609 741 304 
3 616 805 375 
3 623 878 656 
3 630 961 153 
3 638 052 872 
3 645 153 819 
3 652 264 000 
3 659 383 421 
3 666 512 088 
3 673 650 007 
3 680 797 184 
3 687 953 625 
3 695 119 336 
3 702 294 323 
3 709 478 592 
3 716 672 149 
3 723 875 000 
3 731 087 151 
3 738 308 608 
3 745 539 377 
3 752 779 464 
3 760 028 875 
3 767 287 616 
3 774 555 693 
3 781833112 
3 789 119 879 
3 796 416 000 
3 803 721481 
3 811036 328 
3 818 360 547 
3 825 641 144 
3 833 037 125 
3 840 389 496 
3 847 751 263 



38.8844 442 
38.8973 006 
38.9101529 
38.9230 009 
38.9358 447 
38.9486 841 
38.9615 194 
38.9743 505 
38.9871774 
39. 

39! 0128 184 
39.0256326 
39.0884 426 
39.0512483 
39.0640 499 
39.0768 473 
39.0896 406 
39.1024 296 
39.1152144 
39.1279 951 
39.1407 716 
39.1535 439 
39.166312 
39.1790 76 
39.1918 359 
39.2045 915 
39.2173 431 
39.2800 905 
39.2428 337 
89. 25," 5 728 
39.2683 078 
39.2810 387 
39.2937 654 
39,8064 88 
39.3192 065 
39.3319 208 
39.3446 311 
39.3573 373 
39.3700 394 
39.3827 373 
39.3954 312 
39.408121 
39.4208 067 
39.4334 883 
39.4461658 
39.4588 393 
39.4715 087 
39.484174 
39.4968 353 
39.5094 925 
39.5221457 
39.5347 948 
39.5474 399 
39.5600 809 
39.5727 179 
39.5853 508 



11.4775 871 
11.4801169 
11.4826 455 
11.4851731 

11.4876 995 
11.4902 249 
11.4927 491 
11.4952 722 
11.4977 942 
11.5003151 
11.5028 348 
11.5053 535 
11.5078 711 
11.5103 876 
11.5129 03 
11.5154173 
11.5179 305 
11.5204 425 
11.5229 535 
11.525-H'34 
11. 5270 722 
11.5304 799 
11.5329 865 
11.5354 92 
11.5379 965 
11.5404 998 
11.5480 021 
11.5455 033 
11.5480 034 
11.5505 025 
11.5530 004 
11.5554 973 
11.5579 931 
11.5604 878 
11.5629 815 
11.5654 74 
11.5679 655 
11.5704 559 
11.5729 453 
11.5754 336 
11.5779 208 
11.5804 069 
11.5828 919 
11.5853 759 
11.5878 588 
11.5903 407 
11.5928 215 
11.5953 013 
11.5977 799 
11.6002 576 
11.6027 842 
11.6052 097 
11.6076 841 
11.6101 575 
11.6126 299 
11.6151012 



238 



SQUARES, CUBES, AND ROOTS. 



Table— {Continued). 



Number. 


Square. 


Cube. 


Square Root. 


Cube Root. 


1568 


2 45 86 24 


3 855 123 432 


39.5979 797 


11.6175 715 


1569 


2 46 17 61 


3 862 503 009 


39.6106 046 


11.6200 407 


1570 


2 46 49 00 


3 869 883 000 


39.6232 


255 


11.6225 088 


1571 


2 46 80 41 


3 877 292 411 


39.6358 424 


11.6249 759 


1572 


2 47 11 84 


3 884 701 248 


39.6484 552 


11.6274 42 


1573 


2 47 43 29 


3 892 119 517 


39.6610 64 


11. 6299*07 


1574 


2 47 74 76 


3 899 547 224 


39.6736 688 


11.6323 71 


1575 


2 48 06 25 


3 906 984 375 


39.6862 696 


11.6348 339 


1576 


2 48 37 76 


3 914 430 976 


39.6988 665 


11.6372 957 


1577 


2 48 69 29 


3 921 887 033 


39.7114 593 


11.6397 566 


1578 


2 49 00 84 


3 929 352 552 


39.7240 481 


11.6422 164 


1579 


2 49 32 41 


3 936 827 539 


39.7366 329 


11.6416 751 


1580 


2 49 64 00 


3 944 312 000 


39.7492 138 


11.6471329 


1581 


2 49 95 61 


3 951 805 941 


39.7617 907 


11.6495 895 


1582 


2 50 27 24 


3 959 309 368 


39.7743 636 


11.6520 452 


1583 


2 50 58 89 


3 966 822 287 


39.7869 325 


11.6544 998 


1584 


2 50 90 56 


3 974 344 704 


39.7994 975 


11.6569 534 


1585 


2 51 22 25 


3 981 876 625 


39.8120 585 


11.6594 059 


1586 


2 51 53 96 


3 989 418 056 


39.8241 155 


11.6618 574 


1587 


2 51 85 69 


3 996 969 003 


39.8376 686 


11.6643 079 


1588 


2 52 17 44 


4 004 529 472 


39.8497 177 


11.6667 574 


1589 


2 52 49 21 


4 012 099 469 


39.8622 628 


11.6692 058 


1590 


2 52 81 00 


4 019 679 000 


39.8748 04 


11.6716 532 


1591 


2 53 12 81 


4 027 268 071 


39.8873 413 


11.6740 996 


1592 


2 53 44 64 


4 034 866 688 


39.8998 747 


11.6765 449 


1-393 


2 53 76 49 


4 042 474 857 


39.9124 041 


11.6789 892 


1594 


2 54 08 36 


4 050 092 584 


39.9249 295 


11.6814 325 


1595 


2 54 40 25 


4 057 719 875 


39.9374 


511 


11.6838 748 


1596 


2 54 72 16 


4 065 356 736 


3!) .'9499 


687 


11.6863 161 


1597 


2 55 04 09 


4 073 003 173 


39.9624 


824 


11.6887 563 


1598 


2 55 36 04 


4 080 659 192 


39.9749 


922 


11.6911955 


1599 


2 55 68 01 


4 088 324 799 


39.9874 


98 


11.6936 337 


1600 


2 56 00 00 


4 096 000 000 


40. 




11.6960 709 



may be greatly extended by aid 



The uses of the preceding table 
of the following Kules : 

To -Ascertain the Scj_u.are or Cn"be of a higher Number 
th.au is contained, in the Table. 

When the Number is divisible by a Number without having a Remainder. 

Rule. — If the number exceed by 2, 3, or any other number of times, any number 
contained in the table, multiply the square or cube of that number in the table by 
the square of 2, 3, etc., and the product will give the result. 
Example.— Required the square of 170"). 

1700 is 10 times 170, and the square of 170 is 2 SD 00. 
Then 2S300xl02 = 2S90000. 
Ex. 2.— What is the cube of 2400? 

2400 is 2 times 1201, and the cube of 12G0 is 1 728 003 000. 
Then 1 728 000 000x23 = 13 S2400J 000. 

When the Number is an Odd Number. 

Rule. — Take the two numbers nearest to each other, which, added together, make 
rli: i- sum ; then from the sum of the squares or cubes of these two numbers, as per 
table, multiplied by 2, subtract 1, and the remainder will give the result. 



SQUARES, CUBES, AND ROOTS. 239 

Example.— What is the square of 1T45? 

The nearest two numbers are j g™ j — 1745. 

mi * ,, ( 8732 == 76 21 29 

Then, per table, | 8722= : 76 03 84 

1 52 25 13X2 = 3045026 — 1 = 3 04 50 26. 

To Compute tlie Squares or Cubes of Numbers following 
each, otlier in .A-rithmetical Progression. 

Rule. — Take the squares of the two first numbers in the usual way, and subtract 
the less from the greater. Add the difference to the greatest square, with the addi- 
tion of 2 as a constant quantity ; the sum will give the square of the next number. 
Example.— What are the squares of 1001, 1002, and 1003? 
10002 = 10000 00 
9992 — 99 80 01 

19 99 

Add 10002 —100 0000 
Add 2 



100 20 01 Square of 1001. 
Difference, 20 01 + 2 = 20 03 

100 4004 Square of 1002. 
Difference, 2003+2 = 20 05 

100 GO 09 Square of 1003. 

Rule 2 Take the cubes of the two first numbers, and subtract the less from the 

greater ; then multiply the least of the two numbers cubed by 6 ; add the product, 
with the addition of 6, to the difference, and continue this the first series of differ- 
ences. 

For the second series of differences, add the cube of the highest of the above num- 
bers to the difference, and the sum Avill be the cube of the next number. 

Example.— What are the cubes of 1001, 1002, and 1C03 ? 

First Series. 
Cube of 1000 = 1000000000 Second Series. 

Cube of 999 = 997002999 Cul)e of 10 oo = 1000000000 

1997001 Difference. Diff. for 1000, 3003001 

999x6+0= 6000 1003003001= 0w6e of 1001. 

3003001 Diff. of 1000. Diff. for 1001, 3009007 

6000 +6= 6006 1006012008 =Cwfo of 1002. 

3009007 Diff. of 1001. Diff. for 1002, 3015019 

G0C6 +6= 6012 1009027027= Cube of 1003. 

3015019 Diff. of 1002. 

!To Compute the Square or Cube Etoot of a higher Num- 
ber than is contained in the Table. 

When the Number is divisible by 4 or 8 without leaving a Remainder. 

Rule Divide the number by 4 or S respectively, as the square or cube root is re- 
quired; take the root of the quotient in the table, multiply it by 2, and the product 
will give the root required; 

Example. — What are the square and cube roots of 3200? 
3230^-4 = 800, and 3200 -f- 8 = 400. 

Then the square root for 800, per table, is 28.2S42712, which, being X2=56.5 08 54 24, 
the root. 

The cube root for 400, per table, is 7.36S063, which, being X2 = 14.73 Gl 20, the root. 

When the Root (which is taken as the Number) does not exceed 1 GOO. 
The Numbers in the table are the roots of the squares or cubes, 
which are to be taken as the numbers. 
Illustration.— The square root of 6400 is 80, and the cube root of 512000 is SO. 



240 SQUARES, CUBES, AND ROOTS. 

When a Number has Three or more Ciphers at its ri^ht hand. 

Rule. — Point off the number into periods of two or three figures each, according 
as the square or cube root is required, until the remainiug figures come within the 
limits of the table ; then take the root for these figures, and remove the decimal 
point one figure for every period pointed off. 

Example What are the square or cube roots of 1500000 ? 

1500000 = 150, the remaining figure, the square root of which = 12.24745; hence 

1224.745, the square root. 
1500000 = 1500, the remaining figures, the cube root of which = 11.44714 ; hencs 

114.4714, the cube root. 

To Ascertain tlie Cnbe Hoot of any ^N"xT.m"ber over 1GOO. 

Rule. — Find by the table the nearest cube to the number given, and call it the as- 
sumed cube ; multiply it and the given number respectively by 2 : to the product 
of the assumed cube add the given number, and to the product of the given number 
add the assumed cube. 

Then, as the sum of the assumed cube is to the sum of the given number, so is the 
root of the assumed cube to the root of the given number. 

Example— What is the cube root of 224809 ? 

By table, the nearest cube is 216 000, and its root is 60. 

216 000 X 2 + 224 S09 = 656 S09, 
And 224 S09 X 2 + 2 i 6 000 = 665 618. 
Then 656 S09 : 6G5618 : : 60 : 60.S04+, the root. 

To Ascertain tne Square or Cvfbe Hoot of a 3N"u.riii6er con- 
sisting of Integers and. Decimals. 

Exile. — Multiply the difference between the root of the integer part and the root 
of the next higher integer by the decimal, and add the product to the root of the in- 
teger given ; the sum will be the root of the number required. 

This is correct for the square root to three places of decimals, and for the cube 
root to seven. 
Example.— What is the square root of 53.75, and the cube root of S43.75? 

-^S44 = 9.4503 
VS43 = 9.4466 

.0037 
.75 



i 


54 = 
53 = 


7.3484 

7.2S01 






.0683 
75 


V 

V53 


53 = 
.75 = 


.051225 

7.2801 

7.331325 



.002775 
3/S43 = 9.4466 
•?/843.75 = 9.449375 

When the Square Root is required for Numbers not exceeding the Roots 
given in the Table. 

The Numbers in the table are the squares and cubes of the roots. 

Illustration.— The square of 27.313 is 746, and the cube of 11.01925 is 1338. 

Rule. — Find, by the table, in the column of numbers, that number representing 
the figures of the integer and decimals for which the root is required, and point it 
off decimally by places of 2 or 3 figures as the square or cube root is required; and 
opposite to it, in the column of roots, take the root and point off 1 or 2 additional 
places of decimals to those in the root, as the square or cube root is required, and the 
result is the root required. 

Example. — What are the square roots of .15, 1.50, and 15.00 ? 

In the table 15 has for its root 3.87298 ; hence .3S7298 = the square root for .16. 

150 has for its root 12.24745 ; hence 1.224745 =zthe square root for 1.50. 

1500 has for its root S8.729S ; hence 3.8729S = *fce square root for 15. 

Ex. 2._What are the cube roots of .15, 1.50, and 15.00? 

Add a cipher to each, to give the numbers three places of figures. 



SQUARES, CUBES, AND EOOTS. 241 

In the table 150 has for its root 5.3133; hence .53103=: the cube root o/.15. 
1500 has for its root 11.447 ; hence 1.1447 = the cube root of 1.50. 
15 has for its root 2. 4602; and 15.000, by the addition of 3 places of figures, has 
24.602 ; hence 2.4662 = the cube root of 15.00. 

To Ascertain tlie Square or Cube Roots of Decimals alone. 

Rule.— Point off the number from the decimal point into periods of two or three 
figures each, as the square or cube root is required. Ascertain from the table or by 
calculation the root of the number corresponding to the decimal given, the same be- 
ing read off by removing the decimal point one place to the left for every period of 2 
figures if the square root is required, and one place for every period of 3 figures if 
the cube root is required. 
Example. — What are the square and cube roots of .S10, .0S1, and .0081? 

.810, when pointed off — .81, and V =z 9. which becomes .9 ; 

.0S1, " " "=.081, u -/= 2.846 " " .2846; 

.0081, " " " = .00Si, " V = 9. « " .09. 

.810, when pointed off = .SI 6, and V = 9.3217, which becomes .93217 ; 
.081, " " "=.0si, " I/— 4.3267, " " .43267; 

.OftSl, u " u = .0031, " y — 2.00S3, " " .20083. 

To Compute the 4:rh Hoot of a Number, 
Rule. — Take the square root of its square root. 
Example.— What is the ^/ root of 1600? 

y/lto>. =40, and ^40 = 6. 3 245 553. 

To Compute trie 6th. Ptcot of a Number, 

Rule. — Take the cube root of its square root. 
Example— What is the V of 441? 

V411 = 21, and 3/21 = 2.7 589 243. 

To Extract tlie Hoot of any Griven Number of any Power. 

Rule — Divide the given number by any assumed root, raised to the next less 
power to that given ; to the quotient add the assumed root, multiplied by the next 
less power to that given ; divide the sum by the given power for a new root, with 
which repeat the operation if necessary. 

Example.— Ascertain the cube root of 64. 

64-^- 4 2 — ^ — quotient of given number divided by assumed root o/4, raised to the 
next less power (its square). 

4-J-4X2 = 12 —sum of above quotient, and the assumed root multiplied by the 

next less power. 
12 -4- 3 = 4. =z quotient of above sum -4- the given power = the root required. 
Ex. 2.— Ascertain the cube r oot of 216. Assume the root to be 4. . 
216 -4- 4* = 13.5, and 13.5 -f 4x 2 = 21.5, whi ch -f- 3 = 7 .1667, which is too great. 
Then 216 -4- 7.16672 = 4.2054, and 4.2054 -f 7.1667 X 2 = 1S.538S, which 4-3 = 
6.1796, which is also too great. 

Again, 216 -f- 6.7962 = 5.6563, and 5.6563 -f 6.1796x2 =d 18.0155, which -4- 3 = 
6.005?, which is also too great. 
Finally, 216 -4> 6.00522 == 5.99, and 5.99 + 6.0052x2 = IS, which -f- 3 = 6, the root 
Ex. 3 — Ascertain the fifth root of 6436314. 
Assume 20, the fourth power of which = 160000. 

JES? 6 ??f^ "*" 16000 = 40 ' 224 ' and 40 - 2 27 + 20^4 = 120.227, which -4- 5 = 
24.045, which is too great. 

Assume 24, the fourth power of which = 331776. 

Then 6436343 -r- 331776 + 24x1 = 115.4, which -4- 5 = 23.08, which is also too great, 

Assume 23, the fourth power of which = 279841 

Then 64G6343 -4- 279841 = 23, the root. 

X 



242 



FOURTH AND FIFTH POWERS OF NUMBERS. 



Table of the 4th. and. 5th IPowers of INTmnVbers, 



Number. 


4th Power. 


5th Power. 


Number. 


4th Power. 


5th Power. 


1 


1 


1 


65 


17S50625 


11 6: »29()625 


2 


16 


32 


66 


18974736 


1252332576 


3 


81 


243 


67 


20151121 


1350125107 


4 


256 


1024 


68 


213S1376 


1453933568 


5 


625 


3125 


69 


22667121 


1564031349 


6 


1296 


7776 


70 


24010000 


1680700000 


T 


2401 


16807 


71 


25411681 


1S04229 351 


8 


4096 


32768 


72 


26873856 


1934917632 


9 


6561 


59049 


73 


28398241 


2073071593 


10 


10000 


100000 


74 


29986576 


• 2219006624 


11 


14641 


161051 


75 


31640625 


2373046875 


12 


20736 


248832 


76 


33362176 


2535525376 


13 


28561 


371293 


77 


35153041 


27067S4157 


14 


38416 


537824 


7S 


87015056 


2SS717431J8 


15 


50625 


759375 


79 


38950 0S1 


3077056399 


16 


65536 


1048576 


80 


40960000 


3276800000 


17 


83521 


1419857 


81 


43046721 


3486784401 


18 


104976 


1889568 


82 


45212176 


3 707398432 


19 


130321 


2476099 


83 


4745S321 


3939JD40643 


20 


160000 


3200000 


84 


49787136 


4182119424 


21 


194481 


4084101 


85 


52200625 


4437053125 


22 


234256 


5153632 


86 


54703016 


4704270176 


23 


279841 


6436343 


87 


57289761 


4984209207 


24 


331776 


7962624 


88 


59969536 


5277319168 


25 


390625 


9765625 


89 


62742241 


55S4059449 


26 


456976 


11 SSI 376 


90 


65610000 


5904900000 


27 


531441 


14348907 


91 


6S 574961 


6240321451 


28 


614656 


17210368 


92 


71639296 


6590815232 


29 


707281 


20511149 


S3 


74805201 


6956 8S3 693 


30 


810000 


24300000 


94 


78074896 


7339040224 


31 


923521 


28629151 


95 


81450625 


7737809375 


32 


1048576 


33554432 


96 


84034656 


8153726976 


33 


1185921 


39135393 


97 


8S 529 281 


85S7340257 


34 


1336336 


45435424 


98 


92236816 


9039207968 


35 


1500625 


52521875 


99 


96059601 


9509900499 


36 


1679616 


60466176 


100 


100000000 


10000000000 


37 


1874161 


69343957 


101 


104060401 


10510100501 


38 


20S5136 


79235168 


102 


108243216 


11 040 SOS 032 


39 


2313441 


90224199 


103 


112550 8S1 


11592740743 


40 


2560000 


102400000 


104 


1169S5S56 


12166529024 


41 


2825761 


115856201 


105 


121550625 


12762815625 


42 


3111696 


130691232 


106 


126247696 


13382255776 


43 


3418S01 


147008443 


107 


131079601 


14025517307 


44 


374S096 


164916224 


108 


136048896 


1469328076S 


45 


4100625 


1S4528125 


109 


141158161 


1 53S6239540 


46 


4477456 


205962976 


110 


146410000 


16105100000 


47 


4S796S1 


229345007 


111 


151807041 


16S505S1551 


48 


5308416 


254S039G8 


112 


157351036 


IT 623 416 832 


4T) 


5764801 


2S2475249 


113 


163047361 


18424351793 


50 


6250000 


312500000 


114 


168S96016 


19254145824 


51 


6765201 


345025251 


115 


174900625 


201135S18T5 


52 


7311616 


380204032 


116 


181063936 


21003416576 


53 


7S'!04S1 


418195493 


117 


1S73SS721 


21924480357 


54 


8503056 


459165024 


118 


193877776 


22S77577568 


55 


9150625 


503284375 


119 


200533921 


23S63536599 


56 


9834496 


550731776 


120 


207360000 


24883200000 


57 


10556001 


601692057 


121 


21435SSS1 


25937424601 


58 


11316496 


65635676S 


122 


221533456 


270270S1632 


59 


12117361 


714924299 


123 


228SS6641 


28153006843 


CO 


12960000 


777600000 


124 


236421376 


29316250624 


61 


13S45841 


844596301 


125 


244140625 


30517578125 


62 


14776336 


916132832 


126 


552047376 


31757969376 


63 


15752961 


992436549 


127 


260144641 


33038369407 


64 


16777256 


1073 741 S24 


128 


26S435456 


3435973S368 






RECIPKOCALS. 



243 







Tatole— 


{Continued). 




Number. 


4th Powar. 


5th Power. 


Number. 


4th Power. 


5th Power. 


129 


.276922881 


35723051649 


140 


384160000 


53782400o00 


130 


285610000 


37129300000 


141 


395254161 


55730836701 


131 


294499921 


385794S9651 


142 


406586896 


57735339232 


132 


303595776 


40074642432 


143 


41S161601 


59797108943 


123 


312900721 


41615795893 


144 


429981696 


61917364224 


134 


322417936 


43 204003424 


145 


442050625 


64097340625 


135 


332150 615 


44840334375 


146 


454371856 


66338290976 


136 


342102016 


46525874176 


147 


4669488S1 


68641485507 


137 


352275361 


48261724457 


148 


479785216 


71008 211968 


138 


362673936 


50049003 16S 


149 


492884401 


73439775749 


139 


373 30 1 641 


51S8S844699 


150 


506250000 


75937500000 



To .Ascertain the 4th Power of a Number greater than is 
contained, in the Tatole. 

Rule. — Ascertain the square of the number by the preceding table or by calcula- 
tion, and square it ; the product is the power required. 

Example What is the 4th power of 15 ? 

152 = 225, and 2252 = 50 625. 

To .Ascertain the £>th Po^wer of a Number greater than is 
contained, in the Ta"ble. 

Rule. — Ascertain the cube of the number by the preceding table or by calcula- 
tion, and multiply it by its square; the product is the power required. 
Example. — What is the 5th power of 15? 

153 — 3375, w hi c h x 152(225) =759 375. 

To Ascertain the 4th and Stli Powers Toy another IMethod. 

Rule.— Reduce the number by 2 until it is one contained within the table. Take 
the power which is required of that number, and multiply it by 16, 162, an( j 103 re _ 
spectively for each division by 2 for the 4th power, and by 32, 322, 3^3 respectively 
for each division by 2 for the 5th power. 

Example.— What are the 4th and 5th powers of 600 ? 

600 -r- 2 = 300, and 300 -4- 2 = 150. 

The 4th power of 150, per table, — 506 250 000, which X 16 2 (256), the multiplier 
for a second division = 129 600 000 000, the ±th power. 

Again, the 5th power of 150 = 75 937 503 000, which X 322 ( 1024) = 77 760 000 000 000, 
the bih power. 

To Compute the 6th Power of a !N"nrnber. 

Rule. — Square its cube. 
. Example.— What is the 6th power of 2? 

(23)2 = 82, and 82 = 64. 

To Ascertain the 4th or 5th !Root of a Number. 

Rule.— Find in the column of 4th and 5th powers the number given, and the 
number from which that power is derived will be the root required. 
Example.— What is the 5th root of 3200000? 
3200000 in the table is the 5th power of 20 ; hence 20 is the root required. 

RECIPROCALS. 

The Reciprocal of a number is the quotient arising from dividing 1 by the num- 
ber ; thus the reciprocal of 2. is 1 -=- 2 = .5. 
The product of a number and its reciprocal is always equal to 1. ; thus, 2 X .5= 1. 
The reciprocal of a vulgar fraction is the denominator divided by the numerator; 

thus, - = .5. 



244 MENSURATION OF AREAS, LINES, AND SURFACES. 
MENSURATION OF AREAS, LINES, AND SURFACES. 

PARALLELOGRAMS. 

Definition. — Quadrilaterals, having their opposite sides parallel. 

To Compute tlie Area of a Square, a Rectangle, a 
Rhombus, or a Rhomboid.— Figs. 1, S, 3, and 4r. 

Rule. — Multiply the length by the breadth or height. 

Or, Ixb = area, I representing the length, and b the breadth. 
Fig. 1. Fig. 3. 

a\ 



Fig. 2. 




b b 

Examine The sides a b,b c, Fig. 1, are 5 feet 6 ins. ; what is the area? 

5.5X5.5 = 30.25 square feet. 
Note. — The side of a square is equal to the square root of its area. 
2. The opposite angles of a- Rhombus and a Rhomboid are equal. 

GNOMON. 

Definition. — The space included between tha lines forming two similar parallel- 
ograms, of which the smaller is inscribed within the larger, so that one angle in each 
is common to both. 

To Compute tlie Area of* a Grnomon. 

Rule. — Ascertain the areas of the two parallelograms, and subtract 
the less from the greater ; the difference will give the area. 
Or, a — a'= area, a and a' representing the areas. 
Example. — The sides of a gnomon are 10 by 10 and G by G ins. ; what is its area ? 
10X10 = 103, and 6X6 = 36. Then 100 — 36 = 64 square ms. 

TRIANGLES. 
Definition — Plain superficies having three sides and angles. 

To Compute the Area of a Triangle-Figs. 5, 6, and. *7. 
Rule. — Multiply the base by the height, and divide the product by 2. 
Or, — . Or, — — = area, b representing the base, and h the height. 

Note. — The TTypothenuse of a right angle is the side opposite to the right angle. 
2. The perpendicular height of a triangle = twice its area divided by its base. 
Fig. 5. Fig. G. Fig. 7. 




Example. — The base a &, 
Fig. 5, is 4 feet, and the height 
c b, 6 ; what is the area? 
4X6= 24, and 24 +- 2 = 12 
square feet. 






MENSURATION OF AREAS, LINES, AND SURFACES. 245 

To Compute tlie Area of a Triangle by the Length, 
of its Sides-Figs. 6 and 7. 

Rule. — From half the sum of the three sides subtract each side sep- 
arately ; then multiply the half sum and the three remainders contin- 
ually together, and take the square root of the product. 

Or, ^/s(s — a)X(» — ^)X(s — c)— area, a, b, c representing the sides, and s half 
the sum of the three sides. 

When all the Sides are Equal. 

Rule. — Square the length of a side, and multiply the product by 

.433. 

Or, S 2 X.433 =± area, S representing length of a side. 

Exami»le.— The sides of a triangle are 30, 40, and 50 feet ; what is the area? 

on -J- 40 4- 50 120* ^ — 30— 30"| 

!i_^L — JL— — — — 60, or half sum of the sides. 60 — 40 = 20 > remainders, 
2 2 60 — 50 = 10) 

Whence 30x20x10x00 = 360000, and ^300000 = 600 square feet. 

To Compute the Length of one Side of a Right-an- 
gled Triangle, the Length of the other t^vo Sides 
"being given— Fig. 5. 

When the two Legs are given, to Ascertain the Hypothenuse. 

Rule. — Add together the squares of the two legs, and take the 
square root of their sum. 

Or, Tjab* + bc2 — hypothenuse. Or, y/b* -4- H>. 

Example. — The hase a b is 30 ins., and the height be 40; what is the length of 
the hypothenuse ? 

302 _]_ 4Q2 — 2500, and -/2500 = 50 ins. 

To Ascertain the other Leg, When the Hypothe- 
nuse and one of the Legs are given— B^ig. 5. 

Rule. — Subtract the square of the given leg from the square of the 
hypothenuse, and take the square root of the remainder. 

Or, V'»UP.>-\&zk 0r < V^H Zzlt 

Example The ba?e of a triangle is 30 feet, and the hypothenuse 50 ; what is the 

height of it ? 
5 j2 — 302 — 25G0 — COO, and 2500 — 900 — 1600. Then V1600 =40 feet. 

To Compute the Length of a Side, "When the Hy- 
pothenuse of a Right-angled Triangle of equal 
Sides alone is given— ZFigs. S and 9. 

Rule. — Divide the hypothenuse by 1.414213. 

hyp, 

O r i -. . . \ A > o = tfte length of a side. 
1.414213 

Examplk.— The hypothenuse of a right-angled triangle is 300 feet ; what is the 
length of its sides ? 

300 -*- 1.414213 = 212.1321 feet. 
X* 



246 MENSURATION OF AREAS, LINES, AND SURFACES. 

To Compute tlie ^Perpendicular or Height of a Tri- 
angle, "W lie 11 tlie Base and. Area alone are given. 

Rule. — Divide twice the area by its base. Or, 2a^-b — h. 

Example. — The area of a triangle is 10 feet, and the length of its base 5; what U 
its perpendicular ? 

10x2 = 20, ami 20 -^5 = 4: feet. 

To Compute the Perpendicular or Height of a 
Triangle, When the two Sides and the Base are 
given. 

Rule. — As the base is to the sum of the sides, so is the difference 
of the sides to the difference of the divisions of the base. Half this 
difference being added to or subtracted from half the base will give 
the two divisions thereof. Hence, as the sides and their opposite di- 
vision of the base constitute a right-angled triangle; the perpendicular 
thereof is readily ascertained by preceding rules. 

_ bc + ca X becoca 

Or, = bdoo da. 

ba 







a c 2 -f- a b* — b c 2 



•ad; whence V a ° 2 — « a 2 — d c. 



2ab 

Example. — The three sides of a triangle, a b c, Fig. 8, 
are 9.928, 8, and 5 feet ; what is the length of the perpen- 
dicular on the longest side ? 

As 9. 923 : S -f 5 : : S co 5 : 3.928, the difference of the divi- 
sions of the base. 

m 9 928 

Then 3.92S-i-2=if 9G4, which, added to— — = 4.r64-f- 

1 5 1.964 = 6.92S, the length of the longest division of the 

d base* 

Hence we have a right-angled triangle with its base 6.92 Q , and its hypothenuse S; 
consequently, its remaining side or perpendicular is ^(8? — 6.92S2)=4/tet 

When any two of the Dimensions of a Triangle and one of the correspond- 
ing Dimensions of a similar Figure are given, and it is required to as- 
certain ffie other corresponding Dimensions of the last Figure. 

c Fig. 9. Fig. 10. 

Let abc,a'b' ' c\ be two similar triangles, Figs. 9 and 10. 

Then a b : bc.'.a'b': V ' c', or a V :b" c* ::ab :b c. 

Note. — The same proportion holds with respect to the 
similar lineal parts of any other similar figures, whether 
plane or solid. 
b a V a' 

Example. — The shadow of a vertical cone 4 feet in length was 5 fe^t ; at the same 
time, the shadow of a tree on level ground was S3 feet ; what was the height of the tree? 
5 a' b':4b'c'::S3ab: 6G.2-5 be feet. 

TRAPEZIUM. 

Definition. — Quadrilaterals having unequal sides. 

To Compute the Area of a Trapezium-Fig. 11. 

Rule. — Multiply the diagonal by tlie sum of the two perpendiculars 

falling upon it from the opposite angles, and divide the product by 2. 

_ dbXa + c 

Or, = area. 




. 



MENSURATION OF AKEAS, LINES, AND SURFACES. 247 

Fig. 11. 
a 

/T^^^ Example. —The diagonal cl 6, Fig. 11, is 125 feet, and the per- 

J.Z....J T .-Trr^2> pendiculars a and c 50 and 37 feet ; .what is the area? 

125x50 -j- 37 = 10S75, and 10S75-r- 2 = 5437.5 square feeL 



When the two opposite Angles are Supplements to each other, that is, 
when a Trapezium can be inscribed in a Circle, the Sum of its opjto- 
site Angles being equal to two Bight Angles, or 180°. 
Rule.— From half the sum of the four sides subtract each side sev- 
erally ; then multiply the four remainders continually together, and 
take the square root of the product. 

Example.— In a trapezium the sides are 15, 13, 14, and 12 feet; its opposite an- 
gles being supplements to each other, required its area. 

154-13 + 14 + 12 = 54, and -^—^- 

27 27 27 27 
15 13 14 12 
12X14X13X15 = 327G0, and V32TG0 = 180.997 square feet, 

TRAPEZOID. 
Definition.— A Quadrilateral with only one pair of opposite sides parallel. 

To Compute tlie Area of a Trapezoid.— Fig. 12. 

Rule.— Multiply the sum of the parallel sides by the perpendicular 
distance between them, and divide the product. 

^ ab + dcxah^ Q ^ s±^Xh _ — ^ g aud g/ representing the sides# 

Fig. 12. 



Example.— The parallel sides ah, c c7, Fig. 12, are 100 and 132 
feet, and the distance between them 62.5 feet; what is the area? 
100 + 132x62.5 = 14500, and 145C0 ^- 2 = 7250 square feet 
c h d 

POLYGONS. 
Definition. — Plane figures having three or more sides, and are either regular or 
irregular, according as their sides or angles are equal or unequal, and they are named 
from the number of their sides and angles. 

Regular Polygons. 
To Compute tlie^rea of a Regular Polygon— P^ig. 13. 

Rule. — Multiply the length of a side by the perpendicular distance 
to the centre ; multiply the product by the number of sides, and di- 
vide it by 2. 



a h X c e X ti 
Or, — — area, n representing the number of sides. 



Fig. 13. 



Example. — What is the area of a pentagon, the side a b, Fig. 13, 
being 5 feet, and the distance c e Qi feet? 
) 5x43^x5 (n) = 100%" = product of length of a side, the distance to 
the centre, and the number of sides. 

Then 100^-^2 = 53. 125 /erf. 



248 MENSURATION OF AREAS, LINES, AND SURFACES. 



To Compute the Area of a Regular Polygon, When 
tlie Length of a Side only is given. 

Rule. — Multiply the square of the side by the multiplier opposite 
to the name of the polygon in the following table : 









A. 


B. 


C. 


D. 


No. of 
Sides. 


Name of Polygon. 


Area. 


Radius of | 
Circumscribed Lengrth of the 


Radius of 
Circumscrib- 


Radius of 
Inscribed 








Circle. 


Side. 


ing Circle. 


Circle. 


3 


Trigon 


.433013 


2. 


1.732 


.5773 


.2887 


4 


Tetragon 


1. 


1.414 


1.4142 


.7071 


.5 


5 


Pentagon 


1.72047T 


1.239 


1.1756 


.8506 


.6SS2 


6 


Hexagon 


2.59807G 


1.156 


1. 


1. 


.866 


T 


Heptagon 


3.633912 


1.11 


.S677 


1.1524 


1.0383 


8 


Octagon 


4.828427 


1.0S3 


.7653 


1.3066 


1.2071 


9 


Nonagon 


6.1S1S24 


1.064 


.634 


1.4619 


1.3737 


10 


Decagon 


7.694209 


1.051 


.618 


1.618 


1.5388 


11 


In dec agon 


9.36564 


1.042 


.5634 


1.7747 


1.7028 


12 


Dodecagon 


11.196152 


1.037 


.5176 


1.9319 


1.S66 



Example.— What is the area of a square when the length of its sides is 7.0710678 
inches ? 

7.071067S2 = 50, and 50x1. =59 ins. 

To Compute the Radius of a Circle tliat contains 
a Given Polygon, When the Length of a Perpen- 
dicular from the Centre alone is given. 

Rule. — Multiply the distance from the centre to a side of the poly- 
gon by the unit in column A. 

Example. — What is the radius of a circle that contains a hexagon, the distance to 
the centre being 4.33 inches ? 

433X1.156 = 5 ms. 

To Compute the Length of a Side of a Polygon that 
is contained in a Griven Circle, When the Radius 
of the Circle is given. 

Rule. — Multiply the radius of the circle by the unit in column B. 

Example. — Yfhat is the length of the side of a pentagon contained in a circle S.5 
feet in diameter ? 

8.5-7-2 = 425 radius, and 4.25x1.1756 = 5/^. 

To Compute the Radius of a Circumscribing Cir- 
cle, When the Length of a Side is given. 

Rule. — Multiply the length of a side of the polygon by the unit in 
column C. 

Example. — What is the radius of a circle that will contain a hexagon, a side being 
5 inches? 

5x1 = 5 ins. 

To Compute the Radius of a Circle that can "be In- 
scribed in a Given Polygon, When the Length of 
a Side is given. 

Rule. — Multiply the length of a side of the polygon by the unit in 
column D. 

Example. — What is the radius of the circle that is bounded by a hexagon, its sidea 
being 5 inches? 

5x.S66 = 4.33rns. 



MENSURATION OF AREAS, LINES, AND SURFACES. 249 



To Compute tne Length ofa Side and. Radius of a 
Regular Polygon, When the Area alone is given. 

Kule. — Multiply the square root of the area of the polygon by the 
multiplier in column E of the following table for the length of the 
Bide ; by the multiplier in column G for the radius of the circumscrib- 
ing circle ; and by the multiplier in column H for the radius of the 
inscribed circle or perpendicular. 







E. 


G. 


H. 








No. of 


Name of 


Length of 


Radius of 


Radius of 


Angle. 


Angle of 


\ 


Sidss 


Polygon. 


the Side. 


Circumscrib- 


Inscribed 


Polygon. 


Tangents. 








ing Circle. 


Circle. 








3 


Trigon 


1.5197 


.8774 


.4357 


120° 


60° 


.57735 


4 


Tetragon 


1. 


.7071 


.5 


90 


90 


1. 


5 


Pentagon 


.7624 


.6485 


.5247 


72 


108 


1.37638 


6 


Hexagon 


.6204 


.6204 


.5373 


60 


120 


1.73205 


7 


Heptagon 


.5246 


.6045 


.5446 


51 25' 


128 4-7 


2.07652 


8 


Octagon 


.4551 


.5346 


.5493 


45 


135 


2.41421 


9 


Nonagon 


.4022 


.588 


.5525 


40 


140 


2.74747 


10 


Decagon 


.3605 


.5S33 


.5548 


36 


144 


3.07768 


11 


Undecagon 


.3268 


.5799 


.5564 


32 43' 


147 3-11 


3.40568 


12 


Dodecagon 


.2989 


.5774 


.5577 


30 


150 


3.73205 



Example.- 

side? 



-The area ofa square (tetragon) is 16 inches; what is the length of its 
V16 — 4, and 4x1 = 4 ins. 



Additional Uses of the foregoing Table. — The 6th and 7th columns of the table 
facilitate the construction of, these figures with, the aid ofa sector. Thus, if it is re- 
quired to describe an octagon, opposite to it, in column 6th, is 45 ; then, with the 
chord of 60 on the sector as radius, describe a circle, taking the length 45 on the 
same line of the sector ; mark this distance off on the circumference, which, being 
repeated around the circle, will give the points of the sides. 

The 7th column gives the angle which any two adjoining sides of the respective fig- 
ures make with each other ; and the 8th gives the tangent of the angle in column 6th. 

REGULAR BODIES. 

To Compute the Surface or Linear Edge of any 
Regular Solid. Body. 

Rule. — Multiply the square of the linear edge, or the radius of the 
circumscribed or inscribed sphere, by the units in the following table, 
under the head of the dimension used : 



No. of 
Sides. 


Names of Figures. 


Surface. 


Radius of 
Circumscribed 

Circle. 


Radius of 
Inscribed 

Circle. 


Linear Edge 
by Surface. 


4 

6 

8 

12 

20 


Tetrahedron 

Hexahedron 

Octahedron 

Dodecahedron 

Icosahedron 


1.73205 

6. 

3.4641 

20.64578 
S. 60025 


1.63290 
1.1547 
1.41421 
.71P64 
1.05146 


4.S9898 

2. 

2.4494*) 

.S9S06 

1.32317 


.759S4 
.40S25 
.53729 
.22008 
.33981 



Example.— What is the surface ofa hexahedron or cube having sides of 5 inches? 
52 x 6 = 25 X 6 = 150 ins. 

To Compute trie Linear Edge wnen the Surface 

alone is given. 

Rule.— Multiply the square root of the surface by the multiplier 

under the head of Linear Edge by Surface. 

Example.— What is the linear edge of a hexahedron, the surface being 6 inches? 

y/6X 40825 = 1 linear edge. 



250 MENSURATION OF AREAS, LINES, AND SURFACES. 

Irregular Polygons. 
Definition.— Figures with unequal sides. 

To Compute tlie Area of an Irregular Polygon- 
Figs. 14 and. 15. 

Fig. 14. Rule. — Draw diagonals, as Fig. 15. 
*^^^ df, d g, gc, and g b, to divide 

dff^^J^^^f the figures into triangles and - ^zT^—Jl^^a 

/£>^ quadrilaterals : ascertain the (\ sy 

I ^<» areas of these separately, and \ ^^/ 

s ' " 3 take their sum. )?-'--""' \ \ 

Note. — To ascertain the area of mixed or compound fig- ^C~ | ~z?h 

ures, or such as are composed of rectilineal and curvilineal N. j ^^ 

figures together, compute the areas of the several figures ^J^ 

of which the whole is composed, then add them together, 

and the sum will be the area of the compound figure. In this manner any irregular 
surface or field of land may be measured by dividing it into trapeziums and trian- 
gles, and computing the area of each separately. 

When any Part of a Figure is bounded by a Curve the Area may be as- 
certained as follows : Erect any number of perpendiculars upon the base 
at equal distances, and ascertain their lengths. 

Add the lengths of the perpendiculars thus found together, and their 
sum, divided by their number, will give the mean breadth ; then multi- 
ply the mean breadth by the length of the base. 

To Compute the Area of a long Irregular Figure- 
Fig. 16. 

Tvs. li>. Rule. — Take the breadth at several places, and 

0^~">^_L at equal distances apart; add them together, divide 

^. b their sum by the number of breadths for the mean 
breadth, and multiply this by the length of the fig- 

d ure. 



& + &'+&" 
Or, = Bx I = area. 

CIRCLE. 

The Diameter is a right line drawn through its centre, bounded by its periphery. 

The Piadius is a right line drawn from its centre to its circumference. 

The Circumference is assumed to be divided into 360 equal parts, termed degrees; 
each degree is divided into CO parts, termed minutes; each minute into 60 parts, 
termed seconds ; and each second into 60 parts, termed thirds, and so on. 

To Compute the Circumference of a Circle,, 
Rule.— Multiply the diameter by 3.1416. 

Or, as 7 is to 22, so is the Diameter to the Circumference. 
Or, as 113 is to 355, so is the Diameter to the Circumference. 
Example. — The diameter of a circle is \% inches; what is its circumference? 
IX X 3. 1416 = 3. 827 ins. 

To Compute the IDiameter of a Circle. 

Divide the circumference by 3.1416. 

Or, as 22 is to 7, so is the Circumference to the Diameter. 

Note. — Divide the area by .7S54, and the square root of the quotient will give the 
diameter of the circle. 



MENSURATION OF AREAS, LINES, AND SURFACES, 251 

To Compute tlie Area of a Circle. 

Rule. — Multiply the square of the diameter by .7854. 
Or, multiply the square of the circumference by .07958. 
Or, multiply half the circumference by half the diameter. 
Or, multiply the square of the radius by 3.1416. 
Or^p r^ — area, r representing the radius. 

Example The diameter of a circle is 8 inches ; what is the area of it? 

82 or 8X8 = 64, and 64X.7854 = 50.2650 ins. 

USEFUL FACTORS. 
In which p represents the Circumference of a Circle, the Diameter of 



p= 3.1415926535S9 + 
2p= 6.2S31S5307179-}- 
4 p =12.566370614859 + 
J4P — 1.570796326794 + 



which is 1. 

U p= .7S5398163397 -f 
4-3 p = 4.18879 
-% p— .523598 
% p— .392C99 



1-12 p— .261799 

1-360 p— .008726 

XVP = -886226 

36^ = 113.007335 



In which the Diameter of a Circle is 10. 

1. Chord of the arc of the semicircle =10. 

2. Chord of half the arc of the semicircle = 7.071067 

3. Versed sine of the arc of the semicircle = 5. 

4. Versed sine of half the arc of the semicircle = 1.464466 

5. Chord of half the arc, of the half of the arc of the semicircle = 3.82683 

6. Half the chord, of the chord of half the arc = 3.535533 

7. Length of arc of semicircle = 15.707963 

8. Length of half the arc of the semicircle = 7.853981 

9. Square of the chord, of half the arc of the semieircle (2) = 50. 

10. Square root of versed sine of half the arc (4) = 1.210151 

11. Square of versed sine of half the arc (4) ' = 2.144664 

12. Square of the chord of half the arc, of half the arc of the semicircle (5;= 14.64467 

13. Square of half the chord, of the chord of half the arc (6) = 12. 5 
Note. — In all the calculations p is taken at 3.1416, %p at .7854, % p at .5236; 

and whenever the decimal figure next to the one last taken exceeds 5, oue is added. 
Thus, 3.14159 for four places of decimals is taken as 3.1416. 

To Compute tlie Length, of an JLvc of a Circle, Fig. 
1*7, "\^hen the Nuxriber of IDegrees and. the X-^acLrus 
are given. 

Rule 1. — Multiply the number of degrees in the arc by 3.1416 times 
the radius, and divide by 180. 

2. Multiply the radius of the circle by .01745329, and the product 
by the degrees in the arc. 

If the length is required for minutes, multiply the radius by 
.000290889 ; if for seconds, multiply by .000004848. 
Fig. 17. 

c- Example. — The number of degrees in an arc, o ab* are 

90, and the radius, o b, 5 inches ; what is the length of the 
arc? 

90x (3.1416x5) =1413.72, which -4- 180 = 7.S54 ins. 
Ex. 2. — The radius of an arc is 10, and the measure of its 
\ i /' angle 44° 30' 30" ; what is the length of the arc? 

\1/ 1 OX. '01745329 = .1745329, which X 44 = 7.6794476, the length 

o for 44°. 

10 X-000290S89 = . 00290889, which X30 = .0S72667, the length for 30'. 
10X.000004S4S = .00004S48, which X30 = .0014544, the length for 30". 
Then 7.6794476) 

.0872667 > = 7.7681687, length. Or, reduce the minutes and seconds to tho 

.0014544) decimal of a degree, and multiply by it. 

See Rule, page 4 / >. 30' 30" = .50S3, and .1745329 from above X44.50S3 = 7.76S168. 



252 MENSURATION OP AREAS, LINES, AND SURFACES. 

When the Chord of half the Arc and the Chord of the Arc are given. 

Rule. — From 8 times the chord of half the arc subtract the chord 
of the arc, and one third of the remainder will give the length nearly. 

Or, — - — , c' representing the chord of half the arc, and c the chord of the arc. 

Example — The chord of half the arc, a c, Fig. 17, is 30 inches, and the chord of 
the arc, a b. 48; what is the length of the arc? 

SOxS = 240 = S times the chord of half the arc ; 240 — 4S = 192, and lC2-^3 = 64 ins. 
When the Chord of the Arc and the Versed Sine of the Arc are given. 

Rule. — Multiply the square root of the sum of the square of the 
chord, and four times the square of the versed sine (equal to twice the 
chord of half the arc), by ten times the square of the versed sine ; di- 
vide this product by the sum of fifteen times the square of the chord 
and thirty-three times the square of the versed sine ; then add this 
quotient to twice the chord of half the arc,* and the sum will give the 
length of the arc very nearly. 

Vc2 + 4v2xl0a2 , n , . 

Or, — 2-L33 2 •" 2 c •> v representing the versed sine. 

Example. — The chord of an arc is 80, and its versed sine 30; what is the length 
of the arc ? 

802 — 6400 = square of the chord ; 302 — 900 — square of the versed sine. 

-^(6400 + 900x4) = 100 = square root of the square of the chord and four times the 
square of the versed sine = twice the chord of half the arc. 

Then 100x30 2 x 10 = 900000 = product of ten times the square of the versed sine 
and the root above obtained. 

SO* x 15 = 96000 — 15 times the square of the chord. 
30 2 X 33 = 29700 = 33 times the square of the versed sine. 
125700 

A , 100x302x10 900000 „ ' ... .. . _ <AA , . .. _ , _, .. 

And — tt=^ — -= +^. fe x — * .1539, which, added to 100. or twice the chord of half 
125i00 125(00 

the arc = 107.1599. 

When the Diameter and Versed Sine are given. 

Rule. — Multiply twice the chord of half the arc by 10 times the 
versed sine ; divide the product by 27 times the versed sine subtract- 
ed from 60 times the diameter, and add the quotient to twice the 
chord of half the arc j the sum will give the length of the arc very 
nearly. 

_ 2c'xl0y . _ , 
Or, — — —— 4-2c'=c. 
60d — 2 i v 

Example.— The diameter of a circle is 100 feet, and the versed sine of the arc 25; 
what is the length of the arc ? 

V 25x100 = 50 = chord of half the arc. 
50 X *? X 25 x 10 = 25000 = twice the chord of half the arc by 10 times the versed sine. 
luuxOO — 15x27 = 5325 = 27 times the versed sine subtracted from 60 times the 
diameter. 

Then ?5^ = 4.6948, and 4.694S 4- 50 x2 ~ 104.6948 feet. 
o325 

* The square root of the sum of the square of the chord and four times tlis square of the versed 
sine is equal to twice the chord of half the arc. 



MENSURATION OF AREAS, LINES, AND SURFACES. 253 

To Compute tlio Chord of* an Arc, "When the Chord of" 
half the -A-rc and. the Yersed Sine are given. 
Rule. — From the square of the chord of half the arc subtract the square of the 
.versed sine, and take twice the square root of the remainder. 
Or, V( c ' 2 — v*)x2 = c. 
Example. — The chord of half the arc is 60, and the versed sine 36 ; what is the 
length of the chord of the arc ? 

602 _ 3G2 — 2304, and V2304 X 2 = 96. 

When the Diameter and Versed Sine are given. 

Rule Multiply the versed sine by 2, and subtract the product from the diame- 
ter ; then subtract the square of the remainder from the square of the diameter, and 
take the square root of that remainder. 

Or, V(yX2 — rf)2— d2 — c. 

Example. — The diameter of a circle is 100, and the versed sine of half the arc is 
36 ; what is the length of the chord of the arc ? 

100 — 36x2 = 23; then 1002 — 282 = 9216, and V9216 = 96. 

To Compute the Chord of half an Arc, When the Chord 
of the .A.rc and the "Versed Sine are given. 

Rule 1. — Divide the square root of the sum of the square of the chord of the arc 
and four times the square of the versed sine by two. 

Rule 2 Take the square root of the sum of the squares of half the chord of the 

arc and the versed sine. 



V/ C 2 -L. 4 u3 



€W 



2 — r' 



When the Diameter and Versed Sine are given. 
Rule. — Multiply the diameter by the versed sine, and take the square root of their 
product. 

Or, Vdxv = e\ 

To Compute a Diameter. 
Rule 1. — Divide the square of the chord of half the arc by the versed sine. 

Or, c' 2 -^ v = diameter. 
Rule 2. — Add the square of half the chord of the arc to the square of the versed 
sine, and divide this sum by the versed sine. 

<c + W + * = (t 
1 v 

To Compute the Versed Sine. 
Rule.— Divide the square of the chord of half the arc by the diameter. 

~ c' 2 
Or, — = v. 
d 

When the Chord of the Arc and the Diameter are given. 

Rule. — From the square of the diameter subtract the square of the chord, and 
extract the square root of the remainder ; subtract this root from the diameter, and 
halve the remainder. 

d — Vd* — c2 
Or, - = v. 

When it is greater than a Semidiameter. 

Rule.— Proceed as before, but add the square root of the remainder (of the squares 
of the diameter and chord) to the diameter, and halve the sum. 

d-f. Vrf2_ C 2 

Or, _I_ = v . 



254 MENSUEATION OF AREAS, LINES, AND SURFACES. 

Proportions of the Circle^ its Equal, Inscribed, and Circumscribed Squares, 

CIECLE. 

> xiiii} = Side of an E( * ual S( * uare - 



2. Circumference ) 

3. Diameter X.7071) 

4. Cii-cumference X • 2251 > = Side of the Inscribed Square. 

5. Area X. 9003 -=- diam. ) 

8QUAEE. 

6. A Side Xl.l442=: Diameter of its Circumscribing Circle. 

7. M X 4.443 = Circumference of its Circumscribing Circle. 

8. " X 1.128 = Diameter ) 

e>of a 






9. u X3.545 = Circumference > of an Equal Circle. 

10. Square inches X 1.273 = Round inches j 

Note The square described within a circle is one half the area of one described 

without it. 

SECTOR OF A CIRCLE. 
Definition.— A part of a circle bounded by an arc and two radii. 

To Compute the Area of a Sector of a Circle, "When 
the Degrees in the Arc are given— Fig. 18. 

Rule. — As 360 is to the number of degrees in a sector, so is the 
Fi°- 18 area °^ tne cnx l e °f which the sector is a part to 

^X- — ^^^ the area of the sector. 

°^r ~y b Or, — - = area, d representing the degrees in the arc, and 

\. / a the area of the circle. 

\. / Example. — The radius of a circle, o a, is 5 inches, and the 

\f number of degrees of the sector, a b o, is 22° 30' ; what ia the 

o area ? 

Area of a circle of 5 inches radius == TS.54 inches. 
Then, as 360° : 22° 30' : : 78.54 : 4.90S75 ins. 

When the Length of the Arc, etc., are given — Fig. 17. 
Rule. — Multiply the length of the arc by half the length of the ra- 
dius, and the product is the area. 

Or, bXr^r-2 = area, b representing the arc, and r the radius. 

SEGMENT OF A CIRCLE. 
Definition.— A part of a circle bounded by an arc and a chord. 

To Compute the Area of a Segment of a Circle, 
.Fig. 17, When the Chord, and Versed Sine of the 
Arc, and Ptadins or IDiameter of the Circle are 
given. 

Rule. — When the Segment is less than a Semicircle, as a b c, Fig. 17, 
Ascertain the area of the sector having the same arc as the segment ; 
then ascertain the area of the triangle formed by the chord of the seg- 
ment and the radii of the sector, and take the difference of these areas. 

Note. — Subtract the versed sine from the radius; multiply the remainder by one 
half of the chord of the arc, and the product will give the area of the triangle. 
Or, a — a'=: area, a a' representing areas of the sector and the triangle. 

Rule. — When the Segment is greater than a Semicircle, Ascertain, by 
the preceding rule, the area of the lesser portion of the circle; subtract 
it from the area of the whole circle, and the remainder will give the 
area. 

Or, a — a'=area, a a' representing areas of circle and the lesser portion. 

See Table of Areas, page 205. 



MENSURATION OF AREAS, LINES, AND SURFACES. 255 

Fig. 19. Example.— The chord, a c, is 14.142 ; the diameter, b e, is 

b 20 ; and the versed sine, b d, is 2.929 inches ; what is the area 

J^<^\ — -v. of the segment ? 

a t f-' \ y % e 1 4.142-^-2 = 7.07 1 = half the chord of the arc. 
/ \ d ! / \ V^ 07l2 -+ 2. 9 29 ^ = 7.654 = the square root of the sum of the 
f \ ■ /' \ squares of half the chord of the arc and versed sine, 

; % vif ! which is the chord ab of half the arc a be. 

\ o j By Rule, page 252, 

\ j 7.654X2X10X2.929 = 448.371 = twice the chord of half the 

\ / arc by 10 times the versed sine. 

Xn * --*'' 20X60 — 2.929X27 = 1120.917 = 60 times the diameter sub* 

e traded from 27 times the versed sine. 

Then 448.371 -?- 1120.917 = .400, and .400 added to 7.654X2 (twice the chord of 

half the arc) = 15.708 inches, the length of the arc. By Rule, page 254, 15.708x-«- 

= 78.54= the arc multiplied by half the length of radius, = the area of the sector. 
10 — 2.929 = 7.071 = the versed sine subtracted from a radius, which is the height of 

14.142 
the triangle aoc, and 7.071 X — ~ — = 5° = area °f tJie triangle. 

Consequently, 78.54 — 50 = 28.54 ins. 

When the Chords of the Arc, and of the half of the Arc, and the Versed 
Sine are given. 

Rule. — To the chord of the whole arc add the chord of half the 
arc and one third of it more ; multiply this sum by the versed sine, 
and this product, multiplied by .40426, will give the area nearly. 

Or, c -J- c'-J- -vX .40426 = area nearly. 

Example The chord of a segment, a c, is 28 feet ; the chord of half the arc, a fc, 

is 15 ; and the versed sine, b d, 6 ; what is the area of the segment ? 

15 
2S-j-15-f- — =the chord of the arc added to the chord of half the arc and one third 

of it more. 48xQ = 28S = product of above sum and the versed sine. Then 
288X. 40426 = 116.427 feet. 

When the Chord of the Arc (or Segment) and the Versed Sine only are 

given. 

Rule. — Ascertain the chord of half the arc, and proceed as before. 

SPHERE. 

Definition. — A figure, the surface of which is at a uniform distance from the cen- 
tre. 

To Compute th.e Convex Surface of a Sphere— 
Fig. SO. 

Fig. 20. Rule. — Multiply the diameter by the circumference, 

and the product will give the surface. 

a Or, dc = surface, d representing diameter, and c the circum- 
, } Or, 4 p r 2 = surface. * 
Or, p d 2 = surface. 
Example What is the convex surface of a sphere of 10 inchea 
diameter ? 
d 10X31.416 = 314.16 ins. 

* p or 7r represents in this, and in all cases where it is used, the ratio of the circumference of a cir- 
cle to its diameter, or 3.1416. 



256 MENSURATION OF AREAS, LINES, AND SURFACES. 



SEGMENT OF A SPHERE. 
Definition. — A section of a sphere. 

To Compute tlie Surface of a Segment of a Sphere 
-Fig. 31. 

Rule.— Multiply the height by the circumference of the sphere, 
and add the product to the area of the base. 

Or, h c -f b = surface, h representing the height, and b area of base. 
Or, 2prh=z convex surface alone. 

Fig. 21. Example — The height, b o, of a segment, a be, is 36 inch- 

b es, and the diameter, b e, of the sphere 100 ; what is the con- 

vex surface, and what the whole surface ? 

36x100x3.1416 = 11309.76 = fcd«7^ of segment multiplied 

by the circumference of the sphere. 

i *N<. ! ,.-''' ; Then, to ascertain the area of the base; the diameter and 

; versed sine being given, the diameter of the base of the seg- 

\ J Eient, being equal to the chord of the arc, is, by Rule, page 253, 

100 — 36x2 = 28; ^/1002_2S2=96. 

y' S62 X .7854 = 7238.2464 = convex surface, and 723S.2464 + 

% 't 11309.76 = 1S54S.0064 = convex surface added to area of 

e base = the surface. 

Note. — When the convex surface of a figure alone is required, the area or areas of 
the base or ends must be omitted. 

When the Diameter of the Base of the Segment and the Height of it are 
alone given. 

Rule — Add the square of half the diameter of the base to the square of the height? 
divide this sum by the height, and the result will give the diameter of the sphere. 
Or, d-f-2 -f 7*2 -^ h = diameter. 

SPHERICAL ZONE (OR FRUSTRUM OF A SPHERE). 
Definition.— The part of a sphere included between two parallel chords. 

To Compute the Surface of a Spherical Zone— 

Fig. 22. 

Rule.— Multiply the height by the circumfer- 
ence of the sphere, and add the product to the 
area of the two ends. 

Or, h c -f a -4- a' = surface. 
Or, 2p r h = convex surface alone. 
Example. — The diameter of a sphere, a b, from which a 
zone c g is cut, is 25 inches, and the height of it, c g, is 8 ; 
what is the convex surface ? 

25x3.1416x8 = 62S.32 = height X circumference of sphere 
= convex surface. 

When the Diameter of the Sphere is not given, Multiply the mean length 
of the two chords by half their difference; divide this product by the 
breadth of the zone, and to the quotient add the breadth. To the 
square of this sum add the square of the lesser chord, and the square 
root of their sum will give the diameter of the sphere. 

SPHEROIDS OR ELLIPSOIDS. 

Definition.— Figures generated by the revolution of a semi-ellipse about one of 
its diameters. 

When the revolution is about the transverse diameter they are Prolate, and when 
it is about the conjugate they are Oblate. 




MENSURATION OF AREAS, LINES, AND SURFACES. 257 



To Compute the Surface of a Spheroid— Fig. 33. 

When the Spheroid is Prolate. 
Rule. — Square the diameters, and multiply the square root of half 
their sum by 3.1416, and this product by the conjugate diameter. 



Fig. 23. 



Or, 



", ./ — 2 — x 3 - 1416 X d = surface, d representing 

conjugate diameter. 
Example. — A prolate spheroid has diameters of 10 and 
14 inches ; what is its surface ? 

10 2 + 14 2 = 296 = sum of squares of diameters. 
296 -j- 2 = 148, and y/U8 == 12.1655 — square root of half 

the sum, of the squares of the diameters. 
12.1655x3.1416x10 r=332. 191 ins. = product of root above 

obtained X 3. 1416, and that product by the conjugate 

diameter. 

When the Spheroid is Oblate. 
Rule. — Square the diameters, and multiply the square root of half 
their sum by 3.1416, and this product by the transverse diameter. 




Or, 



•/■ 



tfa + o"/: 



X3.1416xd' = surface, d' representing transverse diameter. 



Example. — An oblate spheroid has diameters of 14 and 10 inches; what is its 
surface ? 

14 2 -4- 10 2 — 296 = sum of squares of diameters. 
296 -4- 2 = 148, and -/148 == 12. 1655 = square root of half the sum of the squares of 

the diameter. 
12.1655x3.1416x14 m 535. 06T9 ins. = product of root above obtained X3.1416, and 
that product by the transverse diameter. 

To Compute the Convex Surface of a Segment of a 
Spheroid— I^gs. 24 and. 25. 

Rule. — Square the diameters, and take the square root of half their 
sum ; then, as the diameter from which the segment is cut is to this 
root, so is the height of the segment to the proportionate height re- 
quired. Multiply the product of the other diameter and 8.1416 by 
the proportionate height of the segment, and this last product will 
give the surface. 



-Xh 



Or, 



-Xd" or dx3.1416 = surface. 



Fig. 24. 



d or d' 

Example.— The height, a 0, of 
a segment, e /, of a prolate sphe- 
roid, Fig. 24, is 4 inches, the di- 
ameters being 10 and 14 inches; 
what is the convex surface of it ? 
Square root of half the sum of the 
squares of the diameters, as by 
previous examples, 12.1655. 
Then 14 : 12.1655 : : 4 : 3.4758 = 
height of segment, proportion- 
ate to the mean of the diameters. 
10x3.1416x3.4758 == 109.1957 ins. = remaining diameter X3.1416, and again by 
proportionate height of segment. 
Ex. 2. — The height, c 0, of a segment of an oblate spheroid, Fig. 25, is 4 inches, 
the diameters being 14 and 10 inches ; what is the convex surface of it? 
214.0272 ins. 
Y* 





258 MENSURATION OF AREAS, LINES, AND SURFACES. 



the surface. 
Fig. 26. 



Fig. 27. 



To Compute tlie Convex Surface of a Frustrum or 
Zone of a Spheroid— Figs. 25 and. 26. 

Rule. — Proceed as by previous rule for the surface of a segment, 
and obtain the proportionate height of the frustrum ; then multiply 
the product of the diameter parallel to the base of the frustrum and 
3.1416 by the proportionate height of the frustrum, and it will give 
Or, d or d' X 3. 141 6 X h = surface. 

Example. — The middle frus- 
trum, o e, of a prolate spheroid, 
Fig. 25, is 6 inches, the diameters 
of the spheroid being 10 and 14 
j inches; what is its convex sur- 
*5 face? 

/ Mean diameter, as per example, 
page '-57, is 12.1(555. 
Diameter parallel to base of frus- 
trum is 10. 
Then 14 : 12.1655 : : 6 : 5.213S 
= proportionate height of frustrum, and 10x 3.1416x5. 213S = 163.7907 ins, 

CIRCULAR ZONE. 
Definition.— A part of a circle included bstween two parallel chords. 
To Compute tlie Area of a Circular Zone— Fig. 98. 
Rule.— To the area of the trapezoid, a b c d, or of the parallelo- 
gram, ah eg, as the case may be, add the area 
of the segments, a b, c d, or a h, c g, and the sum 
will give the area. 

Or, subtract the areas of the segments, a i c, h k g, from 
the area of the circle. 
Or, a -f a'— s, a representing area of trapezoid, or par- 
a allelogram, and a' area of segments. 
See Table of Areas of Zones, page 207. 





Fig.! 



"J 

lK 



V % 



-fy 



When the Diameter of the Circle is not given, 
Multiply the mean length of the two chords by 
* half their difference ; divide this product by the 

breadth of the zone, and to the quotient add the breadth. 

To the square of this sum add the square of the lesser chord, and 
the square root of their sum will give the diameter of the circle. 

Example.— The greater chord, b d, is S 6 inches ; 
breadth of the zone, a e, is 26 ; what is its area ? 



the lesser, a c, is 60 ; and tho 



06 + 60 

2 
7SX18 

26 



= 7S — mean length of chords ; 



96 — 60 



= 1S= half their difference. 



= 54 = product of chords and their difference -h by the breadth of the zone. 

54 -f- 26 = 80 = sum of above quotient and breadth of zone. 
SO 2 -f- 602 — 10000 = sum of square of above sum and lesser chord. 
Then ^10000 = 100 = diameter, and 73x26 = 202S= area of trapezoid. 
To Compute the Area of the Segments, It is necessary, first, to ascer- 
tain the chord of their arcs; second, the versed sine of their arcs. 

To Ascertain the Chord. — The breadth of the zone is the. perpendic- 
ular, a e, of the triangle, of which either chord, a b, c d, is the hvpoth- 
cnuse. Further, half the difference of the chords a c and b doi the 
zone is the length of the base, b e, of this triangle. 



MENSURATION OF AREAS, LINES, AND SURFACES. 259 



Hence, having the base and the perpendicular, the hvpothenuse or 
chord of the arc of the segment is readily computed. 

Thus, 26 = breadth of the zone or perpendicular of triangle ; 96 — 60 -f- 2 = 18 =t 
length of base of triangles. 

Then 182 4. 202 = 1000, and V1000 == 31.6228 == chord of arc of segments a b, c d. 

To Ascertain the Versed Sine. — From the square of the radius subtract the square 
of half the chord, and subtract the square root of the remainder from the radius. 

Thus, 100 -r- 2 = 50, and 502 — 2500 = square of radius , 31.6228 -r- 2 = 15.8114, 
and 15.81142= 250 = square of half the chord, 2500-250 = 2250, and V2250 = 
47.4342 = square root of the difference of the squares of the radius and half the chord. 
Then 50 — 4T. 4342 = 2.5658 s= versed sine. 

Having obtained the versed sine (2.5658), the diameter of the circle (100), then, 
by Rule, page 253, V 10 °X 2.5658 = 16.0181 = chord of half the arc. 

And by Rule, page 252, to compute the length of an arc, 32.1747= length of the 
arc; 32.1747x50 -j- 2 = 804.3G75 = the product of the length of the arc and half the 
radius of the circle = area cf sector. 

And 804.3675 * — — — = 54.3664 == area of the triangle subtracted from 

the area of the sectors area of each segment, 54.3664x2 = 108. 732S = area of seg- 
ments. Area of trapezoid = 2028 = 2136.7328 ins = area of zone. 

CYLINDER. 

Definition. — A figure formed by the revolution of a right-angled parallelogram 
around one of its sides. 

To Compute tlie Surface of a Cylinder— Fig. 89. 
Rule. — Multiply the length by the circumference, and add the prod- 
uct to the area of the two ends. 



Fig. 29. 




Or, /c-f-2a = 5,a representing area of end. 
Note. — When the internal or convex surface alone is wanted, the 
areas of the ends are omitted. 

Example. — The diameter of a cylinder, b c, is 30 inches, and its 
length, a b s 50 inches ; what is its surface ? 
30 X 3.1416 = 94.248 ins. = circumference ; 94.248 X 50 = 4712.4 = 

area of body. 
And 302 x". 7854 = 706.86= area of one end, 706.86x2 = 1413.72 = 
area of both ends. 
Then 4712.4 -f 1413.72 = 6125 .12 ms. 

PRISMS. 

Definition.— Figures the sides of which are parallelograms, and the ends equal 
and parallel. 

Note. — When the ends are triangles, they are termed triangular prisms; when 
they are square, square or right prisms , and when they are pentagon, pentagonal 
prisms, etc., etc. 

To Com.pu.te tlie Surface of a Riglit Prism—Figs. 
30 and 31. 

Rule: — Ascertain the areas of the ends and 
sides, and add them together. 
Or, 2a-fna' = s, a representing area of the ends, and 
a' the area of the sides. 

Example The side, a b, Fig. 30, of a square prism 

is 12 inches, and the length, b c, 30; what is the surface? 
12 X 12 = 14 i — area of one end , 144X 2 = 288 = area of 
both ends; 12x30 = 360 = area of one side ; 300x4 
= 1440 = area of four sides. 
Then 28S -f 1440 = 1T28 ins. 



Fig. 30. 



Fig. 31. 





260 MENSURATION OF AREAS, LINES, AND SURFACES. 



Definition.- 
e Fig. 32. 




DEFINITION. 

parallel. 

To Com 

Fig. 33. 



WEDGE. 

-A wedge is a prolate triangular prism, and its surface is computed 
by the rule for that of a right prism. 

Example. — The back of a wedge, abed, Fig. 32, is 50 by 
2 inches, and its end, ef, 20 by 2 inches ; what is its surface ? 

202 _j_ 2-M = 401 = sum of the squares of half the base, af 
and the height, ef, of the triangle, ef a. 

-^401 — 20.025 — square root of above sum — length ofea. 
Then 20.025x20X2 = 808 — mta of sides. 

And 20x2 = 40 = area of back; and 20X2 + *x2 = 40 = 
area of ends. Hence 801 -4- 40 -f- 40 = S81 = surface. 

PRISMOIDS. 

-Figures are alike to a prism, but having only one pair of their sides 




pute tlie Surface of a Prism oi&— Fig. 33. 
Rule. — Ascertain the area of the ends and sides as 
by the rules for squares, triangles, etc., and add them 
together. 

tf Example. — The ends of a prismoid, efg h and abed, Fig. 33, 
are 10 and 8 inches square, and its slant height 25; what is its 
surface ? 

10x10 = 100= area of base; 8x8 = 64= area of top. 

10 ^" 8 X25 = 225, and 225x4 = 900 = area of sides 

Then 100 + 64+ 900 =1064= surface. 

UNGULAS. 

Definition. — Cylindrical ungulas are the parts (including all or part of the base) 
loft by a plane cutting a cylinder through any portion and at any angle. 

To Compute tlie Curved. Surface of an TJn gula— Figs. 34, 
35, 36, and. 37. 

Rule. — 1. When the Section is parallel to the Axis of the Cylinder, Fig. 34, Mul- 
tiply the length of the arc of one end by the height. 

Example. — The diameter of a cylinder from which an ungula is cut 
is 10 inches, its length 50, and the versed sine or depth of the ungula 
is 5 inches ; what is the curved surface ? 

10 ■— 2 = 5 = radius of cylinder. 
Hence the radius and versed sine are equal ; the arc, 
therefore, of the ungula is one half the circumference of 
the cylinder, which is 31.416 H- 2 =15.708, and 15. 70SX 
50 = 785.4 ins. 

Rule. — 2. When the Section passes obliquely through 
the opposite Sides of the Cylinder, Fig. 35, Multiply the 
circumference of the base of the cylinder by half the sum 
of the greatest and least heights of the ungula. 

Example. — The diameter of a cylindrical ungula is 10 inches, and 
the greater and less heights, b d and a c, are 25 and 15 inches : what is its curved 
surface ? 

10 diameter = 31 .416 circumference ; 25 + 15 = 40, and 40 -4- 2 = 20. Hence 31.416 
X 20 = 628.32 ins. 

Rule. — 3. When the Section passes through the Base of the Cylinder and one of 
its Sides, and the Versed Sine does not exceed the Sine, Fig. 36, Multiply the sine, 
a d, of half the arc, d g, of the base, d gf, by the diameter, e g, of the cylinder, and 
from thi3 product subtract the product* of the arc and cosine, a o. Multiply the 

* When the cosine is 0, this product is 0. 



Fig. 34. 




Fig. 35. 




MENSURATION OF AREAS, LINES, AND SURFACES. 261 




Fig. 36. difference thus found by the quotient of the height, gc, divided by the 

# % versed sine, a g. 

( \ Example. — The sine, a d, of half the arc of the base of an ungula ia 

5, the diameter of the cylinder is 10, and the height of the ungula 10 
inches ; what is the curved surface ? 

5x10 == 50 = sine of half the arc by the diameter 
Length of arc, the versed sine and radius being equal, under Rule, 
page 252 = 15.708. 
Again, as the versed sine and the radius are equal, the cosine is 0. 
Hence, when the cosine is 0, the product is 0. 50 — = 50 = the dif- 
ference of the product before obtained and the product of the arc 
and the cosine. 
63xi0-f-5 = 50x2 = 100 = the difference multiplied by the height divided by the 
versed sine, which is the surface. 

Rule.— 4. When the Section passes through the Base of the Cylin- 
der, and the Versed Sine, a g, exceeds the Sine. Fig. 31, Multiply the 
sine of half the arc of the base by the diameter of the cylinder, and to 
this product add the product of the arc and the excess of the versed 
sine over the sine of the base. 

Multiply the sum thus found by the quotient of the height divided 
by the versed sine. 

Example. — The sine, a d, of half the arc of an ungula is 12 inches ; 
the versed sine, a g, is 16; the height, c g, 16; and the diameter of 
the cylinder, hg, 25 inches; what is the curved surface? 
12x25 = 800 = sine of half the arc by the diameter of the cylinder. 




Length of arc of base, Rule, p. 252 



Fig. 3S. 



arc of d hf — circumference of base = 46.392. 
Then 46.392x 16 — 12.5 = 162.3T2 == product of arc and the excess of the versed sine 
over the sine; 300 -f 162.372 =: 462. 3T2 = the sum of the above products ; 16 -f- 16 = 
1 = quotient of height divided by the versed sine; 462.372x1 =462.372 
ms. =z the sum of the products and the height divided by the versed 
sine =z the curved surface . 

Note. — When the sine of an arc is 0, the versed sine is equal to the 
diameter. 

Rule. — 5. When the Section passes obliquely through both Ends of 
the Cylinder, Fig 3S, Conceive the section to be continued till it meets 
the side of the cylinder produced ; then, as the difference of the versed 
sines of the arcs of the two ends of the ungula is to the versed sine ; of 
the arc of the less end, so is the height of the cylinder to the part of 
the side produced. 

Ascertain the surface of each of the ungulas thus found by Rules 3 
and 4, and their difference will give the curved surface. 

LUNE. 

Definition. — The space between the intersecting arcs of two eccentric circles. 

To Compute th.e Area of a Lune-Fig. 39. 
Rule. — Ascertain the areas of the two segments from which the 
lune is formed, and their difference will give 
the area. 

Example.— The length of the chord a c is 20, the height 
e d is 3, and e b 2 inches ; what is the area of the lune ? 

By Rule 2, page 253, the diameters of the circles of 
which the lune is formed are thus ascertained: 




Fig. 39. 




For a d c. 



102 + (3 + 2)2 



: 25. For a e c, 



102 + 22 



= 52. 



5 ""7 -"*-""'» 2 

Then, by Rule for Areas of Segments of a Circle, p. 254, the area of a d c is 70.5577 ins 

" aec " 27.1638 " 
their difference 43. 3939 ins 
Note. — If semicircles be described on the three sides of a right-angled triangle a? 
diameters, two lunes will be formed, and their united areas will be equal to that of 
the triangle. 




262 MENSURATION OF AREAS, LINES, AND SURFACES. 

CYCLOID. 

Definition.— A curve generated by the revolution of a circle on a plane. 

To Compute tlie Area of a Cycloid— Figc 40. 

Fi S- 40 « Rule. — Multiply the area of the gener- 

.„ . N . ating circle, a b c, by 3. 

*\ Example. — The generating circle of a cycloid ha3 
; an area of 115.45 inches; what is the area of the 
/ cycloid ? 

c 115.45X3 = 346.35 ins. 

To Compute tlie Length, of a Cycloidal Curve- 
Fig, 40. 

Rule, — Multiply the diameter of the generating circle by 4. 

Example. — The diameter of the generating circle of a cycloid, Fig. 40, is S inches ; 
what is the length of the curve d s c 

8x4=32 = product of diameter and 4= ins 

Note. — The curve of a cycloid is the line of swiftest descent; that is, a body will 
fall through the arc of this curve, from one point to another, in less time than 
through any other path, 

RINGS. ' 

CIRCULAR RINGS. 
Definition.— The space between two concentric circles. 

To Compute the Sectional Area of a Circular Ring. 

Rule. — From the area of the greater circle subtract that of the less. 
Or, a — a'= area. 

CYLINDRICAL RINGS. 
Definition. — A ring formed by the curvature of a cylinder. 

To Compute the Surface of a Cylindrical Ring- 
Fig. 41. 

Rule. — To the thickness of the ring add the inner diameter; mul- 
Fte 41 tipty tn * s sum ky the thickness, and the product by 

9.8696. 

Or, d-\-d' d 9.8696 = surface. 
Example, — The thickness of a cylindrical ring, a &, is 2 inch- 
es, and the inner diameter, b c, is 18 ; what is the surface of it ? 
2 -f- 18 = 20 = thickness of ring added to the inner diameter. 
20X2X9.8696 = 394.784 ins.— the sum above obtainedXthe 
thickness of the ring, and that product by 9.S696. 

LINK. 

Definition. — An elongated ring. 

To Compute the Surface of a Link-Figs. 42 and 43. 

Rcle. — Multiply the circumference of a section of the body of the 
link by the length of its axis. 

Or, c X I = surface. 
Note. —To Compute the Circumference or Length of the Axes. 







MENSURATION OF AREAS, LINES, AND SURFACES. 263 



Fig. 42. 



Fig. 43. 



When the Ring is Elongated.— To the less diameter add its thickness, and multi- 
ply the sum by 3.1416; multiply the difference of the di- 
ameters by 2, and take the sum of these products. 

When the Ring is Elliptical. — Square the diameters of 
the axes of the ring, and multiply the square root of half 
their sum by 3.1416. 

Example. — The link of a chain, Fig. 42, is 1 inch in di- 
ameter of body, a b, and its inner diameters, b c and ef, 
are 12.5 and 2.5 inches ; what is its circumference? 
2.5-|-lx3.1416 = 10.9956= length of axis of ends. 
12.5 — 2.5 X 2 = 20 = length of sides of body. 
Then 10.9956 + 20 = 30.9956 = length of axis of link, which 
X 3. 1416 (cir. of 1 in.) =91.3758 ins. 

CONES. 

Definition. — A figure described by the revolution of a right-angled triangle 
about one of its legs. 
For Sections of a Cone, see Conic Sections, page 239. 

To Compute tlie Surface of a Cone— Fig. 44. 

Fig. 44. 






Rule. — Multiply the perimeter or circumference 
of the base by the slant height, or side of the cone ; 
halve the product, and add it to the area of the base. 
Or, cXA-^-2-f-a'— surface, c representing perimeter. 

Example. — The diameter, a b, of the base of a cone is 3 feet, 
and the slant height, a c, 15 ; what is the surface of the cone? 

9 4248x15 

Circum. of 3 feet = 9.4248, and -' — = 70.6S6 = sur/ace 

b 4 

of side, and area 3 = 7. 068 = 70. 686 + 7. 068 = 77. 754 sq.feet. 

To Compute tlie Surface of tlie ZFrnstriixii of a 
Cone-Fig. 45. 

Rule. — Multiply the sum of the perimeters of the two ends by the 

slant height of the frustrum ; halve the product, and add it to the 

areas of the two ends. 

-p- a* n c + c'xh ... r 

Fig. 45. Or, - J- a -j-a' = surface. 

a 

Example. — The frustrum, abed, Fig. 45, has a slant height 
of 26 inches, and the circumferences of its ends are 15.71 and 
22 inches respectively ; what is its surface ? 

11 ^^=m.23 = surf>ce of sides; (^) 2 X .7S54 + 




58.119: 



X 7854 = 58.119 : 
r 548. 349 inches. 

PYRAMIDS. 



areas of ends. . Then 490.23 -f- 



Definitiok. — A figure the base of which has three or more sides, and the sides of 
which are plain triangles. 

To Compute th.e Surface of a Pyramid-Figs. 46 
and. 4*7. 

Rule. — Multiply the perimeter of the base by the slant height ; 
halve the product, and add it to the area of the base. 



Or, -y -|-a: 



: surface. 



2G4 MENSURATION OF AREAS, LINES, AND SURFACES. 

Fig. 46. Example. — The side of a quadrangular pyramid, Fig. 4 7. 

c a b, Fig. 4Q, is 12 inches, and its slant height, a c, 40 ; 

what is its surface ? 

12 X 4 = 48 = perimeU r of base. 

4SX40 




2 



- = 960 = area of sides. 




Then 12x 12 + 9C0 = 1104. ins. 

To Compete tlie Surface of tlie Fr^istrniXL of a IPyr- 
amid— Fig. 48. 

Rule. — Multiply the sum of the perimeters of the two ends by the 

slant height or side; halve the product, and add it to the areas of the 

ends. c + c'xh , 

Or, j- a -J- a'= surface. 

Example. — The sides, a b, c d, Fig. 48, of the frustrum of a 
quadrangular pyramid are 10 and 9 inches, and its slant height, 
a c, 20 ; what is its surface ? 

10x4 = 40, and 9x4 = 36 = T6 = sw7n of perimeters. 
15 9 
70x20 = 1520, and -^- = 760 = area of sides; 10x10 = 100, 

and 9x9 = 81. 
Then 100 + 81 -4- 760 = 941 = mm. 

HELIX (SCREW). 

Definition. — A line generated by the progressive rotation of a point around an 
axis and equidistant from its centre. 

To Compute tlie Length, of a Helix— Trig. 49. 

Rule. — To the square of the circumference described by the gen- 
erating point add the square of the distance advanced 
in one revolution, and take the square root of their 
sum multiplied by the number of revolutions of the 
generating point. 
Or, -J(c 2 -\-h 2 )n = length, n representing number of revolutions. 





Fig. 
d 


43. 

< 




















I 




1/ 






X 

-M 


i 









Example. — What is the length of a helical line running 3.5 
times around a cylinder of 22 inches in circumference and ad- 
vancing 16 inches in each revolution ? 

222 _j_ 162 — 740 — sum of squares of circumference and of the 
distance advanced* ThenV?40x 3.5 = 95.21 inches. 

SPIRALS. 

Definition. — Lines generated by the progressive rotation of a point around a 
fixed axis. 

A Plane Spiral is when the point rotates around a central point. 

A Conical Spiral is when the point rotates around an axis at a progressing dis- 
tance from its centre, or around a cone. 

Xo Compute tlie Length, of a 3?lane Spiral Line- 
Fig. 50. 

Rule. — Add together the greater and less diameters ; divide their 
sum by two ; multiply the quotient by 3.1416, and again by the num- 

* When the spiral is other than a line, measure the diameters of it from the middle of the body 
composing it. 






MENSUKATION OF AREAS, LINES, AND SURFACES. 265 



ber of revolutions. Or, when the circumferences are given, take their 
mean length, and multiply it by the number of revolutions. 

Or, d -f- d'-r-2 X 3. 141G n = length of line ; pXn — radius, and 
pr 2 -r- 1 == pitch. 
Example. — The less and greater diameters of a plane spiral 
spring, as a b, c d, Fig. 50, are 2 and 20 inches, and the number 
of revolutions 10 ; what is the length of it? 

-2: 



Fig. 50. 




2 -j- 20 -f- 2 = 11 = sum of diameters - 
= above quotient X 3.1416. 
Then 34.5576x10 == 345.576 inches. 



11X3.1416=34.5576 



Note.— The abova rule is applicable to winding engines where it is required to 
ascertain the length of a rope, its thickness, the number of revolutions, diameter of 
drum, etc. , etc. 

To Compute the Length, of" a Conical Spiral Hiine— 
Fig. 51. 

Rule. — Add together the greater and less diameters ; divide their 
sum by two, and multiply the quotient by 3.1416. 

To the square of the product of this circumference and the number 
of revolutions of the spiral, add the square of the height of its axis, 
and take the square root of the sum. 

Fig. 51. Or, *f{d + ^-^-2x3.1416 n+ I 2) = length of line. 

* Example. — The greater and less diameters of a conical spiral, Fig. 

51, are 20 and 2 inches ; its height, c d, 10 ; and the number of revo- 
lutions 10 ; what is the length of it ? 

20 -f 2 -4- 2 = 11x3.1416 — 34.5576 = sum of diameters -r- 2, and 
X3.1416; 34.5576X10 = 345.570, and 345.5762 = 119422.77 = 
square of the product of the circumference and number of revo- 
lutions. 

Then ^11^422.7? + 102 = 345.72 inches. 

SPINDLES. 

Definition — Figures generated by the revolution of a plane area, when the curve 
is revolved about a chord perpendicular to its axis, or about its double ordinate, and 
they are designated by the name of the arc or curve from which they are generated, 
as Circular, Elliptic, Parabolic, etc., etc. 

CIRCULAR SPINDLE. 
To Compute the Convex Surface of a Circular Spin- 
dle, Zone, or Segment of it— ITig. 52, 53, and. 54. 

Rule. — Multiply the length by the radius of the revolving arc ; 
multiply this arc by the central distance, or distance between the cen- 




Fig. 52. 




tre of the spindle and centre of the revolving arc; 
subtract this product from the former, double the re- 
mainder, and multiply it by 3.1416. 

Or, l r — (\/r 2 — ( r ) ) 2 p = surface, a representing length 
| '•*:' | of arc, and c the chord. 

\ j Example. — What is the surface of a circular spindle, Fig. 

/ 52, the length of it,/c, being 14.142 inches, the radius of its 
y arc, o c, 10, and the central distance, o e, 7.071 ? 
" "* 14.142x10 = 141.42 = length X radius. Length of arc,f a c, 

by Rules, pape 252 == 15.708. 
15.708X7.071=3 111.0713= length of arc X central distance; 141.42 — 111.0713 = 
30.3487 = difference of products. Then 30.3487 X2X3. 1410 = 190.0S7 inches. 

z 



2G6 MENSURATION OF AREAS, LINES, AND SURFACES. 




Flo- 53. Example. — What is the convex surface of the rone of a 

circular spindle, Fig. 53, the length of it, i c, being 7.653 inch- 
es, the radius of its arc 10, the central distance 7.071, and 
x the length of its side or arc, d b, 7.S51 inches ? 
' 7.653xl0r=76.53 = Z<:n^X radius; 7.854x7.071 =55.5350 
= length of arc X antral distance ; 76.53 — 55.5356 = 
20.9944 = difference of products. 
\ i / Then 20.9944x2x3.1416 = 131.912 inehes. 

\'/ Example. — What is the convex j..^ ~, 

surface of a segment of a circular r 

spindle, Fig. 54, the length of it being 3.2495 inches, the 
radius of its arc 10, the central distance 7.071, and the length 
of its side, i d, 3.927 inches ? 
3.2495x10 = 32.495 = length X radius ; S.927 X 7.071 = 

27.767S = length of arc X central distance ; 32.495 — \ "" "j — r T v / 

27.7678 = 4.7272 = difference of products. \ /' 

Then 4.7272x2x3.1416 = 29.702 inches. \ : / 

Genesal Formula. — S = 2(Zr — ac)p = surface, I rep- 
resenting length of spindle, segment, or zone, a the length of its revolving arc, r the 
radius of the generating circle, and c the central distance. 

Illustration The length of a circular spindle is 14.142 inches, the length of its 

revolving arc is 15. 70S, the radius of its generating circle is 10, and the distance of 
its centre from the centre of the circle from which it is generated is 7.071; what is 
its surface ? 



2X(14.142X10— 15.70Sx7.071)X3.1416 = 190.6S7 inches. 
Note. — The surface of a frustrum of a spindle may be obtained by the division of 
the surface of a zone. 

CYCLOIDAL SPINDLE. 

To Compute tlie Convex Surface of a CycloicLal 
Spindle— 3Tig. 55. 

Rule. — Multiply the area of the generating circle by 64, and divide 
it by 3. 

Fig. 55. 



n «X64 

Or, — ; — = surface. 

' 3 J 




Example. — The area of the generating circle, ab c, of 
\ b a cycloidal spindle, d e, is 32 inches ; what is the surface 
; c of the spindle ? 

''' 32 X 64 = 2048 = area of circle X04 ; and 2048 -^ 3 = 
6S2. 667 inches. 
Note. — The area of the greatest section of a cycloidal 
spindle is twice the area of the cycloid. 

ELLIPSOID, PARABOLOID, OR IIYPEPJBOLOID OF REVOLUTION. 

Definition, — Figures alike to a cone generated by the revolution of a conic sec- 
tion around its axis. 

Note. — These figures are usually known as Conoids. 

When they are generated by the revolution of an ellipse, they are called Ellipsoids, 
and when by a parabola, Paraboloids, etc., etc. 

The revolution of an arc of a conic section around the axis of the curve will give a 
segment of a conoid. 

ELLIPSOID. 

To Compute tlie Convex Surface of* an Ellipsoid— 
IPig. 5G. 

Rule. — Add together the square of the base and four times the 
square of the height ; multiply the square root of half their sum by 
3.1416, and this product by the radius of the base. 



Or, 



?: 



b* + 4/i2 



3.1416 r = surface, h representing height of the ellipsoid. 



MENSURATION OF AREAS, LINES, AND SURFACES. 267 



To Compnte tlie Convex Surface of a Segment, 
Frustrum, or Zone of an Ellipsoid. 

Fig. 56. See Kules for the Convex Surface of a Segment, 

Frustrum, or Zone of a Spheroid or Ellipsoid, pages 
257, 258. 

Example.— The base, a b, of an ellipsoid, Fig. 56, is 10 inches, 
and the vertical height, c d, 7 ; what is its surface ? 

10 2 -f- 7 2 X4 = 296 = sum of the square of the base and 4 times 
the square of the height ; 296 -5- 2 = 148, and V14S = 12.1655 
= square root of half the above sum. 

Then 12.1655x3.1416x^ = 191.0957 inches. 

PARABOLOID. 

To Compute the Convex Surface of a Paraboloid.— 

iF'ig. 57. 

Eule. — From the cube of the square root of the sum of four times 
the square of the height, and the square of the radius of the base, sub- 
tract the cube of the radius of the base; multiply the remainder by 
the quotient of 3.1416 times the radius of the base divided by six 
times the square of the height. 





Or, (V4/ i 2 +r 2 ) 3_r3xgf 2 : 



: surface. 



Example.— The axis, b d, of a paraboloid, Fig. 57, is 40 inch- 
es : the radius, a d, of its base is 18 inches ; what is its convex 
surface ? 

402x4 = 6400 = 4 times the square of the height; 6400 -f 182 
= 6724= sum of the above product and the square of the radius 
of the base; (^6724)3 — IS 3 =545536 = the remainder, of the cube 
of the radius of the base subtracted from the cube of the square 
root of the preceding sum; 3.1416x18 -4- (6x402) =.0058905 = 
the quotient of 3.1416 times the radius of the base ■+■ 6 times the 
square of the height. 
Then 5 15536 X- 0058905 = 3213.48 inches. 

ANY FIGURE OF REVOLUTION. 

To Ascertain the Convex Surface of any THignre of 

Revolution— Figs. 58, 59, and 60. 

Rule.— Multiply the length of the generating line bv the circum- 
ference described by its centre of gravity. 

Or, I2rp— surface^ r representing radius of centre of gravity. 



Fig. 5S. 



Illustration. — If the generating line, a c, of the cylinder, 
a c df 10 inches in diameter, Fig. 5S, is 10, then the centre of 
gravity of it will b e in Z>, the radius of which is b r = 5. 

Hence 10x5x2x3.1416 = 314.16 inches = the convex surface 
of the cylinder. 

Again. If the generating line is e a c g. and it is (e a = 5, a c 
— 10, and c g = 5) = 20, then the centre of gravity, o, will be in 
the middle of the line joining the centres of gravity of the trian- 
f gles e a c and a c g = 3.75 from r. 

Hence 20 X 3.75 X 2 X 3.1416 = 471.24 inches = the entire sur- 
face of the cylinder. 

p nnn j Convex surface as above 314.16 

Jrwuu *' \ Area of each end, lO 2 x. 7854 = 78.54, and 78.54x2 = 157.03 

471.24 





\ / 






)\ ' 




b 

c 


/ \ 


r 



268 MENSURATION OF AREAS, LINES, AND SURFACES. 



p. £3 Illustration 2. — If the generating elements of a cone, 

s ' ' Fig. 59, are a d =10, dc = 10, and a c the generating line 

a = 14.142, the centre of gravity of which is in o, and or = 5, 



Fig. 60. 



b ox- 



J, 



/ j \ Then 14. 142 x 5x2x3. 1416 = 444 2S5 = 

O/L I t» \ 2 

/ i ' \ the convex surface of the cone, and 10x2 

/: \ X -TS54 == 314. 16 = area of base. 

C " d '~ "' v Hence 444.285 + 314.16 = T5S.445 = the 

entire surface of the cone. 
Illustration 3. — If the generating elements of a sphere, Fig. 
60, are ac = 10, a b c will be 15. 70S, the centre of gravity of which 
is in o, and by Rule, page 339, or~ 3.183. 

Hence 15.708 X 3. 1S3X 2x3.1416 = 314.16 inches— the surface of 
the sphere. 

To Ascertain tlie Area of an Irregular Figure. 

Rule. — Take a uniform piece of board or pasteboard, weigh it, cut 
out the figure of which the area is required, and weigh it ; then, as 
the weight of the board or pasteboard is to the entire surface, so is the 
weight of the figure to its surface. 

CAPILLARY TUBE. 

To Compute tlie Diameter of a Capillary- Tube, 

Rule. — Weigh the tube when empty, and again when filled with 
mercury ; subtract the one weight from the other ; reduce the differ- 
ence to Troy grains, and divide it by the length of the tube in inches, 
Extract the square root of this quotient, multiply it by .0192245, and 
the product will give the diameter of the tube in inches. 

Or, /— X .0192245 = diameter, w representing difference in weights inTroy grah 
and I the length of the tube. 

Example. — The difference in the weights of a capillary tube when empty and 
when filled Avith mercury is 90 grains, and the length of the tube is 10 inches ; what 
is the diameter of it ? 

90 -r- 10 = 9 = weight of mercury -r- length of tube; V 9 = 3, and 3 X. 0192245 = 
.0576735 = the square root of the above quotient X. 0192245 inches = diameter of 
tube. 

Proof. — The weight of a cubic inch of mercury is 3442.75 Troy grains, and the 
diameter of a circular inch of equal area to a square inch is 1.12S (page 254). 

If, then, 3442.75 grains occupy 1 cubic inch, 90 grains will require .0201419 cubic 
inch, which, -4- 10 for the height of the tube == .00261419 inch for the area of the sec- 
tion of the tube. 

Then y/ . 0026141 9 =.051129= side of the square of a column of mercury of this 
area. 

Hence .051129x1.128, which is the ratio between the side of a square and the di- 
ameter of a circle of equal area = .0576735. 

ADDITIONAL RULES FOR ELEMENTS OF REGULAR POLYGONS. 

To Compute tlie Radius of tlie Inscribed, or Cir- 

cum scribed. Circles. 

Rule. — When the Radius of the Circumscribing Circle is given, Mul- 
tiply the radius given hy the unit in column E, in the following Table, 
opposite to the figure for which the radius is required. 

Rule. — Wlien the Radius of the Inscribed Circle is given, Multiply 
the radius given by the unit in column F, in the following Table, op- 
posite to the figure for which the radius is required, 



: 



ad 

I 






MENSURATION OF AREAS, LINES, AND SURFACES. 269 

To Compete the Area, "When tlie Radii of tlie In- 
scribed or Circnmscri"bing Circles are given. 

Rule. — Square the radius given, and multiply it by the unit in col- 
umns G or H, in the following Table, and opposite to the figure for 
which the area is required. 

To Compute tlie Length of* a Side, When the Ra- 
dins of the Inscribed or Circumscribing Circle 
is given. 

Rule.— Multiply the radius given by the unit in column K, in the 
following Table, and opposite to the figure for which the length is re- 
quired. 







E. 


F. 


G. 


H. 


I. 


K. 






Radius of 


Radius of 


Area. 


Area. 


Area. 


Length of 
Side. 


No. of 




Inscribed 


Circumscr. 


By Radius 


By Radius 


By Length 


Sides. 


Name of Polygon. 


Circle. 


Circle. 


of Inscrib'd 


of Circum- 


of Side. 


By Radius 






By Cir- 


By Inscrib- 


Circle. 


scribing 




of Tnscrib'd 






cumscrib. 
Circle. 


ing Circle. 




Circle. 




Circle. 


3 


Trigon 


.5 


2. 


5.19615 


1.29904 


.43301 


3.4G41 


4 


Tetragon 


.70711 


1.41421 


4. 


2. 


1. 


2. 


5 


Pentagon 


.80902 


1.23607 


3.63272 


2.S7T04 


1.72048 


1.45308 


6 


Hexagon .... 


.S6602 


1.1547 


3.4641 


2.59808 


2.59808 


1.1547 


7 


Heptagon . . . 


.90097 


1.10992 


3.37102 


2.73641 


3.63391 


.96315 


8 


Octagon 


.923S8 


1.0S239 


3.31371 


2.82842 


4.81843 


.82S43 


9 


Nonagon 


.93969 


1.06418 


3.27573 


2.89254 


6.18282 


.72794 


10 


Decagon .... 


.95106 


1.05146 


3.2492 


2.93893 


7.69421 


.64984 


11 


Undecagon . . 


.95949 


1.04222 


8.22983 


2.97353 


9.36564 


.58725 


12 


Dodecagon .. 


.96593 


1.03528 


3.21539 


3. 


11.19615 


.53K) 



z* 



270 



MENSURATION OF SOLIDS. 




MENSURATION OF SOLIDS. 

CUBES AND PARALLELOPIPEDONS. 

CUBE. 
Definition. — A solid contained by six equal square sides. 

To Compute the Volume of* a Cube- Fig. 61. 

1 Rule. — Multiply a side of the cube by itself, and that 

product again by a side. 

Or, s 3 = V, s representing the length of a side, and V the volume. 

Example. — The side a b, Fig. 61, is 12 inches ; what is the vol- 
ume of it ? 

12x12x12 = 1728 cubic inches. 

PARALLELOPIPEDON. 

Definition. — A solid contained by six quadrilateral sides, every opposite two of 
which are equal and parallel. 

Pi- 02. 
To Compute tlie Volume of a Paral- 
lelopipedon— Fig. 62. 

Rule. — Multiply the length by the breadth, and 
that product again by the depth. 
Or, lbd=zV. 

PRISMS, PEISMOIDS, AND WEDGES. 

PRISMS. 

Definition Solids, the ends of which are equal, similar, and parallel planes, and 

the sides of which are parallelograms. 

Note. — When the ends of a prism or prismoid are triangles, it is called a triangu- 
lar prism or prismoid ; when rhomboids, a rhomboidal prism, etc. ; when squares, 
a square prism, etc. ; when rectangles, a rectangular prism, etc. 

Fig. 63. To Compute tlie Volume of a Prism- 





Fig. 64. 




Figs. 63 and. 64. 

Rule. — Multiply the area of the base by 
the height. 

Or, aXh = Y. c 

Example. — A triangular prism, abc de, Fig. 64. has 
sides of 2.5 feet, and a length, c d, of 10 feet ; what is 
its volume ? 

By Rule, page 245, 2.52x.433= 2.T0G25 = area, of 
end abc, and 2.706-5x10 = 27.0625 cubic feet d 

PRISMOIDS.* 

To Compute tlie Volume of a Prismoid.— Fig. 65. 

Rule. — To the sum of the areas of the two ends add four times the 
area of the middle section, parallel to them, and multiply this sum by 
% of the perpendicular height. f 

* An excavation or embankment of a road, when terminated by parallel cress see'tions, is a rectan- 
gular prismoid. 

f Thia is the general rule, and applies equally to figures of proportionate or dissimilar ends. 






MENSURATION OF SOLID3. 



271 




Or, a-\- a'-\- 4m — h^-6z= V, a and a' representing areas of 
ends, and m area of middle section. 

Example. — What is the volume of a rectangular prismoid, 
Fig. 65, the lengths and breadths, e g and g h, and a b and b d, 
of the two ends being 7x6 and 3X2 inches, and the height 15 
feet ? 

1X6 + 3X2 = 42 + 6 = 48 = sum of the areas of the two ends > f 
T rK®.-^" 2 == E> = length of the middle section ; 6-)-2-^-2=rr4=: 
breadth of the middle section; 5x4x4 = 80 =zfour times the 
area of the middle section. 



Then 4S + S0x^i?:= 128X30; 



: 3840 cubic inches. 



Fig. 66. 



Note. — The length and breadth of the middle section are respectively equal to 
half the sum of the lengths and breadths of the two ends. 

2. — Prismoids, alike to prisms, derive their designation from the figure of their 
ends, as triangular, square, rectangular, pentagonal, etc. 

WEDGE. 
To Coinpnte tlie Volume of a WecLge— Fig. 66. 

Rule. — To the length of the edge add twice the length of the back ; 
multiply this sum by the perpendicular height, and then by the breadth 
of the back, and take % of the product. 

Or, (l+Fx2Xhb) + 6 = V. 
Example. — The back of a wedge, a b c d, is 20 by 2 inches, 
and its height, ef 20 inches; what is its volume? 

20-}- 20x2 = 60 = length of the edge added to twice the 
length of the back ; 60 X 20 X 2 = 2400 = above sum multiplied 
by the height, and that product by the breadth of the back. 
Then 2400 -4- 6 = 400 cubic inches. 

Note. — When a wedge is a true prism, as represented by 
Fig. 66, the volume of it is equal to the area of an end mul- 
tiplied by its length. 

REGULAR BODIES (POLYHEDRONS). 

Definition. — A regular body is a solid contained under a certain number of simi- 
lar and equal plane faces,* all of which are equal regular polygons. 

Note. — The whole number of regular bodies which can possibly be formed is five. 

2. — A sphere may always be inscribed within, and may always be circumscribed 
about a regular body or polyhedron, which will have a common centre. 




Fig. 67. 



Fig. 68. 



Fig. 69. 



Fig. TO. 




1. The Tetrahedron, or Pyramid, Fig. 6T, which has four triangular faces. 

2. The Hexahedron, or Cube, Fig. 61, which has six square faces. 

3. The Octahedron, Fig. 68, which has eight triangular faces, 

4. The Dodecahedron, Fig. 69, which has twelve pentagonal faces* 

5. The Icosahedron, Fig. 70, which has twenty triangular faces. 



* The angle of the adjacent faces of a polygon is called the diodral angle. 



272 



MENSUKATIOX OF SOLIDS. 




ffl 

c3 

r-t 
Pj 

& 

PS 


B 

S 
a; 






r-> 








•siaqdg 
i> paquosui jo enipwjj Ag 
" • 9.3 pa JBauiq 


4.89898 
2. 

2.44949 

.89806 

1.32317 


>_• -eoEjing .\g 


i^. O rH CO CO 

HOCCOhiO 

io co t^ oc co 

o o o CO © 


•9i9qdg Sutquos 
^ -ranahojo smpt?a Ag 

*9^P3 JB8UJ1 


1.63299 
1.1547 
1.41421 
.71364 
1.05146 


•ajaqds 
c_J paquosui jo siupug A"g 
•9iun[o^ 


13.85641 

8. 

6.9282 
5.55029 
5.05406 


# '9.i9qdg paquosnj A*g 
" -quosuiruuiQ jo snipug 


3. 

1.73205 
1.73205 

1.25841 
1.25841 


•9J9qdg Sniquoa 
02 -umaiio jo snip«a Ag 
•©(.unio^ 


CO t^iO 

CO CO CO rH rH 

co cs co io co 

ri CO cr. GO CO 
iO "O CO L- O 

" rH rH CO CO 


•aiaqd^ 
• Sniquosinnoiio .£g 
w ' -aioqdg 

paquosaj jo snipcjj 


W iO iO >o iO 
CO CO CO CO CO 

CO L^ t- -Hi -Hi 

co t^ t^ cs cs 

CO O iO N t» 


•eSp3 njgmi Ag 
Ph '9iun[o^ 


.11785 
1. 

.4714 
7.66312 
2.1817 


•aranf© a. ^9 
p^ -aigqdg 

paquDsuj jo srupug 


TH CO rH 
CO l^XN 

co ^ t* co 

rH CO CO 00 

H iO iO iO <o 


. -9iunTo A Xg 

Zjr •eoujing 


7.20502 

6. 

5.7191 

5.31161 

5.14835 


•euinjOA A"g 

ryi -aiaqdg Sm 

-quosmnojfQ jo smptjjj 


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MENSURATION OP SOLIDS. 273 

To Compute the Elements of any Regular Body- 
Figs. 67, 6S, 61, 69, and 70. 

To Compute the Radius of a Sphere that will Circumscribe a given Regu- 
lar Body, and the Radius of one also that may be Inscribed within it. 
Rule. — When the Linear Edge is given, Multiply it by the multiplier 
opposite to the body in the columns A and B in the preceding Table, 
under the head of the element required. 

Example.— The linear edge of a hexahedron or cube, Fig. 61, is 2 inches ; what 
are the radii of the circumscribing and inscribed spheres ? 

2 X-S6602 = 1.13204 inches = radius of circumscribing sphere; 2X-5 == 1 inch = 
radius of inscribed sphere. 
Rule.— -When the Surface is given, Multiply the square root of it by 
the multiplier opposite to the body in columns C and D in the preced- 
ing Table, under the head of the element required. 

Rule. — When the Volume is given, Multiply the cube root of it by 
the multiplier opposite* to the body in columns E and F in the preced- 
ing Table, under the head of the element required. 

Rule. — When one of the Radii of the Circumscribing or Inscribed 
Sphere alone is required, the other being given, Multiply the given radius 
by the multiplier opposite to the body in columns G and H in the pre- 
ceding Table, under the head of the other radius. 

To Compute the Linear Edge of a Polyhedron. 

Rule. — When the Radius of the Circumscribing or Inscribed Sphere 
is given, Multiply the radius given by the multiplier opposite to the 
body in columns I and K in the preceding Table. 

Rule. — When the Surface is given, Multiply the square root of it by 
the multiplier opposite to the body in column L in the preceding Table. 

Rule. — When the Volume is given, Multiply the cube root of it by 
the multiplier opposite to the body in column M in the preceding Table. 

To Compute the Surface of a Polyhedron. 

Rule. — When the Radius of the Circumscribing Sphere is given, Mul- 
tiply the square of the radius by the multiplier opposite to the body in 
column N in the preceding Table. 

Rule. — When the Radius of the Inscribed Sphere is given, Multiply 
the square of the radius by the multiplier opposite to the body in col- 
umn O in the preceding Table. 

Rule. — When the Linear Edge is given, Multiply the square of the 
edge by the multiplier opposite to the body in column P in the pre- 
ceding Table. 

Rule. — When the Volume is given, Extract the cube root of the vol- 
ume, and multiply the square of the root by the multiplier opposite to 
the body in column Q in the preceding Table. 

To Compute the "Volume of a Polyhedron. 

Rule.— When the Linear Edge is given, Cube the linear edge, and 
multiply it by the multiplier opposite to the body in column R in the 
preceding Table. 



274 



MENSURATION OF SOLIDS. 



Eule. — When the Radius of the Circumscribing Sphere is given, Mul- 
tiply the cube of the radius given by the multiplier opposite to the 
body in column S in the preceding Table. 

Rule. — When the Radius of the Inscribed Sphere is given, Multiply 
the cube of the radius given by the multiplier opposite to the body in 
column T in the preceding Table. 

Rule. — When the Surface is given, Cube the surface given, extract 
the square root, and multiply the root by the multiplier opposite to the 
body in column U in the preceding Table. 

Fig. 71. CYLINDER. 

To Compute tlie Volume of a Cylinder— 
Fig. 7T_. 

Rule. — Multiply the area of the base by the height. 
Or, aXh = Y. 

Example. — The diameter of a cylinder,* c,is 3 feet, and its length, 
a b, 7 feet ; what is its volume ? 

Area of 3 feet = 7.06S. Then 7.06SxT = 49.176 cubic feet. 

CONE. 

Fig. 72. To Compute trie Volume of a Cone — 

c Fig. 73. 

Rule. — Multiply the area of the base by the per- 
pendicular height, and take one third of the product. 
Or, a h -h 3 — V. 

Example. — The diameter, a b, of the "base of a cone is 15 inch- 
es, and the height, c e, 32.5 inches ; what is its volume? 





Area of 15 inches = 17G.714G. 
cubic inches. 



Then 



17C.T15 x 32. 



-1014.4115 



To Compixte trie Yolume of trie Frustrum or a 
Cone— Fig. 73. 

Rule. — Add together the squares of the diameters of the greater 
and lesser ends and the product of the two diameters; multiply their 
sum by .7854, and this product by the height; then divide this last 
product by three. Or, add together the squares of the circumferences 
of the greater and lesser ends and the product of the two circumfer- 
ences; multiply their sum by .07958, and this product by the height; 
then divide this last product by three. 

Or, rf2_|_rf'2_|_rfxrf/ x .7S54/i-^-3 = V. 
Or, c2+c'2-LcXc / X.07P5SA-H3 = V. 

Example. — "What is the volume of the frustrum of a cone, 
the diameters of the greater and lesser ends, b d,a c. Fig. 73, be- 
ing respectively 5 and 3 feet, and the height, c o, fcet ? 

52_|_32_|_5x3 = 49 =the sum of the squares and the product of 
the diameters; 4Hx.7S54 = 33.4540 = Me above sum %.735'i. 

35 4346x9 ,.^ . Mn 

- =zl.5.45oS cubic feet. 

PYEAMID. 

Note.— The volume of a pyramid is equal to one third of that of a prism having 
cqu:\l bases and altitude. 



Fig. 73. 




MENSURATION OF SOLIDS. 275 

To Compute th.e "Voluxxie of a IPyramid— Fig. 74. 

Fig. 74. Rule. — Multiply the area of the base by the perpendic- 

Ca ular height, and take one third of the product. 

Jk Or, ah-+-3=zV. 

/' Wk Example — What are the contents of a hexagonal pyramid, Fig. 74, 

JIJ mk a side, a b, being 40 feet, and its height, 60 feet ? 

a/jl Wk 402x 2.59S1 (tabular multiplier, page 248) r= 4156.96 = area of base. 

/i- Wk 4156.96X60 ^ nn . 

\T~Wr o — 83139 - 2 cubic feet 

b rf 

To Compute tlie Yolume of tlie Frustrum of a 
IPyraiXLicl— Fig. 75. 

Rule. — Add together the squares of the sides of the greater and 
lesser ends and the product of these two sides ; multiply the sum by 
the tabular multiplier for areas in Table, page 248, and this product 
by the height ; then divide the last product by three. 

Or, sS-j-^-f-sxs'X tab. mult. X/i-^3 = V, s and s' representing the lengths of 
the sides. 
When the areas of the ends are known, or can be obtained without reference to a 
.p. 7K tabular multiplier, use the following. 

c d Oi\ a-\-a' -\--faXa'Xh-±-3 = V, a and a' representing areas of 

the ends. 
Example. — What are the contents of the frustrum of a hexago- 
nal pyramid, Fig. 75, the lengths of the sides of the greater and 
lesser ends, cd,ab, being respectively 3.75 and 2.5 feet, and its per- 
pendicular height, e o, 7.5 feet ? 

3.752 -f- 2.52 — 20.3125 — s um of th e squares of sides of greater 

and lesser ends; 20. 3125 -f- 3.75x2.5 = 29.6875 = above sum added 

to the product of the two sides; 29. 6875x2.5981x7.5 = 57 8. 4S = 

the last sum X tab. mult., and again by the height, which, -^-3 = 

" w 192.83 cubic feet. 

When the Ends of a Pyramid are not those of a Regular Polygon, or 
ivhen the Areas of the Ends are given. 

Rule. — Add together the areas of the two ends and the square root 
of their product ; multiply the sum by the height, and take one third 

of the product. 

Or, a-j-a'4- A /«X« / X/iH-3 = V. 
Example. — What are the contents of an irregular-sided frustrum of a pyramid, 
the areas of the two ends being 22 and 88 inches, and the length 20 inches. 
22 -4- S3 = 110 = sum of areas of ends; 22x88 = 1936, and ^193Q= 44 = square 

110-4-44x20 
root of product of areas. Then = 1026.66 cubic inches. 

SPHERICAL PYRAMID. 

A Spherical Pyramid is that part of a sphere included within three or more ad- 
joining plane surfaces meeting at the centre of the sphere. The spherical polygon 
defined by these plane surfaces of the pyramid is called the base, and the lateral 
faces are sectors of circles. 

To compute the Elements of Spherical Pyramids, see Docharty and Hackley's Ge- 
ometry. 

CYLINDRICAL UNGULAS. 

Definition.— Cylindrical Ungulas are frustra of cylinders. Conical Ungulas are 
frustra of cones. 



276 



MENSTJKATION OF SOLIDS. 



To Compute tlie Yolume of* a Cylindrical XJngnla— 
Fig. 76. 

1. When the Section is parallel to the Axis of the Cylinder. 
Fig. 76. Rule. — Multiply the area of the base by the height of 

a the cylinder. 

Or, aXh — V. 
Example. — The area of the base, d ef, Fig. .16, of a cylindrical un- 
gula is 15.5 inches, and its height, e a, 2l>; what is its volume? 
15.5x20 = 310 cubic inches. 



Fig. 77. 



/*■** 


V 

l : ' r 


---;\ 





f 



2. When the Section passes Obliquely through the opposite 
sides of the Cylinder — Fig. 77. 
Rule. — Multiply the area of the base of the. 
cylinder by half the sum of the greatest and 
least lengths of the ungula. 

Or, aX/lT'^ 2 ^^ 
Example.— The area of the base, d ef, of a cylindrical ungula, Fig. 
77, is 25 inches, and the greater and less heights of it, ad, b e, are 15 
and 17 inches ; what is its volume ? 



25x 



15 + 17 



= 400 cubic inches. 




Fig. 78. 



3. When ihe Section passes through the Base of the Cylinder and one of- 

its Sides, and the Versed Sine does not exceed the Sine — Fig. 78. 

Rule. — From % of the cube of the sine of half the arc of the base 

subtract the product of the area of the base and the cosine* of the 

half arc. Multiply the difference thus found by the quotient 

arising from the height, divided by the versed sine. 

• 2«3 — h „ 7 

Or, — acX—=\,vs representing the versed sine. 

Example. — The sine, a d, of half the arc, d ef, of the base of an un- 
gula, Fig. 78, is 5 inches, the diameter of the cylinder is £0, and the 
height, e g, of the ungula 10 ; what are the contents of it ? 
% of 5 3 = S3. 333 = two thirds of the cube of the sine. As the versed 
sine and radius of the base are equal, the cosine is 0. Hence area 
of base X cosine = 0. 

83.333 — 0x10-^-5 = 106.666 cubic inches. 

4. When the Section passes through the Base of the Cylinder, and the 

Versed Sine exceeds the Sine — Fig. 79. 
Rule. — To % of the cube of the sine of half the arc of the base 
add the product of the area of the base and the cosine. Multiply 
the sum thus found by the quotient arising from the height, divided by 
the versed sine. 

2 S 3 , h 




Fig. 79. 




Example.— The sine a d of half the arc of an ungula, Fig. 70, is 12> 
inches, the versed sine a g is 16, the height g c 10, and the diameter 
of the cylinder 25 inches ; what is its volume ? 

% of 123 — il$2 — two thirds of cube of sine of half the arc of the 
base. Area of base = 331.78; 1152 + 331.78x16 — 12.5= C313.23 
zzzsum of y s of the cube of the sine of half the arc of the base, 
and product of area of base and cosine. 
Then 2313. 23 X 2)^-16 = 2831 5375 cubic inches. 

• When the cosine 13 0, the product is 0. 






MENSURATION OF SOLIDS. 



277 




5. When the Section passes Obliquely through both Ends of the Cylinder 
—Pig. 80. 
Rule.— Conceive the section to be continued till it meets the side 
of the cylinder produced ; then, as the difference of the versed sines 
of the arcs of the two ends of the ungula is to the versed sine of the 
arc of the less end, so is the height of the cylinder to the part of the 
side produced. 

Ascertain the volume of each of the ungulas by Rules 3 and 4 and 
take their difference. 
v' h 
° r -> v — v > — h i v and v ' representing the versed sines of the arcs of 

the two ends, h the height of the cylinder, and h' the height of the 
part produced. 

Example.— The versed sines, ae,dc, and sines, e k and c, of the area 
of the two ends of an ungula, Fig. 80, are assumed to be respectively 
8.5 and 25, and 11.5 and inches, the length of the ungula, b c, within 
the cylinder, cut from one having 25 inches diameter, d c,is 20 inches • 
what is the height of the ungula produced beyond the cylinder, and 
what is the volume of it ? 

25 co S.5 : 8.5 : : 20 : 10.303 = height of ungula produced beyond the 
cylinder. 

Greater unguln, the sine c being 0, the versed sine = the diameter. Base of ungula 
being a circle of 25 inches diameter, area = 490.875. The versed sine and diameter 

of the base being equal (25), the sine = 0. 490. 8T5X 25 co ~ = 6135. 9375 = product 

°f %l ea °f} ase and cosine, or excess of versed sine over the sine of the base. 30.303 
-r- 25 = 1.21212 = quotient of height -5- versed sine. 

Then 6135.9375x1.21212 = 7437.4926 cubic inches; and by Rules 3 and 4, volumes 
of less and greater ungulas = 515.444, and 6922.0486 = 7437.4926 cubic inches. 

SPHERE. 
Definition.— A solid, the surface of which is at a uniform distance from the cen- 

Fig. 81. To Compute trie Volume of a Sphere— 

Fig. 81. 

Rule.— Multiply the cube of the diameter by .5236. 

Or, rf 3 x.5236 = V, d representing the diameter. 

Example.— What is the volume of a sphere, Fig. 81, its diame- 
ter, a b, being 10 inches ? 

103 = 1000, and 1000x.5236 = 523.6 cubic inches. 

SEGMENT OF A SPHERE. 
Definition — A section of a sphere. 

To Compute tlie Volume ofa Segment of a Sphere 
—Fig. 83. 

Rule 1. — To three times the square of the ra- 
dius of its base add the square of its height ; mul- 
tiply this sum by the height, and the product by 

.523G. 

Or, 3r*-f /»2fc X .5236 = V. 
2. — From three times the diameter of the sphere 
subtract twice the height of the segment; multiply 
this remainder by the square of the height, and the 
product by .5236. 

Or, §d"-=2*X.5236 = V. 
Aa 





273 



MENSURATION OF SOLIDS. 



Example.— The segment of a sphere, Fig. 82, has a radius, a o, of 7 inches for its 
base, and a height, b o, of 4 inches ; what is its volume? 

7 2 x3-4-42 = 163 = ^e sum of three times the square of the radius and the squan 
of the height ; 163X4X.5236 = 331.3872 cubic inches. 

SPHERICAL ZONE (OR FRUSTRUM OF A SPHERE). 

Definition. — The part of a sphere included between two parallel chords. 

To Compute tlie Volume of* a Spherical Zone- 
Fig. 83. 

Definition.— The part of a sphere included between two parallel planes. 

Rule. — To the sum of the squares of the radii of the two ends add 
X of the square of the height of the zone; multiply this sum by the 
height, and again by 1.5708. 



Fig. S3. 




Or, rS-f-r's + Zia-r- 3X^X1.5708 = 
Example. — What are the contents of a spherical zone, Fig. 
83, the greater and less diameters, fh and d e, being 20 and 
15 inches, and the distance between them, or height of the 
zone, c g, being 10 inches. 

102-|-7.52 = 156.25 = su m of th e squares of the radii of the 
two ends; 156.25 -f lua h- 3 = 189.5S = the above sum 
added to one third of the square of the height. 
Then 1S9.55X 10x1.5708 = 2977.9226 cubic inches. 

' SPHEROIDS (ELLIPSOIDS). 

Definition. — Solids generated by the revolution of a semi-ellipse about one of its 
diameters. "When the revolution is about the transverse diameter they are Prolate, 
and when about the conjugate they are Oblate. 

To Compute tlie Volume of a Splneroid— Fig. 84. 
Rule. — Multiply the square of the revolving axis by the fixed axis, 
and this product by .5236. 

Fig 84. ^ r ' ^ 2a 'X-5236 = V^a and a re-presenting the revolving 

and fixed axes. 
Or, 4-f-8X3.1416r 2 r / = V, r and r' repres'gthe semi-axes. 
Example.— In a prolate spheroid, Fig. 84, the fixed axis, 
a b, is 14 inches, and the revolving axis, c d, 10 ; what is 
its volume ? 

10 2 xl4 = 1400 =. product of square of revolving axis and 
fixed axis. Then 1400X-5236 = 733.04 cubic inches. 
Note. — The volume of a spheroid is equal to % of a cyl- 
inder that will circumscribe it. 

SEGMENTS OF SPHEROIDS. 

To Compute tlie Volume of tlie Segment of a 

Spheroid. 

When the Base, ef, is Circular, or parallel to the revolving Axis, as c d } 

Fig. 85, or as efto the Axis a b, Fig. 86. 

Kule. — Multiply the fixed axis by 3, the height 
of the segment by 2, and subtract the one prod- 
uct from the other ; multiply the remainder by 
the square of the height of the segment, and the 
product by .5236. Then, as the square of the 
fixed axis is to the square of the revolving axis, so 
is the last product to the volume of the segment. 

3a — 2AX/t 2 X.5236Xa /2 _^ r 
Ul, — — V. 






MENSURATION OF SOLIDS. 279 

Example. — In a prolate spheroid, Fig. 85, the fixed or transverse axis, a b, is 100 
inches, the revolving or conjugate, c d, €0, and the height of the segment, a o, 10 
inches ; what is its volume ? 

100x3 — 10x2 = 2S0 = twice the height of the segment subtracted from three times 
the fixed axis ; 2S0xl0 2 X. 5236 = 14660. 8 inches = product of above remainder, 
the,square of the height, and .5236. Then 1002 : 602 . . 14669.8 : 5277.88S cubic 
inches. 

When the Base, ef, is Elliptical, or perpendicular to the revolving Axis 
a b, Fig. 85, or ef Fig. 86, to the Axis c d. 

Rule. — Multiply the revolving axis by 3, and the height of the seg- 
ment by 2, and subtract the one from the other; multiply the remain- 
der by the square of the height of the segment, and the product by 
.5236. Then, as the revolving axis is to the fixed axis, so is the last 
product to the volume of the segment. 

Fig. S6. da'— 2/EXfr 2 X.5236xa _ v 

x^rrr^^ Example. — The diameters of an oblate spheroid, Fig. 86, 

<* --sS^K f are 100 and 60 inches, and the height of a segment thereof 
is 12 inches ; what is its volume ? 

Ab 100x3 — 12x2 = 276= twice the height of the segment 

[ I subtracted from three times the revolving axis ; 276 X 

122x .5236 = 20S09.9584 = product of above remainder ; 
the square of the height, and 5236. 
d Then 100 : 60 : : 20S09.95S4 : 12485.975 cubic inches. 

FRUSTRA OF SPHEROIDS. 

To Compute trie Volume of trie Middle Frustrmin 
of a Spheroid. 

When the Ends, efandgh, are Circular, or parallel to the revolving 
Axis, as c d, Fig. 87, or a b, Fig. 88. 

Rule. — To twice the square of the revolving axis add the square 

of the diameter of either end ; multiply this sum 

Fig. 87. ky t h e length of the frustrum, and the product by 

~j — -£ - 2618 - Or, 2a'2-f rf2 X z.2618 = V. 

; ;'i\ \ Example. — The middle frustrum of a prolate spheroid, 

* «£ i o, Fig. 87, is 36 inches in length, the diameter of it being, 

,j. ) in the middle, c d, 50 inches, and at its ends, ef and g h, 

•Am ■ ^ ' what is its volume * 

S'ljiUM'''' 5 ° 2 X 2 + 4 ° 2 — 6600 = sum of twice the square of the mid- 

^■^atMMv&^h di e diameter added to the square of the diameter of the 

d ends. Then 6600 X 36 X 2618 = 62203.68 cubic inches. 

When the Ends, ef and g h, are Elliptical, or perpendicular to the re- 
volting Axis a b, Fig. 87, or ef and g h to the Axis c d, Fig. 88. 

Fig. 88. Rule.— To twice the product of the transverse 

c and conjugate diameters of the middle section 

add the product of the transverse and conjugate 
<•' of either end ; multiply this sum by the length 
of the frustrum, and the product by .2618. 
Or, dd'x2 + dd"'X/X.261S=V. 
^ . . -_ Example.— In the middle frustrum of a prolate spheroid, 

— -4-^2^^ Fig. 88, the diameters of its middle section are 50 and 30 
" d " inches, its ends 40 and 24 inches, and its length, o i, IS 

inches; what is its volume? 





280 



MENSURATION OF SOLIDS. 



50x30x2 = 3000 -. 



twice the product of the transverse and conjugate diameters ; 
3000 + 40x24 = 3960 == sum of the above product and the product of the trans* 
verse and conjugate diameters of the ends. 
Then 3960 X IS X. 2618 = 18661.104 cubic inches. 

CYLINDRICAL RING. 

Definition. — A ring formed by the curvature of a cylinder. 

To Compute tlie Volume of a Cylindrical R-ing— i 
Fig. 89. 

Rule. — To the diameter of the body of the ring add the inner di- 
ameter of the ring ; multiply the sum by the square of the diameter 
of the body, and the product by 2.467-1. 

Or, d-\-d'Xd 2 2.4674 = V, d and d' representing the diameter 

of the body and inner diameter. 
Or, axZ = V, a representing area of section of body, and I the 

length of the axis of the body. 
Example. — What is the volume of an anchor ring, Fig. 89, 
the diameter of the metal, a b, being 3 inches, and the inner di- 
ameter of the ring, b c, S inches ? 
3 + Sx3 2 = 99 — product of sum of diameters and the square of 

diameter of body of ring. 
Then 99X2.4674 = 244.2726 cubic inches. 



Fig. 89. 




LINKS. 

Definition. — Elongated or Elliptical rings. 

ELONGATED OR ELLIPTICAL LINKS. 

To Compute tlie Volume of an Elongated, or El- 
liptical JLiinli— Eigs. 90 and 91. 

Rule. — Multiply the area of a section of the body of the link by its 
length, or the circumference of its axis. 
Or, aXl = Y. 
Note — By Rule, page 263, the circumference or length of the axis of an Elonga- 
ted link = the sum of 3.1416 times the sum of the less diameter added to the thick- 
ness of the ring, and the product of twice the remainder of the less diameter sub- 
tracted from the greater. 
Fig. 90. Also, the circumference or length of the axis of an Elliptical ring = 

a the square root of half the sum of the diameters added to the thickness 

of the ring or the axes squared X 3. 1416. 

Example. -.-The elongated link of a chain, Fig. 90, is 1 inch in diam- 
eter of body, a b, and its inner diameters, b c and ef are 10 and 2.5 
inches ; what is its volume? 

Area of 1 inch = .7854; 2.5+1x3.1416 = 10.9956 = 3.1416 times the 
sum of the less diameter and thickness of the ring = length of 
axis of ends ; 10 — 2.5x2 = 15 = twice the remain- 
der, of the less diameter subtracted from the greater Fig. 91. 
= length of sides of body. a 

Then 10.9 " 56 + 15 = 25.9956 = length of axis of link. 
Hence .7S54x25.9956 = 20.417 cubic inches. 

Ex 2.— The elliptical link of a chain, Fig. 91, is of the same dimen- 
sions as the preceding; what is its volume? 

2 2 /133 ^5 

2.5-fl-f-10 + l = 133.25 = diam. of axes squared ; ./ ' " X 3. 1416 

= 25. 643 = square root of half sum of diam eters squared X 3. 1416 
= circumference of axis of ring. Area of 1 inch = .7^54. 
Then 25. 64bX-7S51= 20.14 cubic inches. 





MENSURATION OF SOLIDS. 



281 



Fig. 92. 




SPHERICAL SECTOR. 

Definition. — A figure generated by the revolution of a sector of a circle about a 
straight line through the vertex of the sector as an axis. 

Note. — The arc of the sector generates the surface of a zone, termed the base of 
the sector of a sphere, and the radii generate the surfaces of two cones, having a ver- 
tex in common with the sector at the centre of the sphere. 

To Compute tlie Volume of a Spherical Sector- 
Fig. 9S. 

Rule. — Multiply the surface of the zone, which is the base of the 
sector, by X of the radius of the sphere. 

Or, aXr-^-3:=V, a representing the area of the base. 

Example. — What is the volume of a spherical sector, Fig. 92, generated by the 
sector, c a h, the height of the zone, abed, being, 
a o, 12 inches, and the radius, g h, of the sphere 15 
inches ? 

12 X 94. 248 = 1130.976 = height of zone X circumfer- 
ence of sphere ■=. surface of zone (see page 256). 
d 1130.976x30-^6 (= % of radius) = 5654.8S cub. ins. 
Note. — The surface of a spherical sector = the sum 
■^ of the surface of the zone and the surfaces of the two 
cones. 

SPINDLES. 

Definition. — Figures generated by the revolution of a plane area bounded by a 
curve, when the curve is revolved about a chord perpendicular to its axis or about 
its double ordinate, and they are designated by the name of the arc from which they 
are generated, as Circular, Elliptic, Parabolic, etc. 

CIRCULAR SPINDLE. 

To Compute the "Volume of a Circular Spindle— 
Fig. 93. 

Rule. — Multiply the central distance by half the area of the re- 
volving segment; subtract the product from }{ of the cube of half the 
length, and multiply the remainder by 12.5664. 

— ( c Xr )xl2.5GG4 = V, a repressing the area of the revolving segment. 

Example. — What is the volume of a circular spindle, Fig. 

93, when the central distance, o e, is 7.07106T inches, the length, 

f"c, 14.14213, and the radius, o c. 10 inches? 

7.0710G7X 14.27 = 100.9041 — central distance X half area of 

7.071 67 3 

revolving segment; 100.C041 = 16.947 = rc- 

• \\/ ; mainder of above product and % of cube of half the length. 

\ O J Then 16.497x12.5664= 212.9628 cubic inches. 

Note The area of the revolving segment,/ e, being = tlie 

side of the square tha,t can be inscribed in a circle of 20, is 
"*•- -"' 202X.7S54 — 14.142132 ^4 = 2S.54. 

FRUSTRUM OR ZONE OF A CIRCULAR SPINDLE.* 

To Compute the "Volume of* a ITrvi strum, or Zone 

of a Circular Spindle— Fig. 94. 

Rule. — From the square of half the length of the whole spindle 
take }i of the square of half the length of the frustrum, and multiply 

* The middle frustrum of a Circular Spindle is one of the various forma of oasks. 
A A* 




282 MENSURATION* OF SOLIDS. 

the remainder by the said half length of the frustrum ; mnltiply the 
central distance by the revolving area which generates the frustrum ; 
subtract this product from the former, and multiplv the remainder by 
6.2832. 




Or, ZH-2 H—X-q — (cXa)X6.2S32 = V, I and V representing the lengths of the 

spindle and of the frustrum, and a area of the revolving section of frustrum. 
Note. — The revolving area of the frustrum can be obtained by dividing its plane 
into a segment of a circle and a parallelogram. 

Example. — The length of the middle frustrum of a circular spindle, i c, Fig. 94, 
is 6 inches ; the length of the spindle, fg, is 8 inches ; the central distance, o e, is 3 
inches ; and the area of the revolving or generating seg- 
Fig. 94. ment is 10 inches ; what is the volume of the fr-ustrum ? 

( S -4- 2) 2 _ ^jj-^ = 13 and 13 X 3 = 39 = product of half 

the length of the frustrum, and the remainder of% 
the square of half the length of the frustrum sub- 
tracted from the square of half the length of the spin- 
\ \ / die; 39 — 3 X I s ' = 9 = product of the central distance 

\J / and the area of the segment subtracted from preced- 

** in ** product. 

Then 9 X 6.2832 — 56.54SS cubic inches. 

SEGMENT OF A CIRCULAR SPINDLE. 

To Compute tlie "Volume of a Segment of a Circu- 
lar Spindle— Fig. 95. 

Rule. — Subtract the length of the segment from the half length of 
the spindle ; double the remainder, and ascertain the volume of a mid- 
dle frustrum of this length. Subtract the result from the volume of 
the whole spindle, and halve the remainder. * 

Or, C — c-^-2 = V, C and c representing the volume of spindle and middle frustrum. 

Example. — The length of a circular spindle, i a, Fig.95, is 14.14213 ; the central 

distance, o e, is 7.07107; the radius of the arc, o a, is 10; 

I lg. 95. am | t h e i en gth of the segment, i c, is 3.53553 inches : what 

is its volume? 

14 14^13 

— — 3.53553x2 = 7.07107 == double the remainder, 

a 

of the length of the segment subtracted from half the 
length of the spindle == length of the middle frustrum. 

\ ; / Note. — The area of the revolving or generating seg- 

» ment of the whole spindle is 2S.54 inches, and that of the 

middle frustrum is 19.25. 

The volume of the whole spindle is 212.962S cubic inches. 

" " middle frustrum is ... . 162.S9S2 " " 
Hence 50.0046 -4- 2 = 25.0323 cubic inches. 

CYCLOIDAL SPINDLE.t 

To Compute the Volume of a Cycloidal Spindle— 

Fig. 96. 

Rule. — Multiply the product of the square of twice the diameter of 
the generating circle and 3.927 by its circumference, and divide this 
product by 8. 

* This rule is applicable to the segment of any Spindle or any Conoid, the volume of the figure and 
frustrum being first obtained. 
f The volume of a Cycloidal Spindle is equal to % of its circumscribing cylinder. 






MENSURATION OF SOLIDS. 



283 



Or, 



2 

2d X3.927xdX3.1416 




:V, d representing the diameter of the circle, or half 
width of the spindle. , \iO- V/ 

Fig. 96. Example. — The diameter of the generating circle, a be, 

of a cycloid, Fig. 16, is 10 inches; what is the volume of 
b the spindle, d e ? "' ■-,-__' 

r - 2 

lux 2 X 3. 927=1570.8 = product of twice the diameter 
squared and 3.927. 

Then 1570.8x10x3.1416-=- 8 = 616S.5316 cubic inches, 
ELLIPTIC SPINDLE. 

To Compute tlie Contents of* an Elliptic Spindle— 
Eig. 97. 

Kule. — To the square of it's diameter add the square of twice the 
diameter at % of its length ; multiply the sum by the length, and the 
product by .1309. f 

2 

Or, d 2 -\-2d' X^.1309 = V, d and d' representing the diameter as above. 
v . Q7 Example. — The length of an elliptic spindle, a b, Fig. 97, 

.big. y7. j s 75 i nc h ei s(, its diameter, c d, 35, and the diameter, ef at % 

of its length, 25 ; what is its volume ? 




j 35 2 --f- 25 x 2 = 3725 = sum of squares of diameter of spindle 
and of twice its diameter at }£ of its length;' 3725x75 
-■} = 279375 = above sum X length of the spindle. 

f Then 279375X.1309 = 36570.1875 cubic inches. 

Note. — For all such solid bodies this rule is exact when 
the body is formed by a conic section, or a part of it, revolv- 
ing about the axis of the section, and will always be very near when the figure re- 
volves about another line. 

To Compute tlie Volume of* the Middle Erustrum 
ox* Zone of* an Elliptic Spindle— Eig. 9S* 

Rule. — Add together the squares of the greatest and least diame- 
ters, and the square of double the diameter in the middle between the 
two ; multiply the sum by the length, and the product by .1309.* 

2 

Or, d 2 -J- d' 2 -j- 2 d" X'«1309 = V, d, d\ and d" representing the different diameters. 
Example. — The greatest and least diameters, a b and c d, 
of the frustrum of an elliptic spindle, Fig. 98, are 68 and 50 
inches, its middle diameter, g h, 60, and its length, ef 75 ; 
what is its volume ? 

2 

CS2-J- 502-j- 60X2 = 21524 = sum of squares of greatest and 
least diameters and of double the middle diameter. 
Then 21524X75X.1309 = 211311.87 cubic inches. 

To Compute the "Volume of a Segment of* an Ellip- 
tic Spindle— Eig. 99. 

Rule. — Add together the square of the diameter of the base of the 
segment and the square of double the diameter in the middle between 
the base and vertex ; multiply the sum by the length of the segment, 
and the product by .1309.* 

2 

Or, d 2 -\~ 2 d" X IX .1309 = V, d and d" representing the diameters. 
* See Note above. 





284 MENSURATION OF SOLIDS. 

Fig. 99. Example.— The diameters, c d and g h, of the segment of 

g c .- -— :- — - ^ an elliptic spindle, Fig 99, are 20 and 12 inches, and the 

length, o e, is 16 inches ; what is its volume ? 

202 _j_ 12x2 = 976 = sum of squares of diameter at base 
h ^r .,-- and in the middle. 

« a - j. ' Then 976 X 116 X. 1303 ttc 2044.134 cubic inches. 

PARABOLIC SPINDLE. 

To Compute tlie Volume of* a IParaloolic Spindle- 
Fig. lOO. 

Rule 1. — Multiply the square of the diameter by the length, and 
the product by. 41888.* 

Or, d2/ x .41S8S = V. 
Rule 2. — To the square of its diameter add the square of twice the 
diameter at }£ of its length ; multiply the sum by the length, and the 

product by .1309. f 2 

Or, rf2 + 2 d' X IX .1309 = V. 
Fig. 100. Example. — The diameter of a parabolic spindle, a b, Fig. 

c 100, is 40 inches, and its length, c d, 10; what is its vol- 

ume ? 

402x10 = 16000= square of diameter X the length, 
b Then 1G000X .41SSS = 6702.0S cubic inches. 

Again, If the middle diameter at X of its length is 30, 
then, by Rule 2, 402 + 30x2 x40x.1309 = 6S06.S cubic ins. 

To Compute tlie "Volume of* tlie Middle Frustrum 
of* a iParaoolie Spindle— Fig. lOl. 

Rule I. — Add together 8 times the square of the greatest diameter, 
3 times the square of the least diameter, and 4 times the product of 
these two diameters; multiply the sum by the length, and the prod- 
uct by .05236. _ 

Or, d* 8 +d'23 -f d d' X 4 I. X 05236 = V. 

Rule 2. — Add together the squares of the greatest and least diame- 
ters and the square of double the diameter in the middle between the 
two ; multiply the sum by the length, and the product by .1309. 
O:-, d2 _|_ ^'2 _|_ 2 d /; 2xZx.l309 = V, d" representing the diameter between the two. 

-r,. 1m Example. — The middle frustrum of a parabolic spindle, 

i-ig. iui. Fig 1Q1 ^ hag diameter;?i a b an d e f y f 40 an d 30 inch s, 

e ,- ■-" ~.~c_ ~~" - -./ and its length, c d, is 10 inches ; what is its volume? 

a^--------3^'~ : ^*\ f 402x8 + 302x3 + 40X30X4 = 20300 = /A* sum /S times 

\~ £" *""" ' ~"~^ b the square of the greatest diameter, 3 times the square 

X^^::::^ :^t^y ^ of the least diameter, and 4 times the product of these. 

'--..A..--' ■ Then 20300X10X-05236 = 10629.GS cubic inches. 

To Compute the Volume of* a Segment of a Para- 
bolic Spindle-Fig. 102. 

Rule. — Add together the square of the diameter of the base of the 
segment and the square of double the diameter in the middle between 
the base and vertex; multiply the sum by the length of the segment, 
and the product by .1309. 

Or, d* + d"*X Z.X.I 309 = V. 

* 8-15 of .7854. t S** Note . P a S e 2S3 - 






Fig. 
/ i 


102. 

: ;>< 


\ 


y 



MENSURATION OF SOLIDS. 285 

Example.— The segment of a parabolic spindle, Fig. 102, 
has diameters, e f and g h, of 15 and 8. 75 inches, and the 
length, c d, is 2.5 inches ; what is its volume ? 



152 _L. 8.75x2 = 531.25= sum of square of base and of dou- 
ble the diameter in the middle of the segment. 
Then 531.25x2.5x.l309 = 173.852 cubic inches. 

HYPERBOLIC SPINDLE. 

To Compute the Volume of a Hyperbolic Spindle— 
IFlg. 103. 

Kule, — To the square of the diameter add the square of double the 
diameter at X of its length ; multiply the sum by the length, and the 
product by. 1309.* 

2 

Or, d* + 2d' XZX.1309=:V. 
Example. — The length, a b, Fig. 103, of a hyperbolic spin. 
die is 100 inches, and its diameters, c d and ef are 150 and 
110 inches ; what is its volume ? 



Fig. 103. 



N %tg==sL==ar 1502 -(-110x2 x 109 = 7090000 = product of the sum of the 

^^SH§]j^^^ squares of the greatest diameter and of twice the diame- 

^^K^^^ ter at X of the length of the spindle and the length. 

* Then 7090060x.l309 = 9iS081 cubic inches. 

To Compute the Volume of the Middle Frustrum 
of a Hyperbolic Spindle— Fig. 104. 

Rule. — Add together the squares of the greatest and least diame- 
ters and the square of double the diameter in the middle between the 
two ; multiply this sum by the length, and the product by .1309.f 

Or, d* _j_ d'2 _|_ (2 rf // ) 2 xZX.1309 = V. 
Fig. 104. Example. — The diameters, a b nnd c d, of the middle frus- 

,,''"e~'^ trunvof a hyperbolic spindle, Fig. 1"4, are 150 and 110 incli- 

./^S Sj-TiTpv^ sK es > tne diameter, g h, 14J inches; and the lengch, ef 50; 
/ ' I "~~\ what is its volume ? 



£\_ : '0h 1502 _|_ 1102 _|_ 140X2 = 113000 = sum of squares of greatest 

N^ -f Z? d and least diameters and of double the middle diameter. 

""• *" Then 113000X50X.1309 = 739585 cubic inches. 

To Compute tlie "Volume of a Segment of a H"y 
peroolic Spindle— IFig. lOS. 

Rule. — Add together the square of the diameter of the base of the 
segment and the square of double the diameter in the middle between 
the base and vertex ; multiply the sum by the length of the segment, 
and the product by .1309. 

Or, rf2_j_d//2z x .i309=:V. 

ELLIPSOID, PARABOLOID, AND HYPERBOLOID OF REVOLU- 
TION J (conoids). 

Definition. — Figures like to a cone, described by the revolution of a conic sec- 
tion around and at a right angle to the plane of their fixed axes. 

* See Note, page 283. f H>i»l. 

$ These figures have been known as Conoids. For the definition of a Conoid, see llasweWt Ifew 
titration, page 233. 



286 



MENSURATION OF SOLIDS. 



Fig. 105. Example. — The segment of a hyperbolic spindle, Fig. 105, has 

a diameters, ef and g h, of 110 and €5 inches, and its length, a b, 

£^^: — ^ 25 ; what is its volume ? 

£ \ 1102 _j_ (55 x 2 — 29000 cz sum of squares of diameter of base and 

. ..i ^j, foy^fo tf le middle diameter. 

Then 29C00x25x.l309 = 94902.5 cubic inches. 



Definition. - 




ELLIPSOID OF REVOLUTION (SPHEROID). 
-An ellipsoid of revolution is a semi-spheroid. (See page 257.) 

PARABOLOID OF REVOLUTION.* 

To Compute tlie Volume of a Paraboloid of Revo- 
lution-Fig. 106. 

Rule. — Multiply the area of the base by half the 
altitude. Or,ax/t-H2 = V. 

Note. — This rule will hold for any segment of the paraboloid, 
whether the base be perpendicular or oblique to the axis of the 
solid. 

Example. — The diameter, a b, of the base of a paraboloid of 
revolution, Fig. 106, is 20 inches, and its height, d c, 20 inches ; 
what is its volume ? 

Area of 20 inches diameter of base =314.16. 
c Then 314.16x20-^-2 = 3141.6 cubic inches. 

FRUSTRUM OF A PARABOLOID OF REVOLUTION. 

To Compute trie Volume of a Frustrum of a Para- 
boloid, of Revolution— Fig. 107. 

Rule. — Multiply the sum of the squares of the 
diameters by the height of the frustrum, and this 
product by .3927. 

Or, d*+d'2xhX. 3927 = Y. 

Example. — The diameters, a b and d c, of the base anu 
vertex of the frustrum of a paraboloid of revolution, Fig. 107, 
are 90 and 11.5 inches, and its height, ef 12.6; what is its 
volume ? 

202 -j_ 11.52 — 539.95 — sum of squares of the diameters. 

Then 532.25xl2.6x.392T = 2633.5837 cubic inches. 

SEGMENT OF A PARABOLOID OF REVOLUTION. 

To Compute tlie Volume of tlie Segment of a Par- 
aboloid, of Re volution— Fig. 108. 



Fig. 107, 





-Multiply the area of the base by half the 
-2 = V. 



Rule. 

altitude. 

Or, aX h 

Note. — This rule will hold for any segment of the parabo- 
loid, whether the base bj perpendicular or oblique to tlie axis 
of the solid. 

Example. — The diameter, a b, of the base of a segment of a 
paraboloid of revolution, Fig. 10S, is 11.5 inches, and its height, 
ef is 7.4; what is its volume? 

Area of 11.5 inches diameter of base = 103 869. 

Then 103. S69 X 7. 4 -r- 2 = 3S4.315 cubic inches. 



* The volume of a Paraboloid of Revolution is = % of its circumference. 



1 



MENSURATION OF SOLIDS. 



287 



HYPERBOLOID OF REVOLUTION. 

To Compute tlie Volume of a Hyperooloid of 

Revolution-Fig. 109. 
Rule. — To the square of the radius of the base add the square of 
the middle diameter; multiply this sum by the height, and the prod- 
FI-.109. net by .5236. 

° f Or, r 2 -f &X h X-5236 = V, d representing middle di- 

<L ameter. 

Example.— The base, a b, of a hyperboloid of revolu- 
tion," Fig. 109, is 80 inches ; the middle diameter, c d, 
66 ; and the height, ef, 60 ; what is its volume? 





80 -7- 2 + 662 — 5956 — sum of square of radius of base 
and middle diameter. 
Then 5956 X 60 X .5236 = 87113.7 cubic inches. 

FRUSTRUM OF A HYPERBOLOID OF REVOLUTION. 

To Compute the Volume of the Frustrum of a 
Hyperholoid. of Revolution— Fig. HO. 

Rule. — Add together the squares of the greatest and least semi- 
diameters and the square of the diameter in the middle of the two ; 
multiply this sum by the height, and the product by .5236. 

0r ' (^) 2 +(!y + d " 2xhxMZQ = V > rf > d '' and d " rep- 
resenting the several diameters. 
Example. — The frustrum of a hyperboloid of revolu- 
tion, Fig. 110, is in height, e i, 50 inches ; the diame- 
ters of the greater and lesser ends, a b and c d, are 110 
and 42 ; and that of the middle diameter, g h, is 80 ; 
what is tlie volume? 

110 -4- 2 = 55, and 42 -4- 9 = 21. Hence 1102 _f_ 212 _j_ 
80 2 = 9866 = sum of the squares of the semi-diam- 
eters of the ends and of the middle diameter. 
Then 9S66x50x.5236 = 25329 l.SS cubic inches. 

SEGMENT OF A HYPERBOLOID OF REVOLUTION. 

To Compute trie Volume of tlie Segment of a Hy- 
perboloid of Re volution, as Fig. 109. 

Rule. — To the square of the radius of the base add the square of 
the middle diameter ; multiply this sum by the height, and the prod- 
uct by .5236. 

Or, r 2 -\- d 2 XhX .5236 = V, r representing radius of base. 

Example. — The radius, a e, of the base of a segment of a hyperboloid of revolu- 
tion, as Fig. 109, is 21 inches; its middle diameter, c rf, is 30; and its height, ef 
15 ; what is its volume? 

212 -J- 302x15 = 20115 = £Ae product of the sum of the squares of the radius of the 
base and the middle diameter multiplied by the height. 

Then 20115X-5236 = 10532.214 cubic inches. 

ANY' FIGURE OF REVOLUTION. 

To Compute trie Volume of any Figure of Revo- 
lution — Fig. 111. 

Rule. — Multiply the area of the generating surface by the circum- 
ference described by its centre of gravity. 

Or, a 2rp = V, r representing radius of centre of gravity. 



2S8 



MENSURATION OF SOLIDS. 



Fig. 111. 



~-s/A ^i'/_ ^ 






Fig. 112. 



Illustration — If the generating surface, ab c d, of the 
cylinder, bed/. Fig. Ill, is 5 inches in width and 10 in 
height, then will ab = 5 and b d ~ 10, and the centre of 
gravity will be in o. the radius of which is r o = 5 — 2 = 2.5. 

Hence 10x5 = 50 = area of generating surface. 

Then 50x2.5x2x3.1416 = 785.4= area of generating sur- 
face X circumference of its centre of gravity = the volume 
of the cylinder. 

Pkoof. — Volume of a cylinder 10 inch- 
es in diameter and 10 inches in height. 
102x .TS54=TS.54, and 78.54X10=185.4. 
Illustration 2 — If the generating surface of a cone, Fig. 
112, is a e = 10, d e = 5, then will a d = 11.18, and the area of 
the triangle = 10x5 -=- 2 ; = 25, the centre of gravity of which is 
in o, and o r, by Rule, page 339 = 1.666. 

Hence 25x1.666x2x3.1416 = 261.8 = area of generating sur- 
face X circumference of its centre of gravity 
— the volume of the cone. 

Illustration 3. — If the generating sur- 
face of a sphere, Fig. 113, is ab c, and a c = 10, a b c will be 

^ J x3 39.27, the centre of gravity of which is in o, and 

by Rule, page 339, orz: 2.122. 

Hence 39.27x2.122x2x3.1416 = 523.6= area of generating 
surface X circumference of its centre of gravity = the volume 
of the sphere. 





l W^0k 



To Compute th.e Volume ofan Irregular Body. 

Kule. — Weigh it both in and out of fresh water, and note the dif- 
ference in pounds; then, as 62.5* is to this difference, so is 1728f to 
the number of cubic inches in the body. Or, divide the difference in 
pounds by 62.5, and the quotient will give the volume in cubic feet. 

Note. — If salt-water is to he used, the ascertained weight of a cubic foot of it, or 
64, is to he used for 62.5. 

Example. — An irregular-shaped body weighs 15 pounds in water, and 30 out % 
what is its volume in cubic inches ? 

30 — 15 = 15 = difference of weights in and out of water. 

62.5 : 15 : : 1728 : 414.72 = volume in cubic inches. 

Or, 15 ^-62.5 = .24, and .24x1728 = 414.72 = volume in cubic inches. 



* The weight of a cubic foot of fresh water. 
t The number of inches ia a cubic foot. 



CONIC SECTIONS. 



289 




Fig. 3. 




CONIC SECTIONS. 

A Cone is a figure described by the revolution of a right-angled tri- 
angle about one of its legs, or it is a solid having a circle for its base, 
and terminated in a vertex. 

Conic Sections are the figures made by a plane cutting a cone. 
The Axis is the line about which the triangle revolves. 

The Base is the circle which is described by the revolving base of the triangle. 
Notes. — If a cone is cut by a plane through the vertex and base, the section will 
be a triangle. 
If a cone is cut by a plane parallel to its base, the section will be a circle. 

An Ellipse is a figure generated by an oblique plane cutting a 
cone above its base. 

The transverse axis or diameter is the longest right line that 
can be drawn in it, as a b, Fig. 1. 

The conjugate axis or diameter is a line drawn Fig. 2. 

through the centre of the ellipse perpendicular to < 

the transverse axis, as c d. 

A Parabola is a figure generated by a plane 
cutting a cone parallel to its side, as a b c, Fig. 2. 
The axis is a right line drawn from the vertex 
to the middle of the base, as b o. 
Note. — A parabola has no conjugate diameter. 

A Hyperbola is a figure generated by a plane 
cutting a cone at any angle with the base greater c 
j^IZ—.^-.q than that of the side of the cone, as a b c, Iig. 3. 

The transverse axis or diameter, o b, is that part of the axis, e b } 
which, if continued, as at o, would join an opposite cone, ofr. 
„V' The conjugate axis or diameter is a right line drawn through the 

A " centre, g, of the transverse axis, and perpendicular to it. 

The straight line through the foci is the indefinite transverse 
axis ; that part of it between the vertices of the curves, as o b, is 
the definite transverse axis. Its middle point, g, is the centre of 
/ i\\ the curve. 

/„ "41-lk. The eccentricity of a hyperbola is the ratio obtained by dividing 

t.'- Alfip the distance from the centre to either/ocws by the semi-transverse 

axis. 

The asymptotes of a hyperbola are two right lines to which the 
curve continually approaches, touches at an infinite distance, but does not pass; 
they are prolongations of the diagonals of the rectangle constructed on the extremes 
of the axes. 

Two hyperbolas are conjugate when the transverse axis of the one is the conjugate 
of the other, and contrariwise. 

General Definitions An Ordinate is a right line from any point of a curve to 

either of the diameters, a e and d o, Fig. 4; a b and df are dou- 
ble ordinates. 

An abscissa is that part of the diameter which is contained 
between the vertex and an ordinate, as c e, g o. 

The parameter of any diameter is equal to four times the dis- 
tance from the focus to the vertex of the curve ; the parameter 
b of the axis is the least possible, and is termed the parameter of 
g the curve. 

The parameter of the curve of a conic section is equal to the 
chord of the curve drawn through the focus perpendicular to the 
axis. 

The parameter of the transverse axis is the least, and is termed 
the parameter of the curve. 

The parameter of a conic section and the foci are sufficient ele- 




ments for the construction of the curve. 



Bb 



290 



CONIC SECTIONS. 



Notes.— In the Parabola the parameter of any diameter is a third proportional to 
the abscissa and ordinate of any point of the curve, the abscissa and ordinate being 
referred to that diameter and the tangent at its vertex. 

In the Ellipse and Hyperbola the parameter of any diameter is a third proportional 
to the diameter and its conjugate. 

To Determine tlie Parameter ofan Ellipse or Hyperbola. 
Fig. 5. 



Fig. 6. 






Rule. — Divide the product of 
the conjugate diameter, multi- 
plied by itself, by the transverse, 
and the quotient is equal to the 
parameter. 

In the annexed Figs. 5 and 6, 
of an Ellipse and Hyperbola, the 
transverse and conjugate diame- 
ters, ab, c d, are each 30 and 20. ( 

Then 30 : 20 : : 20 : 13.833 = 
parameter. 
The parameter of the curve = ef, a double ordinate pass- 
ing through the focus, s. 

In a Parabola, Fig. 7. The abscissa, a b, 
and ordinate, c b, are also equal to 30 and 20. 

A Focus is a point on the principal axis where the double ordinate 
to the axis, through the point, is equal to the parameter, as ef in 
the preceding figures. 

It may be determined arithmetically thus : Divide the square of 
the ordinate by four times the abscissa, and the quotient will give 
the focal distances, a s and s, in the preceding figures. 

A Conoid is a warped surface generated by a right line being 
moved in such a manner that it will touch a straight line and curve, 
and continue parallel to a given plane. The straight line and curve 
are called directrices, the plane a plane directrix, and the moving line the generatrix. 
The Directrix of a conic section is a straight line, such that the ratio obtained by 
dividing the distance from any point of the curve to it by the distance from the same 
point to the focus shall be constant. It is always perpendicular to the principal 
axis ; and if the curve is given, it is readily constructed. (See HaswelVs Mensura- 
tion, page 232.) 

Ellipsoid, Paraboloid, and Hyperboloid of Revolution— figures generated by the 
revolution of an ellipse, parabola, etc., around their axes. (See Mensuration of Sur- 
faces and Solids.) 

Note — All the figures which can possibly be formed by the cutting of a cone are 
mentioned in these definitions, and are the five following: viz., a Triangle, a Circle, 
an Ellipse, a Parabola, and a Hyperbola ; but the last three only are termed the 
Conic Sections. 

ELLIPSE. 
To Describe Ellipses. 
When any three of the four following Terms ofan Ellipse are given, viz., the 
Transverse and Conjugate Diameters, an Ordinate, and its Abscissa, to 
A scertain the remaining Terms. 

To Compute tlie Ordinate, t lie Transverse and Conjugate 
Diameters and tlie Abscissa "being given — Eig. S. 

Fig. S. Rule.— As the transverse diameter is to the conju- 

gate, so is the square root of the product of the abscissae 
to the ordinate which divides them. 




-Xy/aX(t — a') = o, t representing the transverse 

diameter, c the conjugate, a' the less abscissa, and o 

the ordinate. 

Example. — The transverse diameter, a ft, of an el- 
lipse, Fig. 8, is 25; the conjugate, c d, is 16; and the 
abscissa, a i, 7 ; what is the length of the ordinate, i e ? 



CONIC SECTIONS. 291 



25 — 7 = 18 — second abscissa ; y/1 X 13 = 11.225 = square root of the abscissas. 
Hence 15 : 1C : : 11.225 : 7.1S4 inches = length of the ordinate. 

To Compute tlie Abscissae, the Transverse and. Conjugate 
Diameters and. the Ordinate "being given — Fig. 8. 

Rule.— As the conjugate diameter is to the transverse, so is the square root of the 
difference of the squares of the ordinate and semi-conjugate to the distance between- 
the ordinate and centre ; and this distance being added to, or subtracted from the 
semi-transverse, will give the abscistae required. 

t // c \ 2 / 2-r-2-|- # = a, \x representing the distance obtained, and 

® r i ~ X / ( ^ ) 2 — °° ^i ~ 2 — x = a',) a a' the greater and lesser abscisses. 

Example.— The transverse diameter, a b, of an ellipse, Fig. 8, is 25 ; the conju- 
gate, c d y 16 ; and the ordinate, i e, T.1S4; what is the abscissa, i b ? 
V8 2 — 7.1S4^ = 3.519943 =z square root of difference of squares of semi- conjugate 
and ordinate. 
Hence as 16 : 25 : : 3.52 : 5.5 = distance between ordinate and centre. 
Then 25 -r- 2 = 12.5, and 12.5 -f- 5.5 = 18 = b i, \ abscissa > 
25-2=12.5, and 12.5-5.5= 7 = al,jf absasscB - 

To Compute tlie Transverse Diameter, the Conjugate, 
Ordinate, and Abscissa "being given — Fig. 8. 

Rule.— To or from the semi-conjugate, according as the greater or lesser abscissa 
is used, add or subtract the square root of the difference of the squares of the ordi- 
nate and semi-conjugate. Then, as this sum or difference is to the abscissa, so is 
the conjugate to the transverse. 

aXc . 

Or,c-2+ ) 



::!!>-©• 



Example.— The conjugate diameter, c d, of an ellipse, Fig. 8, is 16; the ordinate, 
t e, is 7.1S4; and the abscissae, b i i,a, are 18 and 7 ; what is the length of the trans- 
verse diameter ? 



(16 -7- 2) 2 — 7.184 2 = 3.52 = square root of difference of squares of ordinate and 

semi-conjugate. 
16 -l. 2 -f 3.52 : IS : : 16 : 25 ; 16 -r- 2 — 3.52 : 7 : : 16 : 25 = transverse diameter. 

To Compute tlie Conjugate Diameter, tlie Transverse, 
Ordinate, and .A/bscissa "being given — Fig. 8. 

Rule. — As the square root of the product of the abscissae is to the ordinate, so is 
the transverse diameter to the conjugate. 

Or, oXt-^- ^aXa' =zc. 

Example. — The transverse diameter, a 6, of an ellipse, Fig. 8, is 25; the ordinate, 
i e, is 7.1S4; and the abscissae, b i and i a, 18 and 7 ; what is the length of the conju- 
gate diameter ? 

VIS x 7 = 11 .225 = square root of product of abscissce. 

11.225 : 7.1S4 : : 25 : 1Q = conjugate diameter. 

To Compute tlie Circumference of an Ellipse — rTig- 8. 

Rule. — Multiply the square root of half the sum of the squares of the two diame- 
tera by 3.1416. +d , 2 

Or, / X 3. 1416 == circumference. 

Example. — The transverse and conjugate diameters, a b and e d, of an ellipse, 
Fig. 6, are 24 and 20 ; what is its circumference ? 

242 _l 202 . 

■ =48S. and -^483 =± 22.09 = square root of half the sum of the squares of 

the diameters. 
Hence 22.09x3.1416 = 69.398 — the above root X 3. 1416 = area. 



292 



CONIC SECTIONS. 



, 8. 
Or, multiply 

and its con* 
rea. 



To Compute the Area of an Ellipse — Trig. 

Rule. — Multiply the diameters together, and the product by .7S54. 
one diameter by .7654, and the product by the other. 
Or, dXd'X-~tSb4: = area. 
Example.— The transverse diameter of an ellipse, a b, Fig. S, is 12, and its con* 
jugate, c rf, 9 ; what is its area? 

12x9X.7851= S4.S232 = product of diameters and .7854 = 

SEGMENT OF AN ELLIPSE. 

To Compute the Area of* a Segment of an. Ellipse when 
its Base is parallel to either Axis, as eif; Eig. 9. 

Rule. Divide the height of the segment, b i, by the diameter or axis, a b, of 

which it is a part, and find in the Table of Areas of Segments of a Circle, page 205, 
a segment having the same versed sine as this quotient ; then multiply the area of 
the segment thus found and the two axes of the ellipse together. 

Or, h-^r-dX tab. area X d.d' = area. 
Example. — The height, b i\ Fig. 9, is 5, and the axes 
of the ellipse are 30 and 20 ; what is the area of the seg- 
ment ? 

5-^-30 = .lG66 = faZmZar versed sine, the area of which 
(page 205) is .0^554. 
Hence .0S554X 30x20 = 51 324= area. 
Note. — The area of an elliptic segment may also be 
found by the following rule : 

Ascertain the segment of the circle described upon the 
same axis to which the base of the segment is perpendic- 
ular. Then, as this axis is to the other axis, so is the circular segment to the ellip- 
tical segment. 

Illustration. — In the above example, the axis to which the base of the segment 
is perpendicular is the conjugate, 50, and the height of the segment 25. Also, the 

1063.4954 
area of the segment is one half of that of a circle of 50 diameters— — ^ = 

DS1.7477. 
Hence 50 : 70 : : 9S1.75 : 1374.45 = area of elliptic segment. 



Fig. 9. 





PARABOLA. 

To Describe a Parabola, the Base and. 
Height Toeing given — Eig. lO. 

Operation.— Draw an isosceles triangle, as a b d, Fig. 10, the 
base of which shall be equal to, and its height, c 6, twice that 
of the proposed parabola. 

Divide each side, a b, d b, into any number of equal parts ; 
then draw lines. 11, 2 2, 3 3, etc., and their intersection will 
define the Giirve of a parabola. 

To Compnte either Ordinate or Abscissa of 
a Parabola, the other Ordinate and the 
Abscissa?, or the other Abscissa and the 
Ordinates being given — Fig. 11. 

Rule.— As either abscissa is to the square of its ordinate, sC 
is the other abscissa to the square of its ordinate. 

a 
o2xa' 



2. °^ = o>. 



o'*Xa _ 
o'2 o 2 

Or, as the square root of any abscissa is to its ordinate, so is the square root of any 
other abscissa to its ordinate. 



Hence — 



oX\/a' 



■\Ja 



CONIC SECTIONS. 



293 



Fig. 11. 




Example.— The absci-sa, a b, of the parabola, Fig. 11, is 9; 
is ordinate, b c, 6 ; what is the ordinate, de, the abscissa of 
which, a d, is 10? 

Hence 9 : 62 : : 16 : 64, and ^/G4 — 8 — length of ordinate. 

Or, V 9 : 6 : : V 1G : 8 = ordinate. 

Ex. 2.— The abscissae of a parabola are 9 and 16, and their 
corresponding ordinates 6 and 8; any three of these being 
taken, it is required to find the fourth. 

/6 2 Xl6 8 2 X9 

1. v — <j — == 8 = ordinate. 2. V-jtt- == <> == ordinate. 



3. 



62X16 



— 9 = /ess abscissa. 4. 



82X9 



== 16 = abscissa. 



PARABOLIC CURVE. 

To Compute trie Length, of tlie Curve of a 3?ara"bola exit 

oft" V.y a Doviole Ordinate — Fig. 11. 

Rule. — To the square of the ordinate add 4-3 of the square of the abscissa, and the 
square root of this sum, multiplied by two, will give the length of the curve nearly. 

Or, \/(° 2 ~\ — ~ ) 2 — length of curve. 

Example. — The ordinate, d e, Fig. 11, is 8, and its abscissa, a d, 16; what is the 
length of the curve, f a el 

4Xl6 2 
82-] : = 405.333 =zsum of square of the ordinate and 4-3 of the square of the 

abscissa, and ^405. 333 = 20.133, which x2 = 40.267 = length. 



To Compute tlie Area of a IParatoola-- 
Fig. is. 

Rule. — Multiply the base by the height, and take % of the 
product.* 2 

Or, § bxh = area. 

Example. — What is the area of the parabola, a be, Fig. 12, the 
height, b e, being 16, and the base, or double ordinate, a c, 16? 
16X16 = 256 = product of base and height, and % of 256-= 
170.667 = area. 

To Compute tlie Area of a Segixieiit of a I?ara"bola — Fig. IS. 

Rule. — Multiply the difference of the cubes of the two ends of the segment, a c, 
df by twice its altitude, e o, and divide the product by three times the difference of 
the squares of the ends. 
rf3cocT3 X 2fr 

^ d2cod'2xa 

Example.— The ends of a segment of a parabola, a c and df, Fig. 12, are 10 and 
6, and the height, e o, is 10; what is its area? 

103 co 62 x 10 X 2 = 1 56S0 = difference of cubes of the ends X twice the height. 
15GS0-^K)2co62x3 = SI. 667 = preceding product H- 3 times the difference of the 
squares of the ends = area. 

Note — Any parabolic segment is equal to a parabola of the same altitude, the 
base of which is equal to the base of the segment, increased by a third proportional 
to the sum of the two ends avid the lesser end. 

Illustration. — Tn Example 1 the base and end are 10 and 6. 

Then lO-j-6 : 6 : : 6 : 2.25 = third proportional to the sum of the two ends and the 
lesser end. 

Hence 10 -f- 2.25 = 12.25=: sum of length of base of parabola and third propor- 
tional, and the area then, the height being 10 = 8! .667. 




— = area, d and d' representing the lengths of the base and lesser end. 



* Corollary. — A parabola is % of its circumscribing parallelogram. 

Bb* 



294 CONIC SECTIONS. 



HYPERBOLA. 
To Describe a Hyperbola. 

(See HaswelVs Mensuration, page 246.) 

To Compute the Ordinate of a Hj-perbola, tlie Transverse 
and. Conjugate Diameters and tlie .A/bscissae being given 
—Fig. 13. 

Rule — As the transverse diameter is to the conjugate, so is the square root of the 
product of the abscissas to the ordinate required. 

Fig. 13. Or, cX ^ aXa ' = ordinate, 

d y b * 

Example. — The hyperbola, a b c, Fig. 13, has a 
transverse diameter, a t, of 120; a conjugate, df of 
72 ; and the abscissa, a e, is 40 ; what is the length of 
the ordinate, e c? 

40 -(-120 = 160 = sum of lesser abscissa and axis=. 
greater abscissa. 



f \c 120 : 72 : : V (40x160) : 4S = ordinate. 

Note In hyperbolas the lesser abscissa, added to the axis (the transverse diam- 
eter), gives the greater. 

2. — The difference of two lines drawn from the foci of any hyperbola to any point 
in the curve is equal to its transverse diameter. 

To Compute tlie .Abscissas, tlie Transverse and Conjugate 
Diameters and the Ordinate toeing given — Fig. 13. 

Rule. — As the conjugate diameter is to the transverse, so is the square root of the 
sura of the squares of the ordinate and semi-conjugate to the distance between the 
ordinate and the centre, or half the sum of the abscissae. Then the sum of this dis- 
tance and the semi-transverse will give the greater abscissa, and their difference 
the les=er abscissa. 

*a/o2-|-(c-^2)2 a -fa' , 7 „ , • ; , 

or, — ■ == — - — = half the sum of the abscissas. 

a-\-a' , t , a-\- a' t . 

■ — ! — = a, and — ■ = a. 

2 T 2 2 2 

Example.— The transverse diameter, a t, of a hyperbola. Fig. 13, is 120; the con- 
jugate, df 72; and the ordinate, ec, 4S; what are the lengths of the abscissae, te 
and a el 

72 : 120 : : y/±5* 4- (72 -f- 2)2 = 60 : 100 = half the sum of the abscissa. 
100-|- (120 -r- 2) = 160= above sum added to the semi-transverse == the greater ab- 
scissa ; and 
100 — (120 -H 2) = 40 = above sum subtracted from the semi-transverse = the lesser 
abscissa. 

To Compute the Conjugate Diameter, the Transverse Di- 
ameter, tlie .A-hscissoe, and Ordinate "being given-- T^ig. 13. 

Rule.— As the square root of the product of the abscissae is to the ordinate, so is 
the transverse diameter to the conjugate. 

Or, —. conjugate diameter. 

Example. — The transverse diameter, a b, of a hyperbola, Fig. 13, is 120 ; the ordi- 
nate, e c, 4S; and the abscissa), t e and a e, 160 and 40; what is the length of the 

conjugate, df? 

•V/40X100 = SO : 4S : : 120 : 72 = conjugate. 

To Compute the Transverse Diameter, the Conjugate, tlie 
Ordinate, and an A/bscissa "being given — T^ig. 13. 

Rule. — Add the square of the ordinate to the square of the semi-conjugate, and 
extract the square root of their sum. 



CONIC SECTIONS. 295 

Take the sum or difference of the semi-conjugate and this root, according as the 
ereater or lesser abscissa is used.* Then, as the square of the ordinate is to the 
product of the abscissa and conjugate, so is the sum or difference above ascertained 
to the transverse diameter required. 

Or, a or a'XcX(V<* + (c~~W± cT¥) -f- o* = transverse diameter. 
Example— The conjugate diameter, df of a hyperbola, Fig. 13, is 72 ; the ordi- 
nate, e c, 4S; and the lesser abscissa, a e, 40; what is the length of the transverse 
diameter, at? 

^4g2 _|_ (72 -i-2)2 = 60 = square root of the squares of the ordinate and semi-conju- 
gate. l 
60 -|-72-r-2 = 96 = sum of above root and the semi-conjugate (the lesser abscissa 

being used). 
40 xT2=i 2880 =. product of abscissa and conjugate. 
48 2 : 2SSQ : : 96 : 120 = transverse diameter. 

To Compute tlie Length of any Arc of a Hyperbola, com- 
mencing at tlie "Vertex — Fig. 14. 



Ritte— To 19 times the transverse diameter add 21 times the parameter of the 
axis, and multiply the sum by the quotient of the lesser abscissa divided by the 
transverse diameter. *iVJwIii« 

To 9 times the transverse diameter add 21 times the parameter, and multiply the 
«um by the quotient of the lesser abscissa divided by the transverse diameter. 

To each of the products thus ascertained add 15 times the parameter, and divide 
the former by the latter; then this quotient, being multiplied by the ordinate, will 
give the length of the arc nearly. 

^ 14 fXl9 + 2l3<Jx^ + 15Xp 

b * b Or,— -Xo= arc nearly. 

S\ tX 9 + 2lXpX^ + 15Xi? 

Example.— In the hyperbola, a be, Fig. 14, the transverse diameter 
i3 120, the conjugate 80, the ordinate, e c,48, and the lesser abscissa, a e, 
40 : what is the length of the arc, ab? 
129 : 80 : : 80 : 53.3333 ^parameter. 

120 X 19 + 53. 3333 X 21 X^ = 1133.3333 = product of the sum of 19 
times the transverse and 21 times the parameter, by the quotient o) 
the lesser abscissa and the transverse. 
123x9 + 53.3333X21 X-^- = 733.3333 —product of the sum of 9 times the trans- 
verse and 21 times the parameter, by the quotient of the lesser abscissa and 

transverse^ 

1133 333 + 53.333 X 15 -4- (T33.333 + 53.-B3 X 15) = 1.2609 = quotient of former 
product and 15 times the parameter -r- latter product and 15 times the parameter. 
1.2G09X48 = G0.5232 = above quotient X the ordinate — length. 

Note.— As the transverse diameter is to the conjugate, so is the conjugate to the 
parameter. (See Rule, page 290. ) 

To Compnte tlie Area of a Hyperbola, tlie Transverse, 

Con.jngate, and lesser Abscissa being given— Fig. 14. 

Rule.— To tlie product of the transverse diameter and* lesser abscissa add 5-7 of 
the square of this abscissa, and multiply tlie square root of the sum by 21. 

Add 4 times the square root of the product of the transverse diameter and lesser 
abscissa to the product last ascertained, :md divide the sum by 75. ; 

Divide 4 times the product of tlie conjugate diameter and lesser abscissa by the 
transverse diameter, and this last quotient, multiplied by the former, will give the 
area nearly. 



VtX ft + -g- a ,2 X21 + (V<Xa / X4) ^cX^x4_ 
75 



Or, v — X : = area. 



* When the greater abscissa is used, the difference is taken, and contrariwise. 



296 



PLANE TEIGONOMETRY. 



Example. — The transverse diameter of a hyperbola, Fig. 14, is 60, the conjugate 
36, and the lesser abscissa or height, a e, 20 ; what is the area of the figure ? 
60x20 -f | of 202 = 14S5.7143 = swm of the product of the transverse and abscissa 

and ^ of the square of the abscissa. 
V1435. 7 1^3x21 — 809.424 = 21 times the square root of the above sum. 
V6ux 20x4 + 809.424 = 947.988 = sum of above result and the root of 4 times the 

product of the transverse and abscissa. 
947.988 -4- 75 = 12.0393 = quotient of above result -=- 75. 
£0x20x4 
- — — X.12.G39S = 60G. 7 104 — vroduct of 4 times the product of the conjugate 

and abscissa -=- the transverse and the above quotient = area. 



PLANE TRIGONOMETRY. 

By Plane Trigonometry is ascertained how to compute or determine 
four of the seven elements of a plane or rectilinear triangle from the 
other three when one of the given quantities is a side, or the urea. 

The determination of the mutual relation of the Sines, Tangents, 
Secants, etc., of the sums, differences, multiples, etc., of arcs or angles 
is also classed under this head. 

For Explanation of Terms, see Geometry, page 1 63. 

When any three elements of a Plane Triangle are given, one of 
which being a side, or its area, the remaining elements may be de- 
termined ; and this operation is termed solving the triangle. 

RIGHT-ANGLED TEIAXGLES. 

For Solid' on by Lines and Areas, see also Mensuration, page 245. 

In the following figures, A = 90°, B = 45°, C = 45°, Radius = 1, Secant = 1-4142, 
Cosine =.7071, Sin 45° = .7071, Tangent = 1, Area = .25, Figs. 1 and 2, r.nd =.5, 
Figs. 3 and 4. 

By Sin., Tan., Sec, etc., etc., A, B, etc., is expressed the Sine, Tangent, Secant, 
etc., of the angle, A, B, etc. 

To Compute Sides A. C and. 13 A. — Figs. 1 and 2. 



(1.) 



(2.) 






*/ 


; 


f/ 


\ 


1 


*v 


« 


\ ,' 




73 


jj 



c Hyp. B C, (R. 1 : BC 1 : : sin. B .7071 : A C : .7071, Fig. 1. 
Given \ Leg. B A, JTL 1: BO 1 : : sin. C .7071 : B A : 7071, Fig. 1. 

( and Angles (Sin. C.7071 : : sin. B. .7071 : : B A .7071 : A C .7071, Figs. 1 & 2. 

To Compute Sides 13 .A. and 33 C--3Tigs. 3 and 4. 

(AC, (R. 1 : A C 1 : : tan. CI: B A 1, Fig. 3. 
Given 1 and ■{ R. 1 : A C 1 : : sec. C 1 4142 : B C 1.4149, Fig. 3. 
(Angles (Sin. B .7071 : A C 1 : : R. 1 : B C 1 414 2, Fig. 4. 



(a* 



PLANE TRIGONOMETRY. 
B 



297 





Tangent. 

To Compute tlie A.ngles and Side, .A. C — P^igs. 1 and 3. 

n . ( Hyp. BC /B C 1 : R. 1 : : B A .7071 : sin. C .7071— Fig. 1. 
Ulven \and Leg B A \R. 1 : B C 1 : : sin. B .7071 I A C. 7071— Hg. 2. 

To Compute Angles and Side, B C — Fig. 2. 

(A C .7071 : R. 1 : : B A .7071 : tan. C 1— Fig. 2. 

Given both Legs 1 Sin. C .7071 : B A .7071 : : R. 1 : B C 1— Fig. 2. 

(R 1 : A C .7071 : : sec. C 1.4142 : B C 1— Fig. 2. 

To Compute Area— Fig. 1. 
Given B A and A C . B Ax A C -h 2 = .7071X.7071 -4- 2 = area .25. 
'4 : B C2 1 : : sin. 2 C 1 : area .25— Fig. 1. 

: area .25 — Fig. 1. 



I Hyp. B C 
Given < and 
( Angle C 



k AC2 ' " .'5 , 
— Xtap. C = -Xl = 



'BA2 



X cot. C = ^-Xl = area. 25— Fig. 1. 



Giveni^ 1 » ■■■« i ' 



( BA -^ 

2 



.7071 



XV(1+.7071)X1 — .7071 =.3535 



BA)=- 

i id m 

\ x.7071 — area .25— Fig. 1. 

(5.) b Let B A C be a right-angled triangle, in which 

C A is assumed to be radius \ B A is the tan- 
gent of C, and B C its secant to that radius ; or, 
dividing each of these by the base, there is ob- 
tained the tangent and secant of C respectively 
to radius 1. 

Sine dg — .7071. 

Cosine C g or o d = .7071. 

Versed sine g A = .2929. 

Co- versed sine oe- .2929. 

Angle C A B = 90°. 




base 
hyp. 

base 
perp. 

hyp. 

base 

perp. 
hyp. 

perp. 
base 

hyp. 



Radius CA=1. 
Secant CBn 1.4142. 
Tangent A B = 1. 
Co-secant CB^ 1.4142. 
Cotangent e B = 1. 

— tan. angle C = 
= sec. angle C = 
= sin. angle C = 
= tan. angle B = 
= sec. angle B = 
= sin. angle B = x , . 



.7071 


— 


1. 


1.4142 
1 


= 


1.4142. 


1 


= 


.7071. 


1.4142 


1 
1 


= 


1. 


1.4142 
1 


= 


1.4142. 


1 


__ 


.7071. 



298 



PLANE TRIGONOMETRY. 






B A 



5. 



A C 

Sin. C 
(Jos. C 

. 1 — C08. C 
1 

coZ~c — 



= tan. C = 



Formulae — Fig. 5. 

1 AC 

B~C 

2 Area 



-..7071. 

:1. 



-0=™=! 



2. 
4. 

C. 



AC2 
1 



Cot. C 
versin C = 1 — .7071 = .2929. 



= cos. C = 
= tan. C = 
== tan. C = 



1.4142 
.5X2 

13 

1 

1 



.7071. 
= 1. 



ec. C 4 



10. V A( ^ 2 + ^ A2 = 



.7071 

1 
hyp. B G 

1 
~~ .7071" 



1.4142. 



9. 



BC 



B A 

VI + 1 = 1.4142. 
AC 



:sec.C= 1 -^? = 1.4142. 



: 1.4142. 12. 



_hyp.BC = — — =1.4142. 

COS. C .*0;1 



13. 2 v / Are f , = hyp. B C = *?/&£= 1.4142. 

V am. ^g V i- 

14. B C X cos. C = radius = 1.4142 X .7071 == 1. 

15. B A X cot. C == radius =1x1=1. 

16. B C X sin. C = radius = 1.4142 X .7071 = 1. 



17. BAxtan.B = radius = lXl = l. IS. . I* Area = radius = A /-^^= 1. 

V Tan. C V 1. 

19. Sin. C 2 -f Cos. 2 — radius 2 — .70712 -f .70712 = 1. 

20. B C X sin. C = perp. B A =■ 1.4142 X .T071 = 1. 

21. A C X tan. C = perp. BA=lXl = l. 



22. -— — = cosec. C = — L = 1.4142. 
Sin. C .7071 



23, 



Cos. C 



cot. C = — r- = L 

.iOil 



24. 



— — = cot.C = i = l. 
Tan. C 1 



Sin. C 
25. 1 — sin. C = co- versin. = 1 — .7071 = .29:9. 



26. 2 sin. C. cos. C = sin. 2 C = 2 X .7071 X .7071 = 1. 



27. : y/ sin. C- 4- versin. C 2 = sin. - C = — 



.70712 -f .29292 _ 



: .3S2GS. 



2S. Sin. C. cos. B ± sin. B cos. C = sin. (C ± B) = .7071 X .7071 -f .7071 X .7071 
= 1, or (45°— 45°; = 0; .7071X-7071 — .70T1X-70T1 = 0. 

iLLFSTRATiox.—TIie side, B A, of a right-angled triangle is 100 feet: the angle 
C = 30° ; and B = 60° ; what are the lengths of the side^, B C and A C ? 

10( |, n = — = 200 feet, B C ; 200 X cos. 30° = 200X.SGG03 = 173.206 feet, A C. 
Sin. o0° .5 

2.— Side B C = 200 feet, and ansrle C = 30° ; what is the length of side B A? 

200 X sin. 30° = 200 X .5 = 100 feet. 



OBLTQUE-ANGLED TRIANGLES. 
B B 




(7.) 




PLANE TRIGONOMETRY. 299 

Fig. 6. Fig. 7. 

A = 116°.30'; B = 1S°.30'; C = 45°. A — 63°.30'; B = 7l°.30'; C = 45°. 

BA = 1.1174; A C = .5014; BC = 1.4142. 

Area = .2507 ; S hz~% sura of sides = 1.51G5. 
BA = 1.1174; A C = 1 ; BC=: 1.4142. 

Area = .7493 ; S = % sum of sides = 2.0151. 



To Compute 33 .A. and J±. C — ITig. <3. 

r . _ n /Angles and /Sin. A .89493 : B C 1.4142 : : sin. C .7071 : 
^iven j gide B c j gin> A #89493 . B c 1<4142 . . shh B >|>1T3 . 



B A 1.1174. 
A C .5014. 



To Compute Angles 13 and. C and Side A. C — Figs. 6 and T. 

(B C 1.4142 : sin. A .89493 : : B A 1.1174 : sin. C .7071, which 

(A B B C \ an £ ! e added to A and the sum subtracted from 180° =£ angle 

Given -^and AiMeV of B_ Fig - G = ^^.SO', and which added to A and the sum 
lrt\en < auu au & ic <v gubtracfced from 1SQ o _ angle of B __ F ; g> 7 _ 71 o >30 / # 

^ /Sin. A .83493 : B C 1.4142 : : sin. B .3173 : A C .5014— Fig. 6. 

^Sin. A .89493 : B C 1.4142 : : sin. B .94832 : A C 1 .4986— Fig. 7. 

To Compute B C and B A. — Figs. 6 and T'. 
, k C f Sin. B ,3173 : A C .5014 : : sin. A .89493 : B C 1.4142— Fig. 6. 
PivPTi J f»ui J Sin - B - 94832 : A c 1 - 49S6 : : sin - A - 89493 : B c 1-4142— Fig. 7. 

1 A^Ip- 1 Sin - B - 31T3 : A c - 5014 : : sin - c - -™n : B A l.HT4-Fig. 6. 
VAn Q ie & ^ gim B <94832 . A c 14Qm . . gin> c 7Qn . B A 1>1174 __ Fi& 7J 

To Compute Angles B and C and Side IB C — Figs. 6 and T. 

r Subtract half of the given angle, A, from 90° ; the remain- 
der is half the sum of the other angles. 
[A C, A B, Then, as the sum of the sides, A C, A B, is to their differ- 
Given < and Angle 1 ence, so is the tangent of the half sum of the other angles to 
( A the tangent of half their difference, which, added to and sub- 

tracted from the half sum, will give the two angles B and C, 
Jihe greatest angle being opposite to the greatest side. 

Opeeation. 90° — Z_ A -r- 2 = 90° — 116°.30' #■ : 2 = 31°.45' = - — ^ -— 

half sum of the other angles. 

Then A C + A B = . 5014+1.1174 = 1.61SS : A C co A B = 1.1174 — .5014 = .6160 : : 
tan. /_ B + C -=- 2 = 1S°.3U' + 45°-4- 2 = 31°.45' = .61SS2 : tan. /_ B co C -h 2 = 
450 _ 1S c >30 /_i_ 2 — 130. 15/ — .23547; which, being added to 31°.45' (the half sum) 
= 13 .15 / + 31 .45 / = 45 = /_C, opposite to side A B, and 31°.45' — 13°.15' = 18°.30' 
Z_ B, opposite side A C. 

To Compute all tlie Angles — Figs. G and *7. 

Let fall a perpendicular, B d, opposite to the required angle. 
Then, as A C : sum of A B, B C : : their difference : twice d g, the 
Given all ^distance of the perpendicular, B d, from the middle of the base, 
three sides. \ Hence A d, C g are known, and the triangle, A B C, is divided into 
f two right-angled triangles, B C d, B A d ; then, by the rules in right- 
-angled triangles, ascertain the angle A or (J. 

Operation.— A C, Fig. 6, .5014 : A B + B C 1.1174 + 1.4142 = 2.5316 : : A B co 
B C 1.4142 — 1.1174 = .2968 : 2 X d #=1.4986. 
1 4986 .5014 

Hence Ad—dg— A C -4- 2 = ^—— — = .4986, and C d = A d + A c = 1. 

2 2 

Consequently, the triangle B d C is divided into two triangles, B A C and B d A. 

Again, A (Fig. 7) 1.498'J : A B + B C 1.1174 + 1.4142 = 2.5316 : : A B co B C 
1. 4142 — 1. 117 4 = .2908 : 2x d g = 5014. 

Hence A d = dg— AC-f-2 = .5014 -f- 2 — 1.4986 -4- 2 = .4986, and d C = A C — 
A rf=1.49S6 — .4986 = 1. 

Consequently, the triangle A B C is divided into two triangles, B d C and B d A. 
Then, by the preceding rules for right-angled triangles, ascertain the angle A or C. 



300 PLANE TRIGONOMETRY. 



Formulas — Figs. 6 and 7. 
BC ri..B = s . n _ A = 1.4142X^70 • ^ 
A C .5014 

. BC.sin. C . A 1.4142X.T071 __,_. 

2. BA = sm. A = LnT4 = .89493. 

q AC. sin. A . .5014X.S0493 Q ._ 

3. Bc =bulB= La4a = .3U3. 

. AC. sin. C . ■ .5014X.T0T1 -» 

4 BA = sin. B = 11174 = .31,3. 

K 2 Area . M .2507x2 

* AUTB13 = Sm ' C 7 .5014X1.4142 = iWll: 
BA.sin.A 1.1174X.89493 

B(J 1.4142 

T. BA ; SiD - A = hyp. BC i ™%™™ = ,.41431 

sm. (J J * .70.1 

.AC. sin. A ^ .5014X.S9413 *_ 

8. : — - — =c hyp. B C == ^^ = 1.4142. 

sm. B ,31io 



_ / 2 Area. sin. A t ^ ri __ /. 2507 X 2 X- 89493 ,_ L ... 40 

9 - V sin.C.sin.(A + C) = hyP " BC ^V .7071X.3173 = ^ = 1AU2 ' 



10. BC . gi " C ,BA, 1 - 414 Q 2 ^! ^ = 1.H74. 
sm. A .89493 

11 B C • sin - C t*a 14142X.7071 

U * 3E«0+» " .89493 • = 1 ' 11T4 * 

2 Area .2597x2 

12 ' AC.sin.A = BA ^ .5Jl4X.894:3 = 1 - 11<4 ' 



13 / 2 Area . sin. C / .2507x2x.7Q71 _ 

' Vsm. B.sin. (C + B) "~ A_ V .3173X. 89493 ~~ T4 " 

14. AC ;t C -BA^ ' 5014 X- 7071 ^l.H74 
sin. B .3173 ****** 

B C . sin. B 14142X.3173 KA1 ^ 

15 ' sin. a = A ° = -T89493— f - 5 ° 14 

2 Area . „ .2507 X. 2 

16 - EU^ioTo = A C = 1.4U2X.T0.1 = - 5014 - 



17 Z 2 Area ■ 8ln - ( c + A > . P /. 5j7x*X.3173 : 

"• V sin. C . sin. A ~ A C ~ V .T071X.S9403 = - 5014 
M. AC.BC.riD.C _ area = .5014X14142X7071 = 

* 2 

1Q BA. A C . sin. a 1.1174X.5014X. 89493 mm 
19 2" = area = = 2507. 

B C2 . sin. C. sin. B 1.41422*7071 X 3173 
20 ' -T^mTTBW- = area = 2X789493 = ' 250T - 



21. y/S. (S — B (J) (S — B A) (S — A C) = area = V*-5165 X (1.5165 — 1.4142) X 
(1.5105 — 1.1174) X (1.5165 — .5014) = V-062S534S = .2507. 



NATURAL SINES AND COSINES. 



301 







Taole of Natural S 


ines 


and. Cosines. 






o *- 






0° 


1° 


2° 


3° • 


4° 




O fe 

Pu a. 


29 


, 


N.sine. 


N. coa. 


N sine. N cos. 


N.sine. j N. eos. 
03l9~!99~37 


N.sine. 
05.34 


N. cos. 


N.sine. N. cos. 




2 








00000 1. 


01745 99985 


90863 


0u976 99756 


60 


2 





1 


00029 1 . 


01774 99984 


03519 9993S 


05263 99861 


07005 99754 


59 


2 


1 


2 


00058 1. 


01 803 i 99984 


0354S 99937 


.05292 9986 


07084 99752 


58 


2 


1 


3 


000S7 .1. 


01832 99983 


03577 199936 


05321! 99858 


07063 9975 


57 


2 


2 


4 


00116 1. 


01S62 999S3 


03606 99935 


0535 199857 


07092:99748 


56 


2 


2 


5 


00145 1. 


01S9l;99l)S2 


03635 j 99: 34 


05379|99855 


07121 ! 99746 


55 


2 


3 





00175 


1. 


0192 ;99.8> 


03664 '99933 


05408 99854 


0715 |99744 


54 


2 


3 


7 


00204 


1. 


01949 99981 


0361399932 


05437 99852 


07179 99742 


53 


2 


4 


8 


00233 


1. 


01978! 9998 


03723 99931 


C5466I 99851 


07208J9974 


52 


2 


4 


9 


00262 


1. 


02007J 9998 


03752 9993 


05495 91849 


07237 '99738 


51 


2 


5 


10 


00291 


1. 


02036 99979 


03781 


99929 


05524:99847 


07266 99736 


50 


2 


5 


11 


001 


.99999 


02065 99979 


0381 


99927 


05553 99846 


07295:99734 


49 


2 


6 


12 


0/J49 

0037'. 


99999 


02094 99978 


03839 


99926 


05582 


99844 


07324 99731 


4S 


2 


6 


13 


99999 


02123 199977 


03868 


99925 


05611 


99842 


07353 99729 


47 


2 


7 


14 


ooj n 


99999 


02152 199977 


03897 


991 24 


0564 


99841 


07382 : 99727 


46 


2 


7 


15 


0'.430 


99999 


02181! 99976 


03926 


99923 


C5069 


99839 


07411 ! 99725 


45 


2 


8 


16 


00465 


99999 


02211 99976 


03955 99922 


95698 99S38 


0744 |99723 


44 


1 


8 


17 


00495 


99999 


0224 


99975 


03984 1 99(21 


05727 99836 


07409,99721 


43 


1 


9 


IS 


00524 


99999 


02269 


99974 


04013 


99919 


05756 9H 834 


0749SI99719 


42 


1 


9 


19 


00553 


99998 


02298 99974 


04042 


999 IS 


05785 99833 


07527199716 


41 


1 


10 


20 


00582 


99998 


02327:99973 


04071 


99917 


05814 99831 


07556 199714 


40 


1 


10 


21 


00611 


99998 


02356! 99972 


041 


99916 


05S44 ! 99829 


075S5 99712 


39 


1 


11 


22 


0064 


99998 


023S5 : 99972 


041'29 


99915 


05S73 99827 


07614 


9971 


38 


1 


11 


23 


00669 


99998 


02414 99971 


04159 


99913 


05902 


99826 


07643 


99708 


37 


1 


12 


24 


00698 


99998 


02443 9997 


04188 99912 


05931 


99S24 


07672 


99705 


36 


1 


12 


25 


00727 


99997 


02472 99969 


04217 


91)911 


0596 


99822 


07701 


99703 


35 


1 


13 


26 


00756 


99997 


02501 99969 


04246 


9991 


05989 


99821 


0773 


99701 


34 


1 


13 


27 


00785 


99997 


0253 99968 


04275 


99909 


06018 


99819 


07759 


99699 


33 


1 


14 


2S 


00S14 


99997 


0256 199967 


04G04 


99907 


06047 


99S17 


0778S 


99696 


32 


1 


14 


29 


00844 


99996 


025S9, 99966 


04333 


99906 


06076 


99815 


07817 


99694 


31 


1 


15 


30 


00S73 


99996 


02618 99966 


04362 


99905 


06105 


998' 3 


07846 99692 


30 


1 


15 


31 


00902 


99996 


02647199965 


04391 


99904 


06134 


91812 


07S75 99689 


29 


1 


15 


32 


00931 


99996 


02676! 99964 


0442 


99902 


06163 


99S1 


07904 


99687 


28 


1 


16 


33 


0096 


99995 


02705 99963 


04449 '99901 


06192 


99808 


07933 


99685 


27 


1 


16 


34 


00980 


99995 


02734 ! 99963 


04478:999 


06221 


99806 


07962 


99683 


26 


1 


17 


35 


01018 


99995 


02763 1 99962 


04507 I9989S 


0625 


99S04 


07991 


9968 


'25 


1 


17 


30 


01047 


99395 


02792,99961 


04536 9; 897 


06279 


99803 


0802 


99678 


24 


1 j 


18 


37 


01076 


99994 


02S21 1 99.-6 


045L599896 


06308 


99801 


08049 


99676 


23 


1 


18 


38 


01105 


99994 


02S5 99959 


0459419894 


06337 


99799 


08078 


99673 


22 


1 


19 


39 


01134 


99994 


02S79 99959 


04623 If 989 3 


06366 


99797 


08107 


99671 


21 


1 


19 


40 


01164 


99993 


02908 ! 99958 


04653 1 9981 2 


06315 


99795 


08136 


99668 


20 


1 


20 


41 


01193 


99993 


0293S ! 99957 


046S2 9989 


06424 


99793 


08165 


99666 


19 


1 


20 


42 


01222 


99993 


02967 99956 


04711 


99S89 


06453 


99792 


08194 


99664 


18 


1 


21 


43 


01251 


99992 


02996:99955 


0474 


9988S 


064S2 


9979 


0S223 99661 


17 


1 


21 


44 


0128 


99992 


03025 99854 


04769 


99SS6 


06511 


99788 


08252 99659 


16 


1 


22 


45 


01309 


99991 


03054 99953 


0479S 


99SS5 


0654 


99786 


08-281 


99657 


15 


1 


22 


46 


0133S 


99991 


030S3J 99952 


04827 


998S3 


06569 


99784 


0S31 


99654 


14 





23 


47 


01367 


99991 


03112 99952 


04856 


99SS2 


0059S 


99782 


08339 


99652 


13 





23 


48 


01396 


9999 


03141 


99951 


048S5 99881 


06027 


997S 


0836S 


99649 


12 





24 


49 


01425 


9999 


0317 


9995 


04914 99S79 


06656 


9977S 


0S397 99647 


11 





24 


50 


01454 


99989 


03199 


99949 


04943 j 99878 


066S5 


99776 


0S4°6 99644 


10 





25 


51 


01483 


99989 


03228 99948 


04972 99876 


06714 


99774 


08455 


99642 


9 





25 


52 


01513 


99989 


03257 


99947 


05001 


99875 


06743 


99772 


084S4 


99639 


8 





26 


153 


01542 


999S8 


03286 


99946 


0503 


99S73 


06773 


9977 


0S513 


99637 


7 





26 


54 


01571 


999S8 


03316 


99945 


0505) 


99872 


06802 


99768 


0S542 


99635 


6 





27 


55 


016 


99987 


03345 


99944 


0508S9987 


06831 


99766 


0S571 


99632 


5 





27 


66 


01629 


99987 


03374 !99943 


05117 99869 


0686 


99764 


086 


9963 


4 





28 


57 


0165S 


999S6 


03403 


99942 


05146 9f 867 


06889 


99762 


0S023 


99627 


3 





28 


58 


01087 


99986 


03432 


99941 


05175 99866 


069 IS 


9976 


08658 


99625 


2 





29 


59 


01716 


99985 


03461 


9994 


05205 99864 


06947 


99758 


086S7 


99622 


1 





29 


60 


01745 


99985 


0349 


99939 


05234,99863 


06376 


99756 


0S716 


99619 












N.cos. 


N. sine. 


N.oos. N.sine. 


N.cosi N.sine. 


N.cos. 


N.sine. 


N. cos. 


N.sine. 






I 




89° 


8 


S° 


S 


7° 


8t 


>° 


8{ 


)° 







302 



NATURAL SINES AND COSINES. 



Ta"ble-~ (Continued). 



If 

a. c 




E 


o 


• 6 3 


7° 


i S3 


9° 




a. a, 


29 




X. sine. N. cos. 


N sine. N. cos 


N.sine. N.cos'N sine. N. cos. 


N.sine. N cos. 




4 








I.S716 99619 


10453 9 9452 


121o7 9*255 13917 99027 


15643 


98 : 69 


60 


4 





1 


03745 99617 


104S2 99449 


1:216 99251 


13946 99023 


15672 


! S764 


59 


4 


1 


2 


0S774 99014 


105119 446 


12245 99248 


13975 99019 


15701 


9376 


53 


4 


1 


3 


OSS03 9.612 


1054 99443 


12274 '99244 


14"04 99015 


1573 


98755 


57 


4 





4 


0S331 


93609 


10569 9944 


123 .12 9924 


14033 99011 


15753 


93751 


56 


4 


2 


5 


0836 


99607 


105j7 994-37 


12331 99237 


14061 99006 


15787 


98746 


75 


4 


3 


6 


03831 


99604 


10626 99434 


1236 99233 


14 )9 99002 


15316 


18741 54 


4 


8 


7 


03913 


99602 


10655 99431 


12339 9923 


14119 9899S 


15845 


9S737 


53 


4 


4 


8 


08947 


9;533 


106S4 9942S 


12413 99226 


14143 9S994 


15S73 


98782 


52 


3 


4 


9 


88376 


99536 


10713 99424 


12447 99222 


14177 9S99 


15902 


93728 


51 


3 


5 


10 


09005 


995 4 


10742 99421 


12476 99219 


142(5 98986 


15931 


9-728 


50 


3 


5 


11 


09034 


99531 


10771 99418 


1-504 99215 


14234 9S932 


15959 


' 9S71S 


49 


3 


6 


12 


09063 


995S3 


108 99415 


12533 99-11 


14203 93973 


159-3 


98714 


4> 


3 


€ 


13 


03092 


99536 


10329 99412 


1256 J 99203 


14292 9S973 


16017 


98709 


47 


3 


7 


u 


09121 


99533 


10353 99409 


12591 99204 


1432 9S969 


16046 


9S704 


46 


3 


7 


15 


0315 


9953 


10837 99406 


1262 992 


143.9 98965 


16074 


987 


45 


3 


6 


16 


09179 


99578 


10J16 99402 


12649 99197 


14578 9S961 


16103 


93695 


44 


3 


6 


17 


09308 


99575 


10345 99399 


12678 99193 


14407 9835,7 


16132 


9S69 


43 


3 


9 


IS 


09237 


99572 


10373 93396 


12706 991S3 


14436 9S953 


1616 


9S6S6 


42 


3 


9 


19 


09266 


9957 


11002 993 3 


12735 991S6 


14464 9894S 


16189 


98081 


41 


3 


10 


■20 


>2 5 995 J7 


11031 9939 


T-764 991S2 


14493 98944 


16218 


9-676 


40 


3 


10 


21 


03324 


99564 


1106 993S6 


12793 9917S 


14522 9394 


16246 


98371 


39 


3 


11 


22 


09353 


99562 


11039 993S3 


12822 99175 


14551 9S93G 


162; 5 


98667 


38 


3 


11 


23 


03332 


99553 


11118 993S 


12851 99171 


1453 93931 


16304 


98682 


87 


2 


12 


24 


09411 


99556 


11147 99377 


1233 9.167 


14603 98927 


16333 


98657 


36 


2 


12 


•25 


0944 


99553 


11176 99374 


12903 99163 


14637 9S923 


16361 


98652 


35 


2 


13 


26 


0T469 


99551 


11205 9937 


12937 9.M6 


14666 93919 


1639 


9364S 


34 


2 


13 


27 


094 '8 


9354S 


11234 99367 


12966 99156 


14695 9S914 


16419 


9S643 


33 


2 


14 


28 


03527 


99545 


11263 99364 


12995 99152 


14723 9S91 


16447 


9S63S 


32 


2 


14 


29 


03556 


99542 


11231 9936 


13024 991 4S 


14752 98906 


16476 


98633 


81 


2 


15 


30 


03585 


9954 


1132 99357 


13053 99144 


14781 98902 


16595 


9S629 


30 


2 


15 


81 


03614 


99537 


11349 99354 


130S1 99141 


1431 I9SS97 


16533 


5 3624 


29 


2 


35 


32 


642 


99534 


1137S 99351 


1311 99137 


14338 : 9SS93 


1(5 2 


98619 


28 


2 


1C 


33 


09671 


9953 1 


11407 99347 


13139 93133 


14S67 9-339 


1«391 


93814 


27 


2 


16 


34 


097 


9952S 


11436 99344 


13163 99129 


14396 9S334 


1662 


98809 


26 


2 


17 


85 


09729 


99526 


11465 99-341 


13197 93125 


14925 9S33 


16643 


9-604 


25 


2 


17 


36 


09753 


99523 


11494 99337 


13226 99122 


14 54 93876 


16677 


9<6 


24 


2 


18 


87 


03787 


9952 


11523 99334 


13254 9911S 


14982 98871 


10706 


98595 


23 


2 


IS 


38 


09S16 93517 


11552 99331 


13233 99114 


15011 9S367 


16734 


9859 


22 


1 


19 


39 


10845 99514 


115 3 99327 


13312 9911 


1504 I9S863 


16763 


98585 


21 


1 


19 


40 


09874 


99511 


11609 99324 


13341 99106 


15969 9SS53 


16792 


9858 


20 


1 


20 


41 


903 


9! 5 3 


136-38 9932 


1337 99102 


15097 9S854 


1632 


98575 


19 


1 


20 


42 


09932 


99506 


11667 99317 


13399 9903S 


15126 9S649 


16S49 


9857 


18 


1 


21 


43 


03961 


99503 


11696 99314 


13427 9 034 


15155 9SS45 


10^78 


985 5 


17 


1 


21 


44 


0399 


995 


11725 9931 


13456 99991 


151S4, 98841 


16906 


985 51 


16 


1 


22 


45 


10919 


99437 


11754 99307 


13485 990S7 


15212 9S336 


16835 


98556 


15 


1 


22 


46 


10043 


99494 


11783 99303 


13514 99083 


15241 9S332 


16964 


! 8551 


14 


1 


23 


47 


100T7 


99491 


11813 993 


13543 99079 


15 7 98827 


16992 


98540 


13 


1 


23 


4S 


10106 99433 


1184 99297 


13572 99075 


15299 9SS23 


17ii21 


98541 


12 


1 


24 


49 


10135 99435 


11863 99293 


136 99171 


15327 98818 


1705 


93536 


11 


1 


24 


50 


10164 99482 


1189S 9929 


1H629 99067 


15 56 98614 


17073 


98531 


10 


1 


25 


51 


10192 99479 


11927(99283 


13653 99063 


15385 98S09 


17107 




9 


1 


25 


52 


10221 99476 


11956 99283 


136S7 99059 


15414'9SS05 


17136 


98531 


S 


1 


20 


53 


1025 99473 


11985 99279 


13716 99055 


15442 ! 


17164 


98516 


7 





26 


54 


10279 9947 


12014 99276 


18744 99051 


15471 9S796 


17193 


9>511 


6 





27 


55 


10303 93467 


12043 99272 


13773 99i47 


155 93791 


17222 


93506 


5 





'-'7 


55 


10337 99464 


12071 99269 


13802 99043 


15529 9S7S7 


1725 


98501 


4 





2S 


57 


10366 99461 


121 99265 


13831 9903) 


15557 987S8 


17279 


98496 


3 





2S 


■ 


10395 9945 > 


12129 99262 


138.1 99035 


155S6 ! 8773 


173i 3 


98491 


2 





29 


59 


10424 99455 


1215S 99253 


138&1 99031 


15615 ' 9S773 


17380 


98436 


1 





29 


00 


10453 99152 


12187 99255 


13»17 99027 


15S43 98769 


17365 


984S1 








1 


N.sine. 


N. cos. N.sine. N. cos. N.sine. 


N.sine. 


N. ens. .N.sine. 


' 








8* 


1° 


83° 


82° 


trt° 


6C 


>° 







NATURAL SINES AND COSINES. 



303 













Tal 


Die- 


(Continued). 












K 


. 10° 


11° j 12° ] 13° 


14° 




II 


28 


/ 


N.|sine. N. cos. 


N. sine 


■ N cos 


N.sine 


N. cos. 'N. sine. N. cos. 


N.sine.| N. cos. 




6 








17365: 98481 


19081 


98163 


20791 


, 97815 


22495 97-137 


24192 


9703 


60 


6 





1 


17393 1 98476 


1910.) 


98157 


20S2 


| 97809 


22523 


9743 


2422 


97023 


59 


6 


1 


2 


17422 98471 


1913S 


98152 


20S4S 


! 97803 


22552 


97424 


24249 


97015 


58 


6 


1 


3 


17451 ;9S466 


19167 


£8146 


20877 


97797 


2258 


£7417 


24277 


9700S 


57 


6 


2 


4 


17479 |9S461 


19195 


9814 


20905 


97791 


22608 


97411 


24305 


97001 


56 


6 


2 


5 


17508 ! 98455 


19224 


98135 


20933 


97784 


22637 


97404 


24333 


£6994 


55 


6 


3 


6 


17537 


9845 


19252 


98129 


20962 


97778 


22665 


9739S 


24362 


96987 


54 


5 


3 


7 


17565 


98445 


19281 


98124 


2099 


97772 


22693 


97391 


2439 


9698 


53 


5 


4 


8 


17594 


9844 


19809 


9811S 


21019 


1*7766 


22722 


973S4 


24418 


96973 


52 


5 


4 


9 


17623 


98435 


1933S 


98112 


21047 


9776 


2275 


97378 


24446 


96966 


51 


5 


5 


10 


17651 


9843 


19366 


9S1C7 


21076 


97754 


22778 


97371 


24474 


96959 


50 


5 


5 


11 


170S 


98425 


193:' 5 98101 


21104 


97748 


22807 


97365 


24503 


96952 


49 


5 


6 


12 


17708 


9842 


19423 


18096 


21132 


97742 


22835 


97358 


24531 


96945 


4^ 


5 


6 


13 


17737 


98414 


19452 


1S09 


21161 


97735 


22863 


97351 


24559 


96937 


47 


5 


7 


14 


17766 


98409 


19481 


98084 


211S9 


97729 


22892 


97345 


24587 


9693 


46 


5 


7 


15 


17794 


98404 


I. 509 


9S079 


2121S 


97723 


22.12 


9733S 


24615 


96923 


45 


5 


7 


16 


17823 


98399 


19538 


98073 


21246 


97717 


22948 


97331 


24644 


96916 


44 


4 


8 


IT 


17852 


98394 


1! 566 


98067 


21275 


97711 


22977 


97325 


24672 


96909 


43 


4 


8 


18 


1788 


98389 


19595 


9S061 


21303 


97705 


23005 


9731S 


247 


96902 


42 


4 


9 


19 


17909 


98383 


19023 


18)56 


21331 


97698 


23033 


97311 


24728 


96894 


41 


4 


9 


20 


17: 37 


98378 


19652 


98 5 


2136 


97692 


23062 


97804 


24756 


968S7 


40 


4 


10 


21 


17966 


98373 


1968 


98044 


21C8S 


97686 


2309 


9729S 


94784 


9688 


39 


4 


10 


22 


17915 


9836S 


19709 


£8039 


21417 


9768 


23118 


97291 


24813 


96873 


38 


4 


11 


23 


18023 


9S362 


19737 


98033 


21445 


97673 


23146 


972S4 


24S41 


96866 


37 


4 


11 


24 


18052 


9S357 


19766 


9S027 


21474 1 97667 


23175 


97278 


24869 


9685S 


36 


4 


12 


25 


1S0S1 


9S352 


19794 


98021 


21502 


97661 


23203 


97271 


24897 


96851 


35 


4 


12 


26 


18109 


£8347 


19823 


98010 


2153 


97655 


28231 


97264 


24925 


96S44 


34 


3 


13 


27 


18188 


9S341 


19851 


9801 


21559 


97648 


2326 


97i57 


24954 


96S37 


33 


3 


13 


28 


18166 


9S336 


1988 


9S004 


215S7 


97642 


232SS 


97251 


249 S2 


96829 


32 


3 


14 


29 


18195 


98331 


19908 


97998 


21616 


97636 


23316 


97244 


2501 


96S22 


31 


3 


14 


30 


1S224 


98325 


19937 


97992 


21644 


9763 


23345 


97237 


25038 


96S15 


30 


3 


14 


31 


18252 


9S32 


19965 


97987 


21672 


97623 


23373 


9723 


25006 


96S07 


29 


3 


15 


32 


18281 


9S315 


19994 


97981 


21701 


97617 


23401 


97223 


25094 


£68 


28 


3 


15 


33 


18309 


9831 


20022 


97975 


21729 


97611 


23429 


97217 


25122 


£6793 


27 


3 


16 


34 


1S338 


98304 


20051 


97169 


2175S 


97604 


23458 


9721 


25151 


96786 


26 


3 


16 


35 


18367 


982 9 


20079 


97963 


21786 


9759S 


234S6 


£7203 


25179 


96778 


25 


3 


17 


30 


18395 


98294 


2010S 


9795S 


21814 


97592 


23514 


97196 


25207 


96771 


24 


2 


17 


3T 


1S424 


9S2>8 


20136 


97952 


21843 


975S5 


23542 


971S9 


25235 


96761 


23 


2 


18 


SS 


18452 


9S2S3 


20165 


1,7946 


21S71 


97579 


23571 


97182 


25263 


96756 


22 


2 


18 


39 


184^1 


98277 


20193 


1794 


21899 


97573 


235911' 


97176 


25291 


96749 


21 


2 


19 


40 


1S509 


98272 


20222 


97934 


21928 


97566 


23627 


97169 


2532 


£6742 


20 


2 


19 


41 


1853 S 


98267 


2025 


979:8 


21956 


9756 


23656 


97162 


25348 


96734 


19 


2 


20 


42 


18567 


98261 


20279 


97922 


21:85 


97553 


22684 


97155 


25376 


96727 


18 


2 


20 


48 


18595 


98256 


20307 


97916 


22013 


97547 


23712 


9714S 


25404 


96719 


17 


2 


21 


44 


18624 


9S25 


20336 


9791 


22041 


97541 


2374 


97141 


25432 


£6712 


16 


2 


21 


45 


1S652 


98245 


20364 


97905 


2207 


97534 


23769 


97134 


2546 


96705 


15 


2 


21 


46 


1S6S1 


9824 


208193 


97S99 


22098 


97528 


23797 


97127 


2548S 


96697 


14 


1 


22 


47 


1871 


9S234 


20421 


97803 


22126 


97521 


23825 


9712 


25516 


9669 


13 


1 


22 


48 


18738 


98229 


2045 


978S7 


22155 


97515 


23853 


97113 


25545 


966S2 


12 


1 


28 


49 


1S767 


98223 


2047S 


97881 


221S3 


97508 


23SS2 


97106 


25573 


96675 


11 


1 


23 


50 


18795 


9S218 


20507 


97875 


222121 


97502 


2391 


971 


25601 


96667 


10 


1 


24 


51 


1SS24 


98212 


20535 97869 


2224 | 


97496 


23938 


97093 


25629 


9696 


9 


1 


24 


52 


18852 


98207 


20563 97893 


22208 974S9 


23966 


970S6 


25657 


96653 


8 


1 


25 


5; 


18881 


9S201 


20592 


97S57 


22297 97483 


23995 


97079 


256S5 


9C645 


7 


1 


25 


54 


1891 


98196 


2062 


97851 


22325 97476 


240 .3 


97072 


25713 


96638 


6 


1 


26 


55 


18 ; 38 


9S19 


20349 


97845 


SftBffS 19940 


24051 


97065 


25741 


9663 


5 


1 


26 


56 


189(57 


98185 


20677 


97839 


22382|97463 


24079 


97058 


25769 


96623 


4 





27 


57 


18995 


98179 


20706 


97833 


2241 97457 


24108 


97051 


25798 


96615 


3 





27 


58 


19024 


(8174 


20734 


97827 


22438,9745 


24136 


97044 


25826 


9660S 


2 





28 


59 


19052 


'98168 


20763 


97821 


22467 97444 


24164 


97037 


25851 


966 


1 





28 


60 


19081 


98163 


20791 


97815 


22495 97437 


24192 


9703 


25882 


96593 








■i 




N. cos. 


N. sine. 


N.cos. N.sine. 


N.cos. 


N.siue. 




N.cos. 


N.sine. 


N. cos. N .sine. ' 






71 


)° 


7* 


J° 


"77 


76 


a 


75 


° J 







:$04 



NATURAL SINES AND COSINES. 











Table— (Continued). 










H 


— 


15° | 16° 


17° ] 18° | 19° 




H 

£ a 


27 


X. sine. X. cos. 


X sine. 


| N. cos. 


X.sine. N. cos. X. sine. X. cos. 1 X. sine, i X T . cos. 




9 








25882 1*6593 


27564 


90126 


29237 


9563 j 30902 95100 


3-_557 94552 


00 


9 





1 


25)1 965S5 


27592 


96118 


292C5 


95622 J 30929 15)97 


32534 94542 


59 


9 


1 


•2 


S5iaS 9657S 


2762 


9611 


29293 


95613 30. '57 950SS 


32012 94533 


58 


9 


1 


3 


25963 9657 


27643 


96102 


29321 


95605 1 30985 95979 


32639 94523 


57 


9 


2 


4 


25994 96562 


27676 


96024 


29343 


95596 31012 9507 


32667 94514 


56 


8 


2 


5 


26122 96555 


27704 


960S0 


29376 


955SS 3104 95061 


32694 94504 


55 


8 


3 


6 


2605 96547 


27731 


966*8 


29404 


95579 


3106S 95052 


32722 94495 


54 


8 


3 


7 


26079 9654 


27759 


9607 


29432 


95571 


31095 95043 


32749 


944S5 


53 


S 


4 


S 


20107 90532 


27787 


96062 


2946 


95562 


31123 1 5033 


32777 


94476 


52 


8 


4 


9 


20135 96524 


2T815 


96054 


29487 


95554 


31151 15024 


3:804 


94406 


51 


8 


5 


10 


26163 96517 


27343 


96043 


29515 


95545 


31178 95015 


32832 94457 


50 


8 


5 


11 


26191 96509 


27871 


96037 


29543 


15536 


34206 95006 


31859 94447 


49 


7 


5 


12 


26219 96502 


27899 


96029 


29571 


£5528 


31233 94997 


32887 9443 S 


4^ 


7 


6 


13 


26247 964 '4 


27927 


96021 


295 9 


95519 


31261 94 8^ 


32914 94428 


47 


7 


6 


14 


26275 964S6 


27955 


96)13 


29626 


95511 


31289 94979 


32942 94418 


46 


7 


7 


15 


26303 96479 


27933 


96005 


29654 


95502 


31316 9497 


32969 94409 


45 


7 


7 


16 


20331 96471 


28011 


95997 


29682 


95493 


31344 94961 


32997 , 94399 


44 


7 


8 


17 


26359 96463 


2S039 


95S89 


2971 


95485 


31372 94952 


33024 9439 


43 


6 


8 


IS 


203S7 9645 3 


28067 


959 SI 


20737 


95476 


31399 94943 


33051 9438 


42 


6 


9 


10 


26415 96 14S 


23095 


95>72 


29765 


95467 


31427 94933 


33079 9437 


41 


6 


9 


•20 


20443 9644 


28123 


95964 


29793 


95459 


31454 94924 


33106 94361 


40 


6 


9 


21 


26471 j 96433 


2S15 


95953 


29821 


9545 


314S2 94915 


33134 94351 


39 


6 


10 


22 


2 '5 964 5 


2S17S 


9594S 


29849 


95441 


3151 94906 


33161 1 94342 


38 


6 


10 


23 


26528 96417 


23206 


9594 


29S76 


95433 


31537 94897 


33189 94332 


37 


6 


11 


24 


26556 9641 


2S234 95931 


29904 


95422 


31565 94888 


S3216'94322 


36 


5 


11 


•25 


26534 96402 


28262 


95323 


29932 


95415 


31593 94878 


33244 ! 94313 


35 


5 


12 


26 


26012 96394 


2S29 


95915 


2996 


95407 


3162 £4869 


33271 94303 


34 


5 


12 


27 


2604 963S3 


28318 


95907 


299S7 


95398 


31648 9 4S6 


33298 i 94293 


33 


5 


13 


28 


26668 96379 


2S346 


95393 


30015 


953S9 


31075 94S51 


33326 1 94284 


32 


5 


13 


29 


26096 96371 


2 S3 74 


95S9 


30043 


9538 


31703 94842 


33353 i 94274 


31 


5 


14 


30 


20724 96363 


2S402 


95382 


30071 


95372 


3173 94832 


33331 94264 


30 


5 


14 


31 


26752 96355 


28429 


95S74 


30098 


95363 


3175S . 94823 


33408 94254 


29 


4 


14 


32 


267S 96347 


28457 


95865 


30126 


C5354 


31786 ;94S14 


33436 i 94245 


28 


4 


15 


33 


20S0S 9634 


2S4S5 


95857 


30154 


95345 


31S13 : 94S05 


33463 1 94235 


27 


4 


15 


34 


26336 96332 


28513 


95S49 


30132 


95337 


31S41 94715 


3349 ,94225 


26 


4 


16 


35 


26S64 96324 


28541 


95341 


30209 


9532S 


31S68 94730 


33518 94215 


25 


4 


16 


36 


26392 i 96316 


28569 


95S32 


31237 


95319 


31896 94777 


33545 94206 


24 


4 


17 


37 


2092 96303 


2S597 


95324 


30205 


9631 


31923 94718 


33573 94196 


23 


3 


17 


38 


2694S 96301 


28625 


95S16 


30292 


95301 


31951 9475S 


336 94186 


22 


3 


18 


39 


20976 96293 


28652 


95807 


3032 


1 5293 


31979 1 1 4749 


33627 94170 


21 


3 


18 


40 


27004 96285 


23 ;s 


95799 


3034S 


95284 


32006 9474 


33655 94167 


20 


3 


18 


41 


27032 96277 


28708 


95791 


30376 


95275 


32034 9473 


33; S2 94157 


19 


3 


19 


42 


2700 96269 


2S736 


957S2 


30403 


95266 


32061 94721 


3371 194147 


18 


3 


19 


43 


27033 96261 


2 S764 


95774 


30431 


95257 


32089 94712 


33737 94137 


17 


3 


20 


44 


27110 '.6253 


28792 


95768 


3045.) 


95248 


32116 j 947C2 


33764 94127 


16 


2 


20 


45 


27144 96^43 


2S82 


95757 


30436 


9524 


S2144 j 94693 


33792 941 IS 


15 


2 


21 


46 


27172 9623S 


2SS47 


15749 


30514 


95231 


32171 i 94684 


33S19 94108 


14 


2 


21 


47 


272 9623 


28S75 


9574 


30542 


95222 


32199 94074 


33S46 9409S 


13 


2 


22 


48 


2722S 96222 


28903 


95732 


3057 


95213 


32227 | 94665 


33S74 94088 


12 


2 


22 


49 


27256 96214 


28)31 


95724 


30597 


95 '04 


32254 ! 4056 


33981 


9407S 


11 


2 


23 


50 


27284 96206 


2S959 


95715 


30625 


95195 


322S2 94640 


33929 


94068 


10 


2 


23 


51 


27312 96193 


28 S7 


95707 


39653 


95186 


32309 94637 


33956 


94058 


9 


1 


23 


52 


2734 9619 


2: 015 


95698 


3063 


15177 


32337 94627 


339S3 


94049 


8 


1 


24 


53 


2836S 96 132 


29042 


9569 


3070S 


9510S 


32364 946 IS 


34011 


94039 


7 


1 


24 


54 


27396 96174 


2907 


956S1 


30736 


95159 


32392 94609 


34038 


94029 


6 


1 


25 


55 


27424 96160 


2.093 


95673 


30763 


9515 


32419 94599 


34065 


94019 


5 


1 


25 


50 


27-152 96159 


29126 95664 


30791 


95142 


32447 9459 


34093 


94009 


4 


1 


26 


57 


2243 9615 


29154 95656 


30819 


95133 


32474 9458 


3412 


93999 


3 





26 


56 


27503 96142 


29132 95647 


30S43 


95124 


32502 94571 


34147 


93989 


2 





27 


59 


27536 96134 


29209 95639 


30874 


95-15 


32529 94561 


34175 


93979 


1 





27 


00 


27564 96126 


29237 9563 


30902 


95106 


3 557 94552 


34202 


93969 












N. cos. N. sine. 


N. cos. N.sine. 


X. cos. N.sine. 


X.coe. X.sine. 


X. cos. N. sine. 


-1 








74° 


7, 


3° 


n 


o 


71° 


n 


J 





NATURAL SIXES AND COSINES. 



a 05 



Table— (Continued). 



a. a. 


— 


20° 


1 21° 


22° 


23° 


24° 




£2 


27 


N". sine.| N. cos. 


IN sine 


' N. cos. 


N.sine. N. cos.' N. sine |N cos 


N.sine 1 N. cos 


60 


11 








34202 


9396;* 


35337 


i 9335S 


37461 


92718 


39073 


9205 


40674 


91855 


"ff 





1 


34229 


93959 


35S64 ! 9334S 


37438 


92707 


391 


92089 


407 


91343 


59 


11 


1 


2 


34257 


93949 


35 3'. >1 [93337 


37515 


92697 


39127 


92028 


40727 


91331 


58 


11 


1 


3 


34284 


93939 


35918 1 93327 


37512 


92685 


39153 


92016 


40753 


91319 


57 


10 


2 


4 


343 11 


93)29 


35)45 


93316 


3756) 


92675 


3918 


92005 


4078 


91307 


50 


10 


2 


5 


3433) 


93319 


35973 


93306 


37595 


92664 


39207 


91994 


40806 


91295 


r.5 


10 


3 


6 


34366 


93909 


36 


93295 


37622 


92653 


39234 


91982 


40833 


91283 


54 


10 


3 


7 


34393 


93839 


36027 


93285 


3764) 


92642 


3)16 


91971 


4086 


91272 


83 


10 


4 


8 


34121 


93839 


35054 


93274 


87676 


92631 


39287 


91959 


40386 


9126 


52 


10 


4 


9 


34448 


93879 


35031 93264 


37703 


9262 


3.314 


91948 


40913 


91248 


51 


9 


5 


10 


34475 


93869 


36103 93253 


3773 


92 603 


39341 


91936 


40939 


91236 


50 


9 


5 


11 


345 )3 


93859 


36135 93243 


37757 


92598 


39867 


91S25 


40966 


91224 


49 


9 


5 


12 


3453 


93849 


36162:93232 


37784 


92587 


39394 


91914 


40992 


91212 


4^ 


9 


6 


18 


3455 f 


93839 


3619 93222 


37811 


92576 


39421 


91902 


41019 


912 


47 


9 


6 


14 


34584 


93829 


36217193211 


37S38 


925:5 


39448 


91891 


41045 


911S8 


46 


8 


7 


15 


34)12 


93319 


36241193201 


37885 


92554 


89474 


91879 


41072 


91176 


45 


8 


7 


16 


34539 


93303 


36271 i 9319 


37832 


92543 


39501 


9186S 


41098 


91164 


44 


8 


8 


IT 


34666 9379!) 


36298 9313 


37319 


92532 


3952S 


91856 


4ir:5 


91152 


43 


8 


8 


13 


346)4 ! 93789 


36325 9316) 


37946 


92521 


39555 


91845 


41151 


9114 


42 


8 


9 


19 


34721 


93779 


35852 93153 


37973 


9251 


39581 


91833 


41178 


91128 


41 


8 


9 


20 


347+8 


9H76) 


36379 ; 93143 


37999 


92493 


39608 


91822 


41204 


91116 


40 


7 


9 


21 


34775 


93753 


36406 93137 


38025 


924SS 


39635 


9181 


41231 


91104 


39 


7 


10 


22 


34803 


93T43 


36434 93127 


38 .53 


92477 


39C61 


91789 


41257 


"91092 


38 


7 


10 


28 


3433 


93738 


35461 93116 


3803 


9246 5 


3 68S 


91787 


41284 


9108 


87 


7 


11 


24 


34857 


93723 


36483 93106 


3S107 


92455 


89715 


91775 


4131 


91068 


36 


7 


11 


25 


34884 


93718 


36515 93)95 


38134 


92,44 


39741 


91764 


41337 


91056 


35 


6 


12 


26 


34 12 


1*3708 


3J542 930S4 


38161 


92432 


39768 


91752 


41863 


91044 


34 


6 


12 


•27 


3493 ) 


936 i'8 


36559 i 93074 


3S183 


92421 


397; 5 


T1741 


4139 


91032 


33 


6 


13 


28 


34*66 


93688 


36596 93063 


33215 


9241 


39822 


i 1729 


41416 


9102 


32 


6 


13 


29 


34993 


93677 


36623 ! 93052 


38241 


92399 


39848 


91718 


41443 


91008 


31 


6 


14 


30 


35021 


93G67 


3665 j 93042 


3S268 


92388 


39875 


91706 


41469 


80996 


30 


6 


14 


31 


35048 


93657 


36677 ' 93031 


332 5 


92377 


3 902 


91694 


41496 


90884 


29 


5 


14 


32 


35075 


93617 


36704 


9302 


33322 


92366 


399'. S 


91GS3 


41522 


10972 


28 


5 


15 


33 


35102 


93637 


35731 


9301 


38349 


92S55 


898 55 


91671 


41549 


(096 


27 


5 


15 


34 


3513 


93626 


36753 


92999 


3S375 


92343 


39:82 


9166 


415:5 


90948 


26 


5 


16 


35 


35157 


93616 


367S5 


92 83 


3S403 


92332 


40003 


91648 


41692 


90836 


25 


5 


16 


36 


351S4 


93696 


35812 


92978 


3343 


92321 


40035 


91636 


4162S 


10924 


24 


4 


17 


37 


35211 


935)6 


36839 


92967 


33456 


9231 


40062 


9165 


41655 


80911 


23 


4 


17 


38 


35239 


93535 


35867 


92956 


8S483 


92239 


400SS 


91613 


41681 


80889 


22 


4 


18 


39 


35266 


93575 


36*94 


92945 


3851 


92287 


40115 


91C01 


41707 


90837 


21 


4 


18 


40 


852 3 


93)65 


36921 


92935 


33537 


92276 


40141 


9159 


41734 


10S75 


20 


4 


18 


41 


3532 


93555 


36948 


92924 


3S5'4 


92265 


40168 


9157S 


4176 


90863 


19 


3 


19 


42 


35347 


93544 


36975 


92913 


38591 


92254 


40195 


91566 


41787 


90651 


18 


3 


19 


43 


35375 


93534 


37002 


92902 


33617 


92243 


40221 


91555 


41813 


9(839 


17 


3 


20 


44 


35402 


93524 


37029 


92892 


33044 


92231 


40248 


91543 


4184 


10826 


16 


3 


20 


45 


85 m 


93514 


37055 


92381 


33671 


8222 


40275 


91531 


41S66 


90814 


15 


3 


21 


46 


35459 


93503 


37083 


9287 


3809 3 


92209 


4 391 


91519 


41892 


90802 


14 


3 


21 


41 


35434 


93493 


8711 


92859 


33T25 


92193 


4082S 


9150S 


41919 


8079 


13 


2 


22 


48 


35511 


93483 


37137 


928,9 


8S752 


921S3 


40355 


914°6 


41845 


80778 


12 


2 


22 


49 


35538 


93472 


37164 


92388 


38778 


92175 


40381 


911484 


41872 


90766 


11 


2 


23 


50 


355 .5 


93462 


37191 


92827 


38805 


921C4 


4M4-8 


91472 


41998 


80753 


10 


2 


23 


51 


35593 


93452 


3721S 


92816 


83832 


92152 


494'14 


91461 


42024 


80741 


9 


2 


23 


53 


35619 


93441 


37245 


923 .5 


3335) 


92141 


40461 


91449 


42051 


90729 


8 




24 


53 


35647 


93431 


3727 1 


927M 


38886 


9213 


4048S 


91437 


42077 


90717 


7 




24 


54 


35674 


9342 


37299 


927S4 


33912 


92119 


40514 


81425 


42104 


90704 







25 


55 


35701 


9341 


37326 


92773 


B8 39 


92107 


40541 


91414 


4213 


90692 


5 




25 


63 


35728 


934 


37853 


92762 


88966 


920'6 


4056 1 


91402 


42156 


90(58 


4 




26 


57 


8,5755 


93339 


3738 


92751 


3S993 


92085 


40594 


9139 


42183 


90003 


3 




26 


58 


35782 


9837) 


87497 


9274 


3902 


92073 


40621 


91378 


42209 


10(555 


2 





27 


59 


85,81 


933 58 


37434 


92729 


39046 


92062 


4 M 647 


913)56 


42235 


90643 


1 





27 


60 


35337 


93353 


3"451 


'92718 


39073 


9205 


40674 


91855 


42202 


99(59,1 












N. cos. 


N sine. 


N. cos. 


N. sine. j N. cos. N.sine. 


N.C08. N sine 


N. cos. N . sini'. 


' 





cs° 



67° 



C6° 



65° 



306 



NATTJKAL SINES AND COSINES. 











Table— (Continued). 












2 * 

a. a. 




25° 


26° 


27° | 2S° 


29° 


— 


a.4 


20 


' 


N. sine. X. cos. 


N sine. N. cos. 


N.sine. N. cos. N.sine. N. cos. 


N.sine.: N. cos. 


11 








42202 90031 


438^7 89879 


4539 89101 


40947 


88-95 


4S431 


S7402 


00 


14 





1 


422 SS 90018 


43833 89S07 


45425 890S7 


46973 


88281 


48506 


S7448 


59 


14 


1 


2 


42315 90000 


43SS.) 89854 


45451 


8.-074 


40 99 


8S207 


43532 


S74::;4 


53 


14 


1 


3 


42341 90594 


43910 89841 


45477 


S90G1 


47024 


88254 


4^557 


8742 


57 


13 


2 


4 


42307 905S2 


43942 S9S28 


45503 


S904S 


4705 


8824 


48533 


S7406 


50 


13 


2 


5 


42394 905;;9 


43908 S9S1G 


45529 


89035 


47076 


83226 


4800S 


S7391 


55 


13 


3 


6 


4242 90557 


43994 S9S03 


45554 


89021 


47101 


88213 


4S034 


S7377 


54 


13 


3 


7 


42440 90545 


4402 S979 


4553 


89008 


47127 


88199 


4SG59 


87363 


53 


12 


3 


S 


42473 90532 


4404G S9777 


45000 


8S995 


47153 


SS1S5 


4S0S4 


87349 


52 


12 


4 


9 


42499 9052 


44072 S-704 


45032 


SS981 


47178 


88172 


4871 


87335 


51 


M 


4 


10 


42525 90507 


44098 S9752 


48658 


889GS 


47204 


8S153 


48735 


87321 


50 


12 


5 


11 


42552 .0495 


44124 89739 


45084 


8S955 


4722.) 


88144 


48701 


87306 


49 


11 


5 


12 


42578 904S3 


44151 S.1720 


4571 


83942 


4725.5 


8S13 


4S7SG 


S7292 


4s 


11 


6 


13 


42004 1047 


44177 S9713 


45736 


8S92S 


47281 


88117 


4SS11 


87278 


47 


11 


6 


14 


42031 9045S 


44203 S97 


45702 


83915 


47300 


8S103 


4SS37 


87264 


40 


11 


7 


15 


42657 90440 


44229 89087 


457^-7 


88902 


47332 


SS0S9 


43302 


S725 


45 


11 


T 


10 


42GS3 90433 


44255 89674 


45313 


SSS88 


4735S 


8S075 


4SSSS 


S7235 


44 


10 


7 


17 


4270) 90421 


-I42S1 89002 


45339 


8S875 


47383 


88062 


4S913 


S7221 


43 


10 


8 


IS 


42736 90408 


44307 89C49 


45SG5 


SS302 


47409 


S8043 


43938 


S7207 


42 


10 


8 


19 


12702 90390 


44333 ; S.;030 


45891 


88S4S 


47434 


SS034 


4S964 


S7193 


41 


10 


9 


20 


42788 903S3 


44359 89023 


45- 17 


8^S35 


4740 


8302 


43989 


8717S 


40 


9 


9 


21 


42S15 


90371 


443S5 8901 


45942 


88S22 


474SG 


8SOO0 


4 -014 


S7164 


39 


9 


10 


22 


42SH : 


9035S 


44411 8.597 


459GS 


SS808 


47511 


87913 


4904 


S715 


38 


9 


10 


23 


42867 ! 


90346 


44437 ! S95S4 


45994 


S3795 


47537 


8797.) 


49065 


87136 


37 


9 


10 


24 


42S94 


90334 


44464 89571 


4002 


88782 


475G2 


87C65 


4909 


87121 


30 


8 


11 


25 


4212 


90321 


4449 8955S 


4G04G 


8S7GS 


475 -8 


87951 


49116 


87107 


35 


8 


11 


20 


42^46 


90309 


44510 ! 8; 545 


46072 


8S755 


47014 


87;37 


49141 


87o93 


34 


8 


12 


27 


42979 


90296 


41542 j S9532 


40097 


83741 


47039 


87923 


49160 S7 079 


33 


8 


12 


2S 


42999 


90284 


44563 ! 83519 


40123 


8872S 


47005 


87909 


49192 


S7064 


32 


7 


IS 


29 


43025 


90271 


445)4 1 S9506 


40149 


83715 


4769 


87890 


49217 


8705 


31 


7 


13 


BG 


43051 


90259 


4432 | 8^493 


40175 


8S701 


47716 


87SS2 


49242 


87036 


30 


7 


13 


31 


43077 


9024G 


44046 8943 


46201 


SSG^S 


47741 


87S63 


4920S 


87021 


20 


7 


14 


32 


43104 


90233 


44372 i 8. 467 


4G220 


88674 


47767 


87S54 


49293 


87007 


28 


7 


14 


33 


4313 


90221 


4439S j 89454 


40252 


S8601 


47793 


87S4 


4931S 


86993 


27 


6 


15 


34 


43(56 


9020S 


44724 S9441 


4G27S 


8S647 


47818 


87S26 


49344 


S0978 


20 


6 


15 


35 


43 182 


90190 


4475 IS942S 


40304 


83*34 


47844 


87S12 


49369 


S6964 


25 


6 


16 


80 


43209 


901S3 


44776 S '415 


4033 


3362 


47860 


8779S 


49394 


86949 


24 


6 


10 


37 


43235 


90171 


44892 89402 


4G353 


88007 


478'. 5 


87784 


49419 


86935 


23 


5 


10 


88 


43201 


90158 


44S2S 8 3S9 


4;381 


SS593 


4792 


8777 


49445 


S6921 


22 


5 


17 


39 


43287 


90140 


44854 S9370 


40407 


S853 


47946 


87750 


4947 


80900 


21 


5 


17 


40 


43313 


90133 


44SS 89363 


46433 


SS500 


47971 


8774;; 


494' 5 


SGS92 


20 


5 


18 


41 


4334 


9012 


44906 89.5 


4045 - 38553 


47097 


87721 


49521 86878 


19 


4 


18 


42 


43300 


90108 


44932 89337 


4G4S4 3S53:> 


48022 87715 49546 S6S63 


IS 


4 


19 


4: , » 


43392 


90095 


4405S 89324 


4651 SS526 


43048 87701 


49571 SGS49 


17 


4 


19 


44 


43 IIS 


900S2 


449S4 89311 


46536 8S512 


4S073 870S7 


49590 86834 


10 


4 


20 


45 


4344 > 


9007 


4501 89898 


46561 SS499 


4S099 8707. 


49622 86S2 


15 


4 


20 


40 


4)471 


90057 


450-36 392S5 


405S7 88485 


48124 


87650 


4 G4 SGS05 


14 


3 


20 


47 


43497 


90045 


45062 81)272 


46G13 88472 


4815 


87045 


4 "672 S6791 


13 


3 


21 


4S 


43523 


90032 


45088 89259 


40039 88458 


48175 


87031 


496'. 7 86717 


12 


3 


21 


4!) 


4354 i 


90019 


45114,89245 


46J04 S-445 


43.01 


87617 


49723 86762 


11 


3 


22 


50 


43575 


90007 


4514 89232 


4009 .88431 


48-220 


87603 


49748 8074S 


10 


2 


22 


51 


43002 


89994 


45166 8 219 


4G71G 88417 


48252 


87689 


49773 86733 


9 


2 


23 


52 


4302S 


89981 


45192 89206 


40742 SSI 04 


48277 87575 


49J98 S6719 


S 


2 


23 


5 3 


13654 


89908 


45218 89193 


46767 8S39 


48303 87561 


49S24 86704 


7 


2 


23 


54 


4308 


99950 


45243 891S 


407. 3 88377 


48328 87546 


4 S 19 8669 





1 


24 


55 


43706 


89943 


4520) 89107 


46S19 S8363 


48:. 54 87532 


49874 S6675 


5 


1 


24 


56 


43733 


8993 


45295 89153 


4GS44 8S34'' 


4<): S7518 


49899 80001 


4 


1 


25 


57 


43759 


89018 


45321 8914 


4867 88336 


4S40J5 S7504 


49;>24 8064(5 


3 


1 


25 


5S 


43785 


89905 


45347 S9127 


4*06 88322 4843 8749 


4995 80032 


2 





20 


59 


43S11 


89892 


45373 89114 


40921 S330S j 48456 87470 


49975 86017 


1 





20 


SO 


43837 


89S79 


4531)9,89101 


46947 88295 

N. c«is N.sine. 


4S4S1 8740' 


5 86603 












N. cos. 


IN. sine. 


N. cos. N.sine. 


N.cos. N.sine 


N. cos. N. sine. 


' 








1 6- 


1° 


63° 


c 


|° 


1 6 


|/> 


6( 


J° 







NATURAL SINES AND COSINES. 



307 













Table— (Continued). 












ft. ft. 




30° 


31° 


32° 


33° 


34° 






25 


' 


N. sine.| N. cos. 


N sine.| N. cos. 


N.sine. N. cos. 


N.sine. 


N. cos. 


N.sine. I N. cos. 


60 


11 








5 


So003 


51504 


8:>717 


52992 


84805 


54464 


8..S07 


55910 


82904 


16 





1 


50025 


8.:588 


51529 


85702 


53017 


84783 


544S8 


83851 


55943 


82S87 


59 


16 


1 


2 


5005 


86573 


51554 


85687 


53>-tl 


84774 


54513 


83835 


55963 


82871 


58 


15 


1 


3 


50076 


86559 


51579 


85672 


53066 


84759 


54537 


83S19 


55992 


82855 


57 


15 


2 


4 


50101 


80544 


51604 


85657 


53091 


84743 


54561 


83804 


56016 


82839 


56 


15 


2 


5 


50126 


8653 


51628 


85642 


53115 


Si7-S 


54586 


837 88 


5604 


82829 


55 


15 


3 





51)151 


86515 


51653 


85627 


5314 


S4712 


5461 


S3772 


56064 


82806 


54 


14 


3 


7 


50176 


83501 


51678 


85612 


53164 


84697 


54635 


83756 


56088 


8279 


53 


14 


3 


8 


50201 


864S6 


51703 


85597 


53189 


S46S1 


54659 


8374 


56112 


82773 


52 


14 


4 


9 


50227 86471 


51728 


85582 


53214 


84638 


54083 


83724 


56136 


82757 


51 


14 


4 


10 


50252 86457 


51753 


85567 


53238 


8465 


54708 


8370s 


5616 


82741 


50 


13 


5 


11 


50i77 


8J442 


51778 


85551 


53263 


84635 


54732 


83692 


56184 


82724 


49 


13 


5 


12 


50302 


86427 


51S )3 


85536 


53288 


84619 


54756 


83676 


5620S 


82708 


4s 


13 


5 


13 


50327 


S6413 


51S2S 


85521 


53312 


84604 


54781 


83v.6 


5623.. 


82692 


47 


13 


6 


14 


50352 


80338 


51852 


85596 


53337 


84.88 


5.8:5 


83645 


56256 


8675 


46 


12 


6 


15 


50377 


86384 


51S77 


85491 


5-3361 


S4573 


548.9 


83C29 


5628 


82*559 


45 


12 


7 


16 


50403 


S6363 


51902 


S5476 


533 S6 


84557 


54S54 


83013 


56305 


S2643 


44 


12 


7 


17 


50423 


86354 


51927 


85461 


53411 


84542 


54S78 


83397 


56329 


82626 


43 


11 


8 


IS 


50453 


8634 


51052 


85446 


53435 


8 1526 


54002 


835S1 


56353 


8261 


42 


11 


8 


19 


50478 


86325 


51 77 


S5431 


5346 


84511 


54327 


83565 


5i,377 


8.593 


41 


11 


8 


20 


50503 


8631 


52002 


S5416 


53484 


84495 


54951 


83549 


56401 


82577 


40 


11 


9 


21 


50528 


86295 


52026 


85401 


5.. 509 


844S 


54975 


83533 


56425 


8561 


39 


10 


9 


22 


5 '553 


86281 


52051 


S53S5 


53534 


S4464 


54999 


83517 


56449 


82544 


38 


10 


10 


23 


50578 


86266 


52076 


8537 


53558 


84448 


55.): 4 


83501 


56473 


825-' 8 


37 


10 


10 


24 


50603 


8J251 


52101 


85355 


536S3 


84433 


55048 


8--4S5 


56497 


82511 


36 


10 


10 


25 


50628 


86237 


52126 


8534 


53607 


84417 


55072 


8346: t 


5i521 


82495 


35 


9 


11 


26 


50654 


86 222 


52151 


85325 


53632 


84402 


55097 


88453 


56545 


82478 


34 


9 


11 


27 


50679 


89207 


52175 


8531 


53u56 


84386 


55121 


S3437 


50569 


8:462 


33 


9 


12 


2S 


50704 


89192 


522 


85294 


53681 


S437 


55145 


834:i 


56593 


82446 


32 


9 


12 


29 


50729 


80178 


52225 


S5279 


53705 


8455 


55169 


83405 


56617 


82429 


31 


8 


13 


30 


50754 


86163 


52 5 


85264 


5373 


84339 


55194 


83389 


5cj641 


82413 


30 


8 


13 


31 


50779 


86148 


52275 


85243 


53754 


84324 


5521S 


83373 


5 6,5 


82396 


29 


8 


13 


32 


50S04 


86133 


52293 


85234 


53779 


8,308 


55242 


83356 


560S.> 


8238 


28 


7 


14 


33 


50829 


80119 


52324 


85218 


53804 


84292 


55266 


8^34 


56713 


82363 


27 


7 


14 


34 


5)854 


86104 


52349 


85203 


5388 


84277 


552:, 1 


$8324 


56736 


82347 


26 


7 


15 


35 


50879 


86083 


52374 


85188 


53S53 


84261 


55315 


8S308 


5076 


8233 


25 


7 


15 


3l'» 


50*.>04 


86074 


52399 


85173 


5..S77 


84245 


55339 


83 92 


56784 


82314 


24 


6 


15 


37 


50929 


36069 


52423 


85157 


53902 


8423 


55363 


83276 


56S0S 


82297 


23 


6 


16 


"38 


50054 


86045 


52448 


85142 


53926 


S4214 


553>-8 


8326 


508- 2 


S22S1 


22 


6 


16 


39 


50079 


8603 


52473 


851.7 


5395 1 


84198 


55412 


83244 


56856 


82264 


21 


6 


17 


40 


51004 


86015 


52408 


85112 


53975 


8418! 


55436 


832:. 8 


5688 


82248 


20 


5 


17 


41 


51029 


86 


52522 


85096 


54 


84167 


5546 


83212 


509t4 


82231 


19 


5 


18 


42 


51054 


S5985 


52547 


85081 


54024 


84151 


554S4 


83195 


56 28 


82214 


18 


5 


18 


43 


51079 


S597 


52572 


85066 


54043 


84135 


55509 


8317 


56-52 


8-198 


17 


5 


IS 


44 


51104 


85956 


52597 


85051 


54073 


8412 


55533 


83163 


56976 


82181 


16 


4 


19 


45 


51129 


85941 


52621 


85035 


54.97 


84104 


55557 


83147 


57 


82165 


15 


4 


19 


46 


51154 


85926 


52646 


8502 


541 --2 


8408S 


55581 


83131 


57024 


8214S 


14 


4 


20 


47 


51179 


85011 


52671 


85005 


541! 6 


84072 


55605 


83115 


57047 


821-^2 


13 


3 


20 


4S 


51204 


85S96 


52696 


84983 


54171 


84057 


5563 


83098 


57071 


82115 


12 


3 


20 


49 


51229 


858S1 


5272 


84374 


51195 


S4041 


5505 1 


83082 


570:5 


82098 


11 


3 


21 


50 


51254 


85S66 


52745 


84053 


5422 


S4025 


55078 


83066 


57119 


820S2 


10 


3 


21 


51 


51279 


85S51 


5277 


8-1943 


54244 


84009 


55702 


8305 


57143 


82065 


9 


2 


22 


52 


51304 


85S36 


52794 


84928 


5.263 


S39"4 


55726 


83034 


57167 


82048 


8 


2 


22 


63 


513 .9 


85811 


52S19 


84913 


542 3 


8397S 


5575 


83017 


57191 


82032 


7 


2 


23 


r»4 


51354 


85806 


5^814 


84897 


5:317 


83962 


55775 


£1001 


57215 


82015 


6 


2 


23 


55 


51379 


85792 


52869 


84882 


54142 


S3946 


5579!) 


829S5 


572.8 


81999 


5 


1 


23 


5$ 


51404 


S5777 


5 S93 


848 56 


54966 


8393 


55S23 


82! (59 


5 262 


81982 


4 


1 


24 


57 


5142.) 


85762 


5291S 


84851 


54391 


83915 


55S47 


82953 


57286 


81965 


3 


1 


24 


58 


5145 1 


85-47 


5!043!84S36 


54415 


83899 


55S71 


829: :C 


5731 


81949 


2 


1 


25 


59 


51479 


85732 


52967 I 8482 


5444 


83883 


55895 


8292 


57334 


81932 


1 





J5 


60 


51504 


85717 


52992 84S05 


54464 


83867 


5591' » 


82 04 


5 358 


81915 












N. coa. 


N. sine. | N. cos. ! N.sine. N. cos. N.sine. 


N.cos. N.sine 


N. cos. .N.sine.l ' 








5 


J° 


I 5 


P j 


*" 


1° 


bi 


> u 


K 


>° 







308 



NATURAL SINES AND COSINES. 



Table — {Continued). 



6.3 

£ a. 


— 


35° 


36° 


3To , 


3S 


° 1 


39° | 




a- a 


25 


V. sine. X. cos. 


X sine. X. cos; 


X'.sine. X. cos. X. sine. X. cos. |x. sine. N. cos. 


IS 








57368 


81915 


53779 809U2 


60 182. 79864 


61563 


7SS01 6'2932 77715 60 


13 





1 


57331 


S1S99 


53302 S0SS5 


60205 7934 1 


61533 


73733 ! 62. 55 77696 1 59 


13 


1 


% 


57405 


S18S2 


53326 S0S67 


6022S 7.S2) 


61612 


737.5 


62977 .7673 ,5S 


17 


1 


3 


57429 


31365 


53S49 S0S5 


60251 79811 


61635 


73747 


03 7766 


57 


17 


2 


4 


57453 


S1S43 


58873 86833 


60274 79793 


6165S 


73729 


63022 77641 


56 


17 


2 


5 


57477 


S1S32 


533 6 S0S16 


602i'S 79776 


61631 


73711 


63<'45 77623 


55 


17 


2 


6 


57591 


S1S15 


58 2 S0799 


60321 79753 


61704 


786 4 


6306S 77605 54 


16 


3 


7 


57524 


817..S 


53943 S0732 


60314 79741 


01726 


73.76 


6309 77586 


53 


10 


3 


S 


575 13 


31732 


53967 80765 


6)367 79723 


61749 


78653 


63113 77563 


52 


16 


3 


9 


57572 


Si 765 


5S99 S074S 


60-39 79706 


61772 


7S64 


63135 7755 


51 


15 


4 


10 


57596 


S1743 


59014 S0730 


60414 796S3 


61795 


73622 


6315$ 77531 


50 


15 


4 


11 


57619 


31731 


59037 S0713 


60437 79671 


61S1S 


78604 


('313 77513 49 


15 


5 


12 


57643 S1714 


53061 306 i6 


6046 79653 


61341 


7353o 


63203 77494 43 


14 


5 


13 


57667 3169 S 


5 0S1 SO 37.) 


60433 79635 


61364 


78668 


03-25 77476 47 


14 


5 


14 


57691 S10S1 


59108 S0662 


60506 79613 


61S37 


7855 


63:48 77453 46 


14 


6 


15 


57715 SI 664 


5)131 80644 


6052) 796 


6190) 


73532 


63271 77439|45 


14 


6 


16 


57733 31647 


5)154 83627 


60553 79533 


61932 


73514 


63293 77421 ' 44 


13 


7 


17 


57762 


31631 


5)178 S061 


60576 795:55 


61955 


7S496 


63310 77402 43 


13 


7 


IS 


57736 


SI 614 


59201 80593 


605 9 79547 


61973 


73473 


6333S 77334 


42 


13 


T 


19 


5781 


81597 


5)225 S0576 


60622 7953 


62001 


7346 


63361 77366 


41 


12 


8 


20 


57833 


S153 


59243 S0553 


60645 79512 


62024 


73442 


633S3 77347 


40 


12 


8 


21 


57857 


S1563 


59272 S0541 


6066S 79494 


62046 


73424 


63406 7732.- 


39 


12 


8 


22 


57831 


8 : 546 


59295 S0524 


60691 79477 


6206 ) 


7S405 


63428 7731 


33 


11 


9 


23 


57304 


3153 


59318 8)507 


60714 7 45) 


620.-2 


73337 


63451 77292 


37 


11 


9 


24 


57923 


31513 


59342 80439 


60733 79441 


6-2115 


78369 


.3473 77273 


36 


11 


10 


•25 


57952 


31416 


53365 30472 


60761 79424 


62133 


73351 


63496 77255 


35 


11 


10 


20 


57 70 S1479 


5933) 3)455 


60734 79403 


6-16 


73333 


- 77236 


34 


10 


10 


27 


5799 SI 462 


53412 S0438 


60S07 79338 


621S3 


73315 


6354 77213 


33 


10 


11 


28 


5S923 S1445 


5U36 3)42 


6033 79371 


62206 


78207 


63563 77199 


32 


10 


11 


29 


5S947 3142S 


59459 30403 


60S53 7>-53 


62229 


73279 


6 5-5 77131 


31 


9 


12 


30 


53)7 31412 


59432 80383* 


60S76 79335 


62251 


73261 


63608 77162 


30 


9 


12 


31 


53091 81395 


59506 80368 


60399 79313 


62274 


73243 


6363 77144 


29 


9 


12 


32 


53113 Si 373 


5 529 80351 


60922 793 


62297 


73225 


63653 77125 


23 


8 


13 


33 


53 141 31361 


59552 8 334 


60 45 792S2 


623 2 


73206 


63675 771"7 


27 


8 


13 


34 


53165 31344 


5)576 30316 


60363 79261 


62342 


73133 


636. S 77033 


26 


S 


13 


35 


531S9 S1327 


535 9 30239 


60991 7 247 


6265 


7317 


6372 7707 


25 


S 


14 


36 


53! 12 8131 


59622 80232 


61015 792 2) 


62388 


73152 


63742 77051 


24 


7 


14 


37 


532-3(5 31293 


59616 30261 


61038 73211 


62411 


73134 


63765 77033 


23 


7 


15 


38 


5326 S1276 


59669 S0247 


61061 79193 


62433 


73116 


03737 77014 


22 


7 


15 


39 


53233 31259 


59693 S023 


61034 79176 


6245* 


7S03S 


63S1 76916 


21 


6 


15 


40 


53307 S1242 


59716 80212 


61107 79158 


62479 


73 7.1 


63832 76977 


20 


6 


16 


41 


5333 312 5 


5 7-39 SO 195 


6113 7914 


62502 


73061 


63-54 76959 


19 


6 


16 


42 


5335 \ S120S 


5)763 80178 


61153 79122 


62524 


7S043 


63377 76 4 


13 


5 


16 


43 


53373 31191 


597S6 3016 


61176 73105 


62517 


73025 


63S 9 76921 


17 


5 


IT 


44 


534 11 31174 


533 '9 S0143 


61199 790S7 


6257 


70007 


63.22 76.03 


10 


5 


17 


45 


534 5 3H57 


53S32 SH25 


61222 7906' 


62592 


77988 


63944 76334 


15 


5 


IS 


40 


5314 3114 


5 356 30108 


61245 79 51 


62615 


7797 


63 0(j 763 6 


14 


4 


18 


47 


53472 S112; 


5 S7> 30091 


61263 7 033 


(52638 


77 5' 


633S9 76347 


13 


4 


13 


4S 


53496 SI 106 


59 02 S>t>73 


01291 791)10 


ma 


77934 


64011 76S28 


12 


4 


19 


49 


5351 8 039 


59>20 30356 


01314 73993 


62633 


77916 


64033 76-1 


11 


3 


19 


50 


53543 


3107' 


5 949 S0938 


61337 739S 


62706 


77- 7 


64058 7-179 i 


10 


3 


20 


51 


53567 


31055 


53 72 80021 


6136 73 62 


6272S 


77S79 


64073 76772 


9 


3 


20 


52 


5S5 » 


81088 


591)95 30003 


61333 73944 


62751 


77301 


641 76754 


S 


2 


20 


53 


53614 


81021 


6 01) 79986! 61406 73926 


62774 


77843 


64123 76735 


7 


2 


21 


54 


53037 


310 >4 


6 042 79963 ' 6142 9 78908 


62796 


77324 


64145 76717 


6 


2 


21 


55 


5S661 - 


6 005 791 51 61451 7-3 1 


62S19 


7730; 


64167 7669S 


5 


2 


21 


50 


58684 - 


6)039 79 34 61474 78873 


6.342 


77733 


6419 76679 


4 


1 


22 


57 


5ST0S S»953 


60112 79916! 614)7 78355 


62364 


77 "69 


0421 •? 76661 


3 


1 


22 


56 


537 ;l 30936 


60135 793: 9, 6152 7883T 


6 SS7 


77751 


64234 76 142 


2 


1 


23 


59 


53755 30919 


60153 7 331 61543 73319 6290.) 


77733 


64256 76623 


1 





23 | 00 5377 I 80902 


61)132 79364 61566 78901 68 32 


77715 


61279 76604 








|N. cos. N. sin*. IX. cos. X. sine. j N. cos- N.sine.| X.cos. X.sne 


N. cos. N.sine 


' 


„ 



J 1 



54° 



53° 



52° 



51° 



50° 



NATURAL SINES AND COSINESo 



309 













Table— -{Continued). 












p. a 

5- a. 




40° 


41° 


42° 1 43° 


44° 


— 


Oh P. 


22 


/ 


N. sine.i N. cos. 


N sine. N. cos. 


N. sine.j N. cos.' N. sine.i N. cos. 


N.sine.. N. cos. 


19 








64279 


76604 


65606 : 75471 


66913 74314 


682 [73135 


69466 71934 


60 


19" 





1 


64301 


76536 


65628 75452 


66935 ' 74295 | 


63221173116 


694S7 


71914 


59 


19 


1 


2 


C4323 


76567 


6565 75433 


66955 


74276 


6S242 | 73096 


69508 | 


71894 


58 


18 


1 


3 


64346 


76548 


65672 75414 


66978 ! 


74256 


68261! 73076 


69529 | 


71873 


57 


18 


1 


4 


64368 


7653 


65694 ; 75395 


66999 ! 


74237 


68285 i 73056 


69549 


71S53 


56 


18 


2 


5 


6439 


76511 


65716 75375 


67021 j 74217 


68306 I 73036 


6957 


71833 


55 


17 


2 


6 


64412 


76492 


65738 75356 


67043 1 


74198 


68327 


T3016 


69591 


71813 


54 


17 


3 


7 


64435 


76473 


65759 j 


75337 


67064 i 


74178 


6834) 


72996 


69612 


71792 


53 


17 


3 


8 


6 1457 


76455 


65781 


75318 


67086 


74159 


6837 


72976 


69633 


71772 


52 


16 


3 


9 


64479 , 76436 


65S03 ! 


75299 


67107 


74139 


6S391 


72957 


69654 71752 


51 


16 


4 


10 


64501 76417 


05325 


7528 


67129 


7412 


68412 


72937 


69675 71732 


50 


16 


4 


11 


64524 ! 76398 


65347 


75261 


67151 


741 


6S434 


72317 


69696 71711 


49 


16 


4 


12 


64546 


7638 


65369 


75241 


67172 


74 8 


6S455 


72897 


69717 71691 


43 


15 


5 


13 


64538 i 


76361 


65S91 


75222 


67194 


74061 


6S476 


72877 


69737 i 71671 


47 


15 


5 


14 


6459 


76342 


65913 


75203 


67215 


74941 


68497 


72857 


6975S 7165 


46 


15 


6 


15 


64612 


76323 


65935 


75184 


67237 


74022 


68518 


72837 


69779 7163 


45 


14 


6 


16 


64635 


76304 


65956 


75165 


67258 


74002 


68539 


72817 


698 J7161 


44 


14 


6 


IT 


64657 


76286 


65978 


75146 


6728 


73983 


68561 


72797 


69821 7159 


43 


14 


7 


18 


64679 


76267 


66 


75126 


67301 


73963 


685S2 


72777 


69842 ; 71569 


42 


13 


7 


19 


64701 


76248 


66022 


75107 


67323 


73944 


68603 


72757 


69862 ! 71549 


41 


13 


7 


20 


64723 


76229 


66044 


750S8 


67344 


73924 


63624 


72737 


69883 ! 71529 


40 


13 


8 


21 


64746 


7621 


66036 


75059 


67366 


73^:04 


6S645 


72717 


69904 ! 71598 


39 


12 


8 


22 


64.68 


76192 


66088 


7505 


673S7 


73385 


68666 


726 7 


69925 71488 


33 


12 


8 


23 


6479 


76173 


6611)9 


7503 


6740) 


73805 


68688 


72677 


69946171468 


37 


12 


9 


24 


64812 


76154 


66131 


75011 


6743 


73846 


6S709 


7 657 


69966 


71447 


36 


11 


9 


25 


64>34 


76135 


66153 


74992 


67452 


73826 


6873 


72637 


69987 


71427 


35 


11 


10 


26 


64856 


76116 


66175 


74973 


67473 


73S06 


68751 


72617 


7000S 


71407 


34 


11 


10 


27 


64878 


76097 


66197 


74953 


67495 


73787 


68772 


72597 


70029 


71386 


33 


10 


10 


28 


64901 


76078 


6621S ; 74904 


67516 


73767 


68793 


72577 


70049 


71366 


32 


10 


11 


29 


64923 


76059 


6624 ,74915 


67538 


73747 


6SS14 


72557 


7007 


71345 


31 


10 


11 


30 


64945 


76041 


66252 74396 


67559 


73728 


68835 


72537 


70091 


71325 


30 


10 


11 


31 


64967 


76022 


662S4 74S76 


6758 


73708 


68857 


72517 


70112 


71305 


29 


9 


12 


32 


6493) 


76003 


66306 74857 


67602 


7368S 


6SS78 


72497 


70132 


71284 


2S 


9 


12 


33 


65011 


75984 


66327 ; 74338 


6762-3 


73669 


68899 


72477 


70153 


71264 


27 


9 


12 


34 


65033 


75965 


66349 j 74S18 


67645 


73649 


6892 


72457 


70174 


71243 


20 


8 


13 


35 


65955 


75146 


66371 ' 74799 


67666 


73629 


6S941 


72437 


70195 


71223 


25 


8 


13 


36 


£6077 


75927 


66393 7478 


6768S 


7361 


6S962 


7241" t 


70215 


71203 


24 


8 


14 


37 


651 


75908 


66414 7476 


67709 


7359 


68983 


723:7 


70230 


71182 


23 


7. 


14 


GS 


65122 


75389 


66436 ; 74741 


6773 


7357 


69004 


72377 


70257 


71162 


22 


7 


14 


39 


65144 


7587 


66453 


74722 


67752 


73551 


69025 


72357 


7 277 


71141 


21 


7 


15 


40 


65166 


75851 


6643 


74703 


67773 


73531 


69046 


7233^ 


70298 


71121 


20 


6 


15 


41 


65183 


75832 


66501 


74683 


67795 


73511 


69067 


72317 


70319 


711 


19 


6 


15 


42 


6521 


75S13 


66523 


74661 


67816 


73491 


690SS 


7 -2'; 7 


7033:: 


7108 


18 


6 


16 


43 


65232 


75794 


66545 


74544 


67837 


73472 


6)109 


mU 


7036 


71059 


17 


5 


16 


44 


65254 


75775 


66556 


74625 


67859 


73452 


6913 


72257 


70381 


71039 


16 


5 


17 


45 


65276 


75756 


665SS 


74606 


6788 


73432 


69151 


7223 


704)1 


71019 


15 


5 


17 


46 


65293 


75738 


6661 


745S6 


67901 


73413 


6172 


72216 


70422 


709 < 8 


14 


4 


17 


47 


6532 


75719 


66632 


74507 


67923 


73393 


69193 


72196 


70443 


70978 


13 


4 


18 


43 


65342 


757 


66653 


! 74548 


67944 


73373 


69214 


72176 


70463 


70957 


12 


4 


18 


49 


65564 


7563 


66675 


74528 


07965 


73353 


69235 


7215, 


704S4 


70937 


11 


3 


18 


50 


653S6 


75661 


66697 


7450) 


67987 


73333 


69256 


72136 


70505 


70 16 


10 


3 


19 


51 


65408 


i 75642 


66718 


744S9 


68008 


73314 


69277 


72116 


705 5 


70896 


9 


3 


19 


52 


6543 


75*523 


6674 


7447 


6S029 


73294 


69293 


72095 


70541, 


70875 


S 


3 


19 


53 


65452 


i 75604 


66762 


74451 


68051 


73274 


69319 


72075 


70567 


70855 


7 


2 


20 


54 


65474 


75585 


66783 


74431 


68072 


73254 


6934 


72055 


70587 


70S34 


6 


2 


20 


55 


65496 


75566 


66805 


; 74412 


6S093 


73234 


69361 


72035 


7050S 


70S13 


5 


2 


21 


56 


655 IS 


; 75547 


66827 


i 74392 


68115 


73-215 


6)332 


72015 


70623 


70793 


4 


1 


21 


57 


6554 


i 75528 


66848 


74373 


6 s 136 


73195 


69403 


71995 


70549 


7'772 


3 


1 


21 


58 


65562 


! 75509 


6687 


i 7435.3 


GS157 


73175 


09424 


71974 


7007 


70752 


2 


1 


22 


59 


65531 


' 7549 


66891 74334 


6S179 


73155 


6)445 


71954 


7069 


70731 


1 





22 


60 


65606 


175171 


66913 74314 


682 


73135 .6)456 


71934 


70711 


70711 












N. cos. N. sine 


N. cos. N.sine 


N. cos. 


N.sine. I N.cos. N.sine N'.sino. 


N. cos. 


/ 








4 


9° 


1 4 


3° 


4 


7° 


i 4 


6° 


1 4 


5° 







310 NATURAL SINES AND COSINES, 

The preceding Table contains the Natural Sine and Cosine for ev- 
ery minute of the quadrant to the radius of 1, and although the deci- 
mal point is not put in the Table, it is always to be prefixed. 

If the Degrees are taken at the head of the columns, the minutes, 
sine, and cosine must be taken from the head also; and if they are taken 
at the foot of the column, they must be taken from the foot also. 
Illustration.— .3173 is the sine of 18° 30', and the cosine of 71° 30'. 

To CoixLpnte tlie Sine or Cosine for Seconds. 

When the Angle is less than 45°. 
Ascertain the sine or cosine of the angle for degrees and minutes 
from the Table ; then take the difference between it and the sine or 
cosine of the angle next below it. Look for this difference or remain- 
der,* if the Sine is required, at the head of the column of Proportional 
Parts, on the left side ; and if the Cosine is required, at the head of the 
column on the right side; and in these respective columns, opposite to 
the number of seconds of the angle in the column, is the number or 
correction in seconds to be added to the Sine, or subtracted from the 
Cosine of the angle. 

Example. —What is the sine of S° 9' 10" ? 

Sine of 8° 9', per Table = .14177 ; ) , .- ^ noo 

Sine of S° 10', « = .14205 ; \ <*#«™*. .W028. 

In left-side column of proportional parts, under 29, and opposite to 10', is 5, the 
correction for 10', which, being added to .14177 == .14182, the sine. 

Ex. 2.— What is the cosine of S° 9' 10"? 

Cosine of S° 9', per Table — .9S990 ; { ,,;#,„„„ nnn n4 
Cosine of S° 10' " = S9980 ; j *#«"«"*, Mm - 

In right-side column of proportional parts, under 4, and opposite to 10', is I, which, 
being subtracted from .98990 = .58989, the cosine. 

When the Angle exceeds 45°. 

Ascertain the sine or cosine for the angle in degrees and minutes 
from the Table, taking the degrees at the foot of it ; then take the dif- 
ference between it and the sine or cosine of the angle next above it. 

Look for the remainder, if the Sine is required, at the head of the 
column oi Proportional Parts, on the right side; and if the Cosine is 
required, at the head of the column on the left side ; and in these re- 
spective columns, opposite to the seconds in the angle, is the number 
or correction in seconds to be added to the Sine, or subtracted from 
the Cosine of the angle. 

Example.— What is the sine of 81° 50' 50"? 

fine of Sl° 50', per Table = .98983 ;\ dl/Ferenre 00004 
Sine of Sl° 5V, " = .9S9. ();) dl lf erenc ^ - UUUU4 ' 

In right-si 'e column of proportional parts, and opposite to 59', is 3, which, being 
added to .9S98G = .989S9, the sine. 

Ex 2 What is the cosine of Sl° 50' 50"? 

Cosine of 81° 50', per Table = .14205 ;) differ >0 00:5. 

Cosine of 81° 51, u = .141i»;j ^ 
In left-side column of proportional parts, and opposite to 50', is 24, which, being 
subtracted from .14205 == . 14181, the cosine. 

* The Tables in some instances, as in the example given, will give a unit too much, but this, in 
general, is of little importance. 






NATURAL SINES AND COSINES. 311 

To Compete the Number of Degrees, Minutes, and 
Seconds of a given Sine or Cosine. 

When the Sine is given. 
Eule. — If the given Sine is found in the Table, the degrees of it will 
be found at the top or bottom of the page, and the minutes in the 
marginal column, at the left or right side, according as the sine cor- 
responds to an angle less or greater than 45°. 

If the given sine is not found in the Table, take the sine in the Ta- 
ble which is the next less than the one for which the degrees, etc., are 
required, and note the degrees, etc., for it. Subtract this sine from the 
tabular sine next greater than it, and also from the given sine. 

Then, as the tabular difference is to the difference between the given 
sine and the tabular sine, so is 60 seconds to the seconds for the sine 
given. 
Example. — What are the degrees, minutes, and seconds for the sine of .75000? 
The next less sine is .74092, the arc for which is 4S° 35'. The next greater sine 
is .75011, the difference between which and the next less is .75011 — .74992 = 19. The 
difference between the less tabular sine and the one given is .75000 — .74992 z=8. 
Then 19 : 8 : : GO : 25 -f, which, added to 48° 35' = 48° 35' 25". 

When the Cosine is given. 

Rule. — If the given Cosine is found in the Table, the degrees of it 
will be found as in the manner specified when the Sine is given. 

If the given cosine is not found in the Table, take the cosine in the 
Table which is the next greater than the one for which the degrees, 
etc., are required, and note the degrees, etc., for it. Subtract this co- 
sine from the tabular cosine next less than it, and also from the given 
cosine. 

Then, as the tabular difference is to the difference between the given 
cosine and the tabular cosine, so is 60 seconds to the seconds for the 
cosine given. 

ExAiMPLE. — What are the degrees, minutes, and seconds for the cosine of .75000? 

The next greater cosine is .75011, the arc for which is 41° 24'. The next less co- 
sine is .749S2, the difference between which and the next greater is .75011 — 74992 = 
19. The difference between the greater tabular cosine and the one given is .75011 
— .75000 = 11. Then 19 : 11 : : GO : 35 — , which, added to 41° 24' — 41° W 35". 

To Compute tne Yersed Sine of an -A^ngle. 
Subtract the cosine of the angle from 1 . 
Example. — What is the versed sine of 21° 30' ? 

The cosine of 21° 30' is .93042, which. — 1 = .06953, the versed sine. 

To Compnte tne Co-versed. Sine of an .A.ngle. 

Subtract the sine of the angle from 1 . 
Example. — What is the co-versed sine of 21° 30'? 

The sine of 21° 30' is .3665, which, — 1 = .6335, the co-versed sine. 

To Compute tne Chord of an _A.ngle. 

Double the sine of half the angle. 
Example.— What is the chord of 21° 30' ? 

21° SO' 
Sine of — — - = sine of 10° 45' ±= .18652, which, X2 '= .37304, the chord. 



312 



NATURAL SECAXTS AXD CO-SECAXTS. 



Table of ZS atnral Secants and Co-secants. 



Prop. ' 
parts Sf 
to 1". a 



10' 





1 

2 

3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
11 
15 
16 
17 
18 
19 
20 
21 
22 
23 



.004 
.013 
.021 
.03 
.038 
.017 
.056 
.061 
.073 
.082 
.091 
.101 
.11 
.12 
.13 
.139 
.15 
.16 
.171 
.182 
.194 
.205 
.217 
.23 
.243 24 
.256 25 
.27 26 
.285 27 
.3 28 
.315 29 
.332 30 
.349 31 
.366 32 
.385 33 
.404 34 
.425 35 
.446 36 
.469 37 
.493 38 
.518 39 
.545 40 
.573 41 
.603 42 
.634 43 
.668 44 
.704 45 
.742 46 
.783 47 
.827 48 
.874 49 
.925 50 



.00015 

.00061 

.00137 

00244 

06382 

00551 

00751 

00983 

01246 

01543 

01872 

02234 

02631 ; 

03061 i 

03528 

04O3 I 

04569 

05146 

05762 

06418 

07114 

0/853 

08636 

09464 

10338 

1126 

12233 

13257 

14335 

1547 

16663 

17918 

19236 

20622 

22077 

23607 

25214 

26902 

28676 

30541 

32501 

34563 

36733 

39016 

41421 

43956 

46628 

49148 

52425 

55572 



1. 

1.00021 
1.00072 
,1.00153 
1.00265 
1.00408 
1.00582 
1.00787 
1.01024 
1.01294 
1.01595 
1.0193 
1.02298 
1.027 
1.03137 
1.03609 
1.04117 
1.04663 
1.05246 
1.05869 
1.06531 
1.07235 
1.07981 
1.08771 
1.09606 
1.10488 
1.11419 
1.124 
1.13433 
] . 14521 
1.15665 
1.16868 
1.18133 
1.19462 
1.20859 
1.22327 
1.23869. 
1.25489 
1.27191 
1.2898 ' 
1.30861 
1.32838 
1.34917 
1.37105 
1.39409 
1.41834 
1.44391 
1.47087, 
1.49933 
1.52938 
1.56114 



SECANTS. 
20' ! 30' 



40' 



50' 



Prop, 
parte 
tol'. 



.00002 1 

.00027 1 

.00083 1 

.00169 1 

,00287 1 



00435 I 



00614 
,00825 
,01067 
01312 
01649 
01989 1 
02362 1 
0277 1 
03213 1 
03691 1 



.00004 1 

.00034 1 

.00095 1 

.00187 1 

.00309 1 

.00462 1 

.006^7 1 

.00863 1 

.01111 1 

.0139 1 

.01703 1 

.02049 1 

.02428 1 

,02841 1 

.0329 1 

.03774 1 



,00007 1 
00042 1 
.00108(1 

.00205 1 



04205 1 



04757 
05347 



.04295 
.04853 



.05449 1 



.05976' 1.06085 
.06761 
,07479 
08239 



06645 1 
07356 1 
08109 1 
,08907 1 
09749 1 
1064 



.09044 1 



.11579 1 



12568 

1361 

14707 

15861 

17075 

1835 

19691 

21099 1 

22579 1 



24134 
25767 
27483 
29287 



,31183 1 

33177 1 

35274 1 

37481 1 

.39804 1 
42251 
44831 
,47551 

50422 1 

53455 1 

56661 1 



09895 
,10793 
.1174 
12738 
,13789 
,14896 
,16059 
.17283 
.18569 
.1992 
.21341 
.22833 
.244 
.26047 
.27778 
.20597 
.31509 
.33529 
,35634 
3786 



00333 
.00491 
.00681 
.00902 
.01155 
.0144 
.017.58 
.02109 
.02494 
.02914 
.03368 
.03858 
.04385 
.0495 
,05552 
,06195 
,06-78 
,07602 
.0837 
.09183 
,10041 
10947 
,11903 
1291 
,1397 
,15085 
.16259 



.00011 89 

.00051 88 

.00122 87 

.00224 86 

.00357 85 

.0052 84 

.00715 83 

.00942 82 

.012 81 

.01491 80 

.01814 79 

.02171 78 
.02562 
.02987 
.03447 
.03944 
.04477 



05047 , 72 
05657 71 
06306 



.25 
.76 
1.27 
1.78 
2.3 
2.81 
3.34 
3.86 
4.39 
4.94 
5c48 
6 04 
6 6 
7.18 
7.77 
8.37 
8.99 
9.62 
10.26 
70 10.93 



17493 1.17 



,40203 1. 

1.42672 1 

1.45274 1. 
1 



48014 
50916 
53977 
57213 



18789 

20152 

21584 

23089 

24669 

2633 

28075 

29909 

31837 

33864 

35997 

,38242 

40606 

43096 

45721 



06995! 69 1 11.61 
07727 68 j 12.32 
.08502 67 13.04 
09322 66 i 13.79 
10189 65 14.57 
11103 64 ,15.37 
12067. 63:16.21 
13083 62 17.07 
14152 61 17.97 
15277 60 ■ 18.91 
16461 59 1 19.89 
.17704 68 20.91 
19012 57 21.98 
203861 56 23.09 
2183 55 21,. 26 
54 25.49 
53 26.78 
26615 52 28.14 
28374 51 29.57 
30223 50 31.08 
32168 49,32.68 
34212 48,34.37 
47 36.16 
46 38.06 



23347 | 
2494 



36363 
38627 



41012 45,40.08 



43524 44 
16173 43 



,48491 1 
51414 1.51918 41 



54504 1. 
71 1. 



55036 4o 
58333 39 



60' i 50' 



40' 



30' 



i 20' I 10' ,Deg. 



42.24 

44.53 

47. 

49.63 

52.45 

55.49 



CO-SECANTS. 



NATURAL SECANTS AND CO-SECANTS. 



313 



Table— {Continued). 











SECANTS. 










Prop. 






Prop. 


parts 

to 1". 


Q 


0' 


10' 


20' 


30' 


40' 


50' || 


parts 
to r. 


.979 


51 


1.58902 


1.59475 


1.60054 


1.60639 


1.61229 


1.61825; 38 


58.75 


1.038 


52 


1.62427 


1.63035 


1.63648 


1.64268 


1.64894 


1.65526; 37 


62.29 


1.102 


53 


1.66164 


1.66809 


1.6746 


1.68117 


1.68782 


1.69452: 36 


66.1 


1.171 


54 


1.70132 


1.70815 


1.71506 


1.72205 


1.72911 


1.73624! 35 


70.24 


1.246 


55 


1.73445 


1.75073 


1.75888 


1 . 76552 


1.77303 


1. 78062 1 34 


74.74 


1.327 


56 


1.78829 


1.79604 


1.80388 


1.8118 


1.81981 


1.8279 1 33 


79.64 


1.416 


57 


1.83608 


1.84435 


1.85271 


1.86116 


1.8697 


1.87834! 32 


85. 


1.515 


58 


1.88708 


1.89591 


1.90485 


1.91388 


1.92302 


1.93226! 31 


90.87 


1.622 


59 


1.9416 


1.95106 


1.96062 


1.97029 


1.98008 


1.98998 30 


97.33 


1.74 


60 


2. 


2.01014 


2.02039 


2.03077 


2.04128 


2.05191 


29 


104.44 


1.872 


61 


2.0626 


2.07356 


2.08458 


2.09574 


2.10704 


2.11847 


28 


112.33 


2.017 


62 


2.13005 


2.14178 


2.15365 


2.16568 


2.17786 


2.19019 


27 


121.06 


2.18 


63 


2.20269 


2.21535 


2.22817 


2,24116 


2.25432 


2.26766 


26 


130.8 


2.362 


64 


2.28117 


2.29487 


2.30875 


2.32282 


2.33708 


2.35154 


25 


141.72 


2.564 


65 


2.3662 


2.38106 


2.39614 


2.41142 


2.42692 


2.44264 


24 


153.84 


2.797 


66 


2.45859 


2.47477 


2.49119 


2.50784 


2.52474 


2.5419 


23 


167.85 


2.06 


67 


2.5593 


2.57697 


2.59491 


2.61313 


2.63162 


2.6504 


22 


183.6 


2.36 


68 


2.66947 


2.68839 


2.70851 


2.7285 


2.74881 


2.76945 


21 


201.6 


3.705 


69 


2.79043 


2.81175 


2.83342 


2.85545 


2.87785 


2.90063 


20 


222.29 


4.104 


70 


2.9238 


2.94372 


2.97135 


2.99574 


3.02057 


3.04583 


19 


246.25 


4.57 


71 


3.07155 


3.09774 


3.1244 


3.15154 


3.1792 


3.20737 


18 


274.19 


5.117 


72 


3.23607 


3.26531 


3.29512 


3.32551 


3.35649 


3.38808 


17 


307.06 


5.768 


73 


3.4203 


3.45317 


3.48671 


3.52094 


3.55587 


3.59154 


16 


346.09 


6.549 


74 


3.62795 


3.66515 


3.70311 


3.74198 


3.78166 


3.82222 


15 


392.91 


7.496 


75 


3.8637 


3.90612 


3.94952 


3.99393 


4.03938 


4.08591 


14 


449.77 


8.662 


76 


4.13357 


4.18238 


4.23239 


4.28366 


4.33621 


4.39012 


13 


519.74 


0.12 


77 


4.44541 


4.50216 


4.56041 


4.62023 


4.68167 


4.74482 


12 


607.21 


1.975 


78 


4.80973 


4.87649 


4.94517 


5.01585 


5.08863 


5.16359 


11 


718.52 


4.553 


79 


5.24084 


5.32049 


5.40263 


5.4874 


5.57493 


5.66533 


10 


863.21 


7.602 


80 


5.75877 


5.85539 


5.95536 


6.05886 


6.16607 


6.27719 


9 


1056.1 




81 


6.39245 


6.51208 


6.63633 6.76547 


6.8998 


7.03962 


8 






82 


7.1853 


7.33719 


7.49571 


7.6613 


7.83443 


8.01564 


7 






83 


8.20551 


8.40466 


8.61379 


8.83367 


9.06515 


9.30917 


6 






84 


9.56677 


9.83912 


10.1275 


10.4334 


10.7585 


11.1045 


5 






85 


11.4737 


11.8684 


12.2912 


12.7455 


13.2347 


13.7631 


4 






86 


14.3556 


14.9579 


15.6368 


16.3804 


17.1984 


18.1026 


3 






87 


19.1073 


20.2303 


21.4937 22.9256 


24.5621 


26.4504 


2 






88 


28.6537 


31.2576 


34.3823138.2015 


42.9757 


49.1141 


1 






89 


57.2987 


68.7574 


85.9456 


114.593 


171.888 


343.775 











60' 


50' 


40' 


3 ' 


20' 


10' 


Deg. 





CO-SECANTS. 

The preceding Table contains the Natural Secants and Co-secants 
for every ten minutes of the quadrant, to the radius of 1. 

The degrees in the column on the left side and the minutes at the 
head of the page are for Secants, and contrariwise for Co-secants. 

If the degrees are taken in the column on the left side, the minutes 
and seconds must be taken from the head of the page ; and if they are 
taken from the column on the right side, the minutes and co-secant 
must be taken from the foot. 

Illustbation 1.16059 is the secant of 30° 30', and the co-secant of 59° 30'. 

Dn 



314 NATURAL SECANTS AND CO-SECANTS. 

To Compute the Secant or Co-secant for Minutes 
not given at the Head or Foot of the Columns. 

Ascertain from the Table the Secant or Co-secant of the angle for de- 
grees, and the next less number of minutes given in the line opposite to 
the degrees. Take the correction or number for one minute from the 
right-hand column of Proportional Parts, and opposite to the degrees 
given ; multipby it by the number of minutes, and add the product to 
the result for degrees and minutes before obtained, if the Secant is re- 
quired, and subtract it if the Co-secant is required. 

Example.— What is the secant of 25° 25' ? 

Secant of 25° 20', per Table = 1.1064. 

The correction for V over 25° is 15.37, which, multiplied by 5 — 77; and 1.1064 
-f- 77 = 1.10717, the secant. 

Ex. 2 What is the co-secant for 64° 35' ? 

Co-secant of 64° 30', per Table = 1.10793. 

The correction for V over 64° is 15.37, which, multiplied by 5" = 77 ; and 1.10793 

— 77 = 1.10716, the co-secant. 

To Compute the Secant or Co-secant of Seconds. 

Ascertain from the Table the Secant or Co-secant of the angle for de- 
grees and minutes. Take the correction or number for one second from 
the left-hand column of Proportional Parts, and multipl}' it by the num- 
ber of seconds ; add the product to the result for degrees and minutes 
before obtained, if the Secant is required, and subtract it if the Co-secant 
is required. 

Example.— What is the secant of 22° 40' 22"? 

Secant of 22° 40', per Table = 1.0S37. 

The correction for 1" over 22° is .217, which, multiplied by 22" =d 4.77; and 
1.0S370 + 4.77= 1.0S375, the secant. 

Ex. 2 What is the co-secant of 67° 19' 38"? 

Co-secant of 67° 19', per previous Rules = 1.0S3S3. 

The correction for 1" over 67° is .217, which, multiplied by 38" = S.25; and 1.0S383 

— 8.25 = 1 08375, the co-secant. 

To Compute the Secant or Co-secant of any ^ngle 
in Degrees, Minutes, and. Seconds. 

Divide 1 by the cosine of the angle for the Secant, and by the sine for 
the Co-secant. 
Example.— What is the secant of 25° 25'? 

Cosine of this angle = .90321. Then 1 H- .90321 = 1.10716, secant. 
Ex. 2.— What is the co-secant of 64° 35' ? 

Sine of this angle — .90321. Then 1 -4- .90321 = 1.10716, co-secant. 

To Compute the Number of* Degrees, Minutes, and 
Seconds of a given Secant or Co-secant. 

When the Secant is given, 
Proceed as by Eule, page 311, for Sines, substituting Secants for Sines. 
Example,— What i3 the secant for 1.56GS5? 
The next less secant is 1.55036, the arc for which is 49° 50'. 

The next greater secant is 1.5S333, the difference between which and the next less 
is 1.58333 — 1.55036 — 3297. 

Difference between the less tab. sine and the one given is 1.566S5 — 1.55036 = 1649. 
Then 32y7 : 1649 : : 60 : 30, which, added to 49° 50' = 49° 50' "SO", degrees. 

When the Co-secant is given, 
Proceed as by Rule, p. 311, for Cosines, substituting Co-secants for Cosines. 



NATURAL TANGENTS AND COTANGENTS. 



315 



Ta"ble of ^Natural Tangents and. Cotangents. 











TANGENTS. 










Prop- 


. 














bb 


Prop. 


parts 
to \". 


o 


0' 


10' 


20' 


30' 


40' 


50' 


G 


parts 
tol'. 


.485 







.00291 


.00582 


.00873 


.01164 


.01454 


89 


29. 


.485 


1 


'.01745 


.02036 


.02327 


.02619 


.0291 


.03201 


88 


29. 


.486 


2 


.03492 


.03783 


.04075 


.04366 


.04658 


.04949 


87 


29. 


.487 


3 


.05241 


.05532 


.05824 


.06116 


.06408 


.067 


86 


29. 


.488 


4 


.06993 


.07285 


.07577 


.0787 


.08163 


.08456 


85 


29. 


.489 


5 


.08749 


.09042 


.09335 


.09629 


.09923 


.10216 


84 


29. 


.491 


6 


.1051 


.10805 


.11099 


.11394 


.11688 


.11983 


83 


29. 


.493 


7 


.12278 


.12574 


.12869 


.13165 


.13461 


-.13758 


82 


30. 


.496 


8 


.14054 


.14351 


.14648 


.14945 


.15243 


.1554 


81 


30. 


.498 


9 


.15838 


.16137 


.16435 


.16734 


.17033 


.17333 


80 


30. 


.501 


10 


.17633 


.17933 


.18233 


.18534 


.18835 


.19136 


79 


30. 


.505 


11 


.19438 


.1974 


.20042 


.20345 


.20648 


.20952 


78 


30. 


.509 


12 


.21256 


.2156 


.21864 


.2217 


.22475 


.22781 


■77 


31. 


.513 


13 


.23087 


.23393 


.237 


.24008 


.24316 


.24624 


76 


31. 


.517 


14 


.24933 


.25242 


.25552 


.25862 


.26172 


.26483 


75 


31. 


.522 


15 


.26795 


.27107 


.27419 


.27732 


.28046 


.2836 


74 


31. 


.527 


16 


.28675 


.2899 


.29305 


.29621 


.29938 


.30255 


73 


32. 


.533 


17 


.30573 


.30891 


.3121 


.3153 


.3185 


.32171 


72 


32. 


.539 


18 


.32492 


.32814 


.33136 


.33459 


.33783 


.34108 


71 


32. 


.546 


19 


.34433 


.34758 


.35085 


.35412 


.3574 


.36068 


70 


33. 


.553 


20 


.36397 


.36727 


.37057 


.37388 


.3772 


.38053 


69 


33. 


.560 


21 


.38386 


.3872 


.39055 


.39391 


.39727 


.40065 


68 


34. 


.568 


22 


.40403 


.40741 


.41081 


.41421 


.41763 


.42105 


67 


34. 


.576 


23 


.42448 


.42791 


.43136 


.43481 


.43828 


.44175 


66 


34. 


.585 


24 


.44523 


.44872 


.45222 


.45573 


.45924 


.46277 


65 


35. 


.595 


25 


.46631 


.46985 


.47341 


.47698 


.48055 


.48414 


64 


36. 


.605 


26 


.48773 


.49134 


.49495 


.49858 


.50222 


.50587 


63 


36. 


.616 


27 


.50953 


.5132 


.51687 


.52057 


.52427 


.52798 


62 


37. 


.628 


28 


.53171 


.53545 


.53914 


.54296 


.54673 


.55051 


61 


38- 


-64 


29 


.55431 


.55812 


.56194 


.56577 


.56962 


.57348 


60 


38. 


.653 


30 


.57735 


.58123 


.58513 


.58904 


.59297 


.59691 


59 


39. 


.667 


31 


.60086 


.60483 


.60881 


.6128 


.61681 


.62083 


58 


40. 


.682 


32 


.62487 


.62892 


.63299 


.63707 


.64117 


.64528 


57 


41. 


.697 


33 


.64941 


.65355 


.65771 


.66189 


.66608 


.67028 


56 


42. 


.714 


34 


.67451 


.67875 


.68301 


.68728 


.69157 


.69588 


55 


43. 


.731 


35 


.70021 


.70455 


.70891 


.71329 


.71769 


.72211 


54 


44. 


.75 


36 


.72654 


.731 


.73547 


.73996 


.74447 


.749 


53 


45. 


.77 


37 


.75355 


.75812 


.76272 


.76733 


.77196 


.77661 


52 


46. 


.792 


38 


.78129 


.78598 


.7907 


.79544 


.8002 


.80498 


51 


47. 


.814 


39 


.80978 


.81461 


.81946 


.82434 


.82923 


.83415 


50 


49. 


.838 


40 


.8391 


.84407 


.84906 


.85408 


.85912 


.86419 


49 


50. 


.864 


41 


.86929 


.87441 


.87955 


,88472 


.88992 


.89515 


48 


52. 


.892 


42 


.9004 


.90568 


.91099 


.91633 


.9217 


.92709 


47 


53. 


.921 


43 


.92552 


.93797 


.94345 


.94896 


.95451 


.96008 


46 


55. 


.953 


44 


.96569 


.97133 


.977 


.9827 


.98843 


.9942 


45 


57. 


.99 


45 


1. 


1.00583 


1.0117 


1.01761 


1.02355 


1.02952 


44 


59. 


1.02 


46 


1.03553 


1.04158 


1.04766 


1.05378 


1.05994 


1.06613 


43 


61. 


1.06 


47 


1.07237 


1.07864 


1.08496 


1.09131 


1.0977 


1.10414 


42 


63. 


1.1 


48 


1.11061 


1.11713 


1.12369 


1.13029 


1.13694 


1.14363 


41 


66. 


1.15 


49 


1.15037 


1.15715 


1.16398 


1.17085 


1.17777 


1.18474 


40 


69. 


1.2 


50 


1.19175 


1.19*82 


1.20593 


1.2131 


1.22031 


1.22758 


39 

Deg. 


72. 






60' 


50' 


40' 


30' 


20' 


10' 





COTANGENTS. 



316 



NATURAL TANGENTS AND COTANGENTS. 



Ta"ble— {Continued). 

TANGENTS. 



Prop. 














| 




Prop. 


parts 
to 1". 


e 
51 


0' 


10' 


20' 30' 


40' 


50' - 1 par'ts 
1 O 1 tol'. 


'l.25 


1.2349 


1.24227 


1.24969 1.25717 


1.26471 


1.2723 1 38 


75. 


1.31 


52 


1.27994 


1.28764 


1.29541 : 1.30323 


1.3111 


1.31904 37 


78. 


1.37 


53 


1.32704 


1.33511 


1.34323' 1.35142 


1.35968 


1.368 36 


82. 


1,44 


54 


1.37638 


1.38484 


1.39336 1.40195 


1.41061 


1.41934 1 35 


86. 


1.51 


55 


1.42815 


1.43703 


1.44598 1.45501 


1.46411 


1.4733 34 


90. 


1.59 


56 


1.48256 


1.4919 


1.50133 1.51084 


1.52043 


1.5301 33 


95. 


1.68 


57 


1.53986 


1.54972 


1.55966 1.56969 


1.57981 


1.59002 32 


100. 


1.78 


58 


1.60033 


1.61074 


1.62125 1.63185 


1.64256 


1.65337 31 


107. 


1.88 


59 


1.66428 


1.6753 


1.68643 1.69766 


1.70901 


1.72047 30 


113. 


2. 


60 


1.73205 


1.74375 


1.75556 1.76749 


1.77955 


1.79174 29 


120. 


2.13 


61 


1.80405 


1.81649 


1.82906 1.84177 


1.85462 


1.8676 28 


128, 


2.27 


62 


1.88073 


1.894 


1.90741! 1.92098 


1.9347 


1.94858 27 


136. 


2.44 


63 


1.96261 


1.97681 


1.99116 2.00569 


2.02039 


2.03526 26 


146. 


2.62 


64 


2.0503 


2.06553 


2.08094 2.09654 


2.11233 


2.12832 25 


157. 


2.82 


65 


2.14451 


2.1609 


2.17749 2.1943 


2.21132 


2.22857J 24 


169. 


3.05 


66 


2.24604 


2.26374 


2.28167 : 2.29984 


2.31826 


2.33693. 23 


183. 


3.31 


67 


2.35585 


2.37504 


2.39449 2.41421 


2.43422 


2.45451 22 


199. 


3.61 


68 


2.47509 


2.49597 


2.51715 2.53865 


2.56046 


2.58261 21 


217. 


3.95 


69 


2.60509 


2.62791 


2.65109 2.67462 


2.69853 


2.72281 20 


235. 


4.35 


70 


2.74748 


2.77254 


2.79802 2.82391 


2.85023 


2.877 |19 


261. 


4,82 


71 


2.90421 


2.93189 


2.96004 2.98868 


3.01788 


3.04749 18 


289. 


5.36 


72 


3.07768 


3.10842 


3.13972 3.17159 


3.20406 


3.23714 17 


322. 


6.01 


73 


3.27085 


3.30521 


3.34023 3.37594 


3.41236 


3.44951 16 


360. 


6.79 


74 


3.48741 


3.52609 


3.56557 3.60588 


3.64705 


3.68909 15 


407. 


7.73 


75 


3.73205 


3.77595 


3.82083 3.86671 


3.91364 


3.96165 14 


464. 


8.9 


76 


4.01078 


4.06107 


4.11256 4.1653 


4.21933 


4.27471 13 


534. 


10.35 


77 


4.33148 


4.38969 


4.44942 4.51071 


4.57363 


4.63825 12 


621. 


12.2 


78 


4.70463 


4.77286 


4.843 |4.91516 


4.9894 


5.06584 11 


732. 


14.6 


79 


5.14455 


5.22566 


5.30928 5.39552 


5.48451 


5.57638 10 


876. 


17.8 


80 


5.67128 


5.76937 


5.8708 5.97576 


6.08444 


6.197031 9 


1068. 


22.19 


81 


6.31375 


6.43484 


6. 5605516. 69116 


6.82694 


6.96823 


8 


1331. 


28.46 


82 


7.11537 


7.26873 


7.42871 


7.59575 


7.77035 


7.95302 


7 


1708. 


37.83 


83 


8.14435 


8.34496 


8.55555 


8.77689 


9.00983 


9.2553 


6 


2270. 


52.8 


84 9.5144 


9.7882 


10.078 


10.3854 


10.7119 


11.0594 


5 


3168. 


78.8 


85 11.4301 


11.8262 


12.2505 


12.7062 


13.1969 


13.7267 


4 


4728. 


130.1 


86 14.3007 


14.9244 


15.6048 


16.3499 


17.1693 


18.075 


3 


7806. 




87 19.0811 


20.2056 


21.4704 


22.9038 


24.5418 


26.4316 


2 






88 28.6363 


31.2416 


34.3628138.1885 


42.9641 


49.1039 


1 






89 57 29 


68.7501 


85.9398 114.589 


171.885 


343.774 









| CO' 


50' 


40' | 30' 


20' 


10' jDeg. 












COTANC 


rENTS. 











The preceding Table contains the Natural Tangents and Co-tangents 
for every ten minutes of the quadrant, to the radius of 1. 

The degrees in the column on the left side and the minutes at the 
head of the page are for Tangents, and contrariwise for Cotangents. 

If the degrees are taken in the column on the left side, the minutes 
and tangents must be taken from the head of the page ; and if they 
are taken from the column on the right side, the minutes and cotan- 
gents must be taken from the foot. 

Illustration. — .1974 is the ta< gent for 11° 10', and the cotangent for 78° 50'. 



NATURAL TANGENTS AND COTANGENTS. 317 

To Compete tlie Tangent or Cotangent for Minutes 
not given at the Head or Foot ofthe Columns. 

Ascertain from the Table the Tangent or Cotangent of the angle for de- 
grees, and the next less number of minutes given in the line opposite to 
the degrees. Take the correction or number for one minute from the 
right-hand column of Proportional Parts, and opposite to the degrees 
given; multiply it by the number of minutes, and add the product to 
the result for degrees and minutes before obtained, if the Tangent is re- 
quired, and subtract it if the Cotangent is required. 

Example.— What is the tangent of 10° 15'? 

Tangent of 10° 10', per Table — .17933. 

The correction for V over 10° is 30, which, multiplied by 5 (15 — 10) = 150 5 and 
.17933 4- 15) = .1S0S3, the tangent. 

Ex. 2.— What is the cotangent of T9° 45'? 

Cotangent of 79° 40' per Table — .18233. 

The correction for V over 40' is 30, which, multiplied by 5 (15 — 10)=: 150, and 
.18233 — 150 = .1S0S3, the cotangent. 

To Compute the Tangents or Cotangents for Sec- 
onds. 

Ascertain from the Table the Tangent or Cotangent ofthe angle for de- 
grees and minutes. Take the correction or number for one second from 
the left-hand column of Proportional Parts, and multiply it by the num- 
ber of seconds ; add the product to the result for degrees and minutes 
before obtained, if the Tangent is required, and subtract it if the Cotan- 
gent is required. 

Example What is the tangent of 54° 40' 40" ? 

Tangjnt of 54° 40\ per Table = 1.410GL 

The correction for V over 54° is 1.44, which, multiplied by 40" s= 5S, and 1.41061 
-j- 5S = 1.41119, the tangent. 

To Compute the Tangent or Cotangent of any ^n- 
gle in Degrees, Minutes, and. Seconds. 

Divide the Sine of the angle by the Cosine for the Tangent, and the Co- 
sine by the Sine for the Cotangent. 
Example.— What is the tangent of 25° IS'? 

The sine of this angle = .42730; the cosine of this angle = .90403. 

Then ^-^ = .4727, the tangent. 

To Compute the Number ofDegrees, Minutes, and 
Seconds of* a given Tangent or Cotangent. 

When the Tangent is given, 
Proceed as by Rule, page 311, for Sines, substituting Tangents for Sines. 
Example.— What is the tangent for 1.41119? 
The next less tang ;nt is 1.41001, the arc for which is 54° 40'. 
The next greate-t tangent is 1.41934, the difference between which and the next 
less is 1 . 4 934 — 1 .41001 = 873. 

The difference between the less tabular tangent and the one given is 1.410G1 — 
1.4111 I = 58. 

Then 873 : 580 (53x10 for tangent of 10') : : 60 : 40, which, added to 54° 40' = 
54° 40' 40". 

When the Cotangent is given, 
Proceed as by Rule, page 311, for Cosines, substituting Cotangents for 
Cosines. 

Dd* 



318 MECHANICS, 



'MECHANICS. 

Mechanics is the science which treats of and investigates the ef- 
fects of forces, the motion and resistance of material bodies, and of 
equilibrium : it is divided into two parts — Statics and Dynamics. 

Statics treats of the equilibrium of forces or bodies at rest Dy- 
namics of the forces that produce motion, or bodies in motion. 

These bodies are further divided into the Mechanics of solid, fluid, 
and aeriform bodies ; hence the following combinations : 

1 Statics of Solid Bodies, or Geostatics. 

2. Dynamics of Solid Bodi.es, or Geodynamics. 

3. Statics of Fluids, or Hydrostatics. 

4. Dynamics of Fluids, or Hydrodynamics. 

5. Statics of Aeriform Bodies, or Aerostatics. 

6. Dynamics of Aeriform Bodies, Pneumatics or Aerodynamics. 
Forces are various, and are divided into moving forces or resistances ; as. 

Gravity, Heat or Caloric, Inertia, 

Muscular, Magnetism, Cohesion, 

Elasticity and Contractility, Percussio?i, Adhesion. 

STATICS. 
Composition and. Tr&esolntion of Forces. 

When two forces act upon a body in the same or in an opposite direc- 
tion, the effect is the same as if only one force acted upon it, being the 
sum or difference of the forces. 

Hence, when a body is drawn or projected in directions immediately op- 
posite by two or more unequal forces, it is affected as if it were drawn or 
projected by a single force equal to the difference between the two or 
more forces, and acting in the direction of the greater force. 

This single force, derived from the combined action of two or more 
forces, is their Resultant. 

The process by which the resultant of two or more forces, or a single 
force equidistant in its effect to two or more forces, is determined, is termed 
the Composition of Forces, and the inverse operation; or, when the com- 
bined effects of two or more forces are equivalent to that of a single given 
force, the process by which they are determined is termed the Decomposi- 
tion or Resolution of Forces. 

Two or more forces which are equivalent to a single force are termed 
Comj)onents. 

When two forces act on the same point their intensities are represented by 

the sides of a parallelogram, and their combined effect will be equivalent to 

that of a single force acting on the point in the direction of the diagonal of 

the parallelogram, the intensity of which is proportional to' the diagonal. 

Illustration — Attach three cords to a fixed point, c. Fig. 1 ; let c a and c b pass 

over fixed roller?, and suspend the weights A and B therefrom. 

The point c will be drawn by the forces A and B in the direc- 
tions a c and b c. Now, in order to ascertain which single force, 
P, would produce the same effect up>n it, set off the distance c 
m and c n on the cords in the same proportion of length as the 
weights of A and B; that is, so that c m : c n : : A.: B: then 
draw the parallelogram c m o n and the diagonal o c, and it will 
represent a single force, P, acting in its direction, and having 
the same ratio to the weights A or B as it has to the sides c m or 
c n of the parallelogram. Consequently, it will produce the same 
effect on the point c as the combined actions of A and B. 




MECHANICS. 



319 



(2.) 




The parallelogram, which is constructed from the lateral forces, and 
the diagonal of which is the mean force, is termed 
the Parallelogram of Forces. 

Illustration.— Assume a weight, W, Fig. 2, to be 
suspended from a ; then, if any distance, a o, is set off 
in numerical value upon the vertical line, a W, and the 
parallelogram, o r a s, is completed, a s and a r, meas- 
ured upon the scale a o, will represent the strain upon 
a c and a e in the same proportion that a o bears to the 
weight W. 

In like manner, when three or more forces are combined upon one point, 
it follows : 

If several forces act upon the same point, and their intensities taken in or- 
der are represented by the sides of a polygon, except one, a single force pro- 
portioned to and acting in the direction of that one side will be their resultant. 

EJqnilibrrurn of Forces. 

Two bodies which act directly against each other in the same line ar& 
in equilibrium when their quantities of motion are equal ; that is, when 
the product of the mass of one, into the velocity with which it moves or 
tends to move, is equal to the product of the mass of the other, into its act- 
ual or virtual* velocity. 

When the velocities w r ith which bodies are moved are the same, their forces 
are proportional to their masses or quantities of matter. Hence, when 
equal masses are in motion, their forces are proportional to their velocities. 
The relative magnitudes and directions of any two forces may be rep- 
resented by two right lines, which shall bear to each other the relations 
of the forces, and which shall be inclined to each other in an angle equal 
to that made by the direction of the forces. 

Illustration. — Assume a body, W, to weigh 150 lbs., anu 
resting upon a smooth surface, to be drawn by two forces, a 
and 6, Fig. 3-24 and 30 lbs., which make with each othel 
an angle ; oWftz: 105°, in which direction and with what 
acceleration will motion occur? 
Cos. a\Y b = 105°, and cos. 180 c — 105° = cos. 75°, the mean 

force . 

Pr= ^302 -J- 24 2— 2X30 X 24 cos. 75° — "/900 -f- 576 — 1440 
cos. 75° =VU16 — (1440x.'i5S82) = VU03.3 = 33.21 lbs. 
Tg = E21X32 : 166 = 
W 150 J 

The Angle of Repose is the greatest inclination of a plane to the hori- 
zon at which a body will remain in equilibrio upon it. 

Hence the greatest an.de of obliquity of pressure between two planes, 
consistent with stability, is the angle the tangent of which is equal to the 
co-efficient of the friction of the two planes. 

Angles of TScinilibrinrn at which, various Substan- 
ces will Repose, as determined "by- a Clinometer. 

Angle measured from a Horizontal Plane. 

Degrees. 




The acceleration is 



Lime-dust falling from a spout 45 

Wheat flour " " "44 

Malt flour " " u 40 

Saw-dust " u " 44 

Dry sand " " " 40 

Sand less dry " " " 39.6 

Wheat corn"' " * " 37 



Degrees. 

Malt corn fall'g from a spout 37 

Common mold " " 37 

Peas " " 35 

Coarse gravel heaps 35 to 38 

Common gravel 35 to 36 

Large flints 40 to 45 

Flints, half size 35 



* Virtunl velocity is the velocity which a body in equilibrium would acquire were the equilibrium 
to be disturbed. 



320 



MECHANICS. 



Table of Coefficients of Friction, and Angles of 
Repose. 

The Coefficient of Friction is the Tangent of the Angle of Repose measured 
from a Horizontal Plane. 

I Cotangent of 
Angle of Fric- 
Material. Coefficient. Angle. tion, or Expo- 

nent of Stabili- 
; ty of Material.. 



Dry masonry and brick-work 

Timber on stone 

Iron on stone 

Wood on wood 

Wood on stone 

Metals on metals 

Masonry on dry clay 

Masonry on moist clay 

Earth on earth 

Earth on dry sand, clay, and earth 

Earth on earth, damp clay 

Earth on earth, wet clay 

Earth on earth, shingle, or gravel 

Dry earth 

Fine sand 



6 to 


.7 


.4 




3 to 


.7 


2 to 


.0 


2 to 


.6 


15 to 


.25 


51 




33 




25 to 


1. 


38 to 


.75 


1. 




31 




81 tol 


.11 


81 




6 





31° to 35° 

22° 

16° 60' to 35° 

11° 20' to 26° 30' 

11° 20 to 31° 

8° 30 to 14° 

27° 

18° 15' 

14° to 45° 

21° to 37° 

45° 

17° 

39° to 48° 

39° 

31° 



1.67 to 1.43 

2.5 

3.35 to 1.43 

5 to 2 

5 to 1.67 

6 .'67 to 4 

1.96 

3. 

4tol 

2.63 to 1.33 

1. 

3.23 

1.23 to .9 

1.24 

1.67 



Revetment "Walls. 



When a wall sustains the pressure of earth, sand, or any loose material, it is called 
a revetment wall. 

The thrust of earth, etc., upon a wall is caused by a certain portion, in the shape 
of a wedge, tending to break away from the general mass. The pressure thus caused 
is similar to that of water, but the weight of the material must be reduced by a par- 
ticular ratio dependent upon the angle of natural slope, which varies from 45° to o0° 
(measured from the vertical) in earth of mean quality. 

The angle which the liue of rupture makes with the vertical is one half of the an- 
gle which the liue of natural slope, or angle of repose, makes with the same vertical 
line. When the earth is level at the top, the pressure of the earth may be ascer- 
tained by considering it as a fluid, the weight of a cubic foot of which is equal tc 
the weight of a cubic foot of the earth, multipl ed by the square of the tangent of 
half the angle included between the natural slope and the vertical. 

Therefore the squares of the tangents of .5 of -.5° and .5 of G0° = .1716, and .3333 
= .1710, which are the multipliers to be used in ordinary cases to reduce a cubic foot 
of the material to a cubic foot of equivalent fluid, will have the same effect as the 
earth by its pressure upon the wall. 

Pressure of Earth, against Revetment Walls. 

Let A B C D, Fig. 4, be the vertical section of a revet- 
ment wall, behind which is a bank of earth, A D/e; let 
D G represent the angle of repose, the line of rupture, or 
natural slope which the earth would assume but for the 
resistance of the wall. 

In sandy or loose earth the angle G D H is gene-nlly 
30° ; in firmer earth it is 3C° ; and in some instances it is 
45°. 

If the upper surface of the earth and the wall which sup., 
ports it are both in one horizontal plane, then the result- 
ant, I n, of the pressure of the bauk, behind a vertical wall, 
is at a distance D n of % A D. 

EciniliToriurxi of Piers. 

For a Diagram, Formula, and Illustration, see Gregory's Mathematics^ 
p. 220, 221. 







MECHANICS. 321 

Thickness of "Walls, T^oth. Faces "Vertical. 

Wall of Bricks. — Weight of a cubic foot, 109 lbs. avoirdupois, bank of 
vegetable earth behind it, A B = .16 A D. 

Wall of uncut /Stone. — 135 lbs. per cubic foot, bank as before, AB = 
.15 A D. 

Wall of Bricks.— Bank of clay, well rammed, A B = .17 A D. 

Wall of cut Freestone. — 170 lbs. per cubic foot, bank of vegetable earth, 
A B = .i3 A D ; if the bank is of clay, A B = .14 A D. 

Wall of Bricks.— Bank of sand, A B = .33 A D. 

Wall of uncut Stone. — Bank of sand, A B — .3 A D. 

Wall of cut Freestone.— Bank of sand, A B = .26 A D. 

Wall of Bricks.— -Bank of earth and gravel == .19 A D. 

Wall of uncut Freestone. — Bank of earth and gravel = .17 A D. 

Wall of cut Freestone. — Bank of earth and gravel = .16 A D. 

Wall of cut Stone. — Bank of common earth = .13 A D. 

Wall of cut Stone.— Bank of sand = .26 A D. 
The Friction in vegetable earths is .5 ; the pressure : in sand .4. 
When vegetable earths are cut in turfs and well laid in courses, the 
thrust is reduced .66. 

Note. — When the bank is liable to be saturated with water, the thickness of the 
wall should be doubled. 

The Line of Rupture behind a wall supporting a bank of vegetable earth 
is at a distance, A G, from the interior face, AD = .618 the height of it. 

When the bank is of sand, A G =-.677 h ; when of earth and small grav- 
el = .646 h ; and when of earth and large gravel = .618 h. 

The prism, the vertical section of which is A D G, has a tendency to 
descend along the inclined plane, G D, by its gravity; but it is retained 
in its place by the resistance of the wall, and by its cohesion to and fric- 
tion upon the face, G D. Each of these forces may be resolved into one 
which will be perpendicular to G D, and into another which will be par- 
allel to G D. The lines c i. i I represent the components of the force, of 
gravity, which is represented by the vertical line c /, drawn from the cen- 
tre of gravity, ia, of the prism. The lines nr,lr represent the compo- 
nents of the forces of cohesion and friction, which is represented by the 
horizontal line n I. The force that gives the prism a tendency to descend 
is i I, and the force opposed to this is r /, together with the effects of co- 
hesion and friction. 

Thus il = r I + cohesion -f- friction. Consequently the solution of prob- 
lems of this nature must be in a great measure experimental. 

It has been found, however, and confirmed experimentally, that the an- 
gle formed with the vertical, b}' the prism of earth that exerts the great- 
est horizontal stress against a wall, is half the angle which the angle of im- 
pose or natural slope of the earth makes with the vertical. 

Equilibrium of a wall is the condition of moments of wall and of earth 
or pressure against it being equal. 

The condition of equilibrium, therefore, of a vertical Revetment or Re- 
taining Wall exposed to the thrust of a bank of earth is, .5 A D x D C 2 X S 



AD 3 2 ADG . ^ /tan 2 .5ADGxs ^ n , 7 , 7 , 

= — — s tan 2 — - — , or A Dw — =D C or breadth o. 

6 2 v 3 o 



D G . ^ /t; 

-— , or A D^ - 

S and s representing specific weights or gravities of wall and earth. 

Illustration. — A revetment wall, Fig. 4, having a specific gravity of 2000, is 40 
feet in heigh% and it sustains earth of a specific gravity of 1428, having a naturaj 
slope of 52° 2-4' ; what should be the thickness of the wall to have equil.briuui ? 
Tangent.2 .52° 24' -h 2 = .242. 

40 /ff*"f = 40 /!*™ = 40 V.05T0 = 40X.24 = 0.0/tet 
V 3X2000 V ti° u ° 



322 



MECHANICS. 



When the Wall has the Section of a Prismoid, or an Exterior Slope or 
Batter, as B E — Fig. 5. 

(5.) i adt\ 

A B a /3(DC + CE)2 S _AD2(Stan.2±±pj 



- = batter, orCE. 

Illustration. — A trapezoidal embankment, Fig. 5, has a depth of 2C 
feet, and a bottom width of 4 feet ; what should be the width of the crown, 
the weight of the material and of the sustained earth being 200 and 125 
lbs. per cubic foot, and the angle of repose of the bank 45° ? 
D C E Tangents 45° -H 2 = .1716. 




v 7 f 



3 X42X200— 202 X125X-1 7 16 > 
200 
1.74 feet, the top width. 



j = V 5 -l = 2.26 feet. Consequently, 4 — 2.26 = 



When the Wall has the Section of a Prismoid, or an Interior Slope, 

[~\ A D G s AD 

A- D «. /w—z + tan. 2 — - — X a — breadth, or DC,n representing the base of 

\ 6n 2 z » n 



ADG 



the slope, or the - of the height. 
•] n J . 

Note When the co-efficient of friction is known, use it for tan. 2 

Illustration. — The height of a wall is 20 feet, the slope of the base is — of the 

height, the specific gravities of the bank and wall are 14 and 26, and the coefficient 
of the material is .166 ; what must be the thickness of the wall at the crown? 



20 



> V^3xW2 + ,166X 2^~2^ = 20 ^'° 00S83 + ,089885 ~ 1 = 20X ' 3 ^ 

feet ; and the thickness at bottom will be 5. 00S -f — of 20 — 5.008 -f 1 = 6.008 feet. 



Surcharged Revetments. 



(6.) 



When the earth stands above the wall, ABC, Fig. 6, 
/ with its natural slope A/, AB C is termed a Surcharged 

^jjHH^ Revetment, C g being the line of rupture, and therefore 
£rfe=sfe- AfffC is the part of the earth that presses upon the wall, 
which part must be taken into the calculation, with the 
exception of the portion ABe, which rests upon the wall ; 
that is, the calculation must be for the part C efg, which 
must be reduced to its equivalent quantity of fluid by 
multiplying the weight of a cubic foot of it by the square 
of the tangent of the angle eQ,g — the angfe of the line 
of rupture, or half the angle which the natural slope makes 
with the vertical, and then proceed as in the previous cases. 

EnLOankinents and "Walls. 

To Compute the Conditions of Equilibrium, of Emhanlz- 
ments or ^Walls sustaining Water. 

When both Faces are Vertical — Fig. 7. 

Assume the Perpendicular embankment or wall, A B 
C D, to sustain the pressure of the water, BCe/. 

Let Hbea vertical line passing through 0, the centre 
of gravity of the wall, c the centre of pressure of the wa- 
ter, the distance C c being = XBC. Draw c I perpen- 
dicular to AD; then, since the section A C of the wall 
is rectangular, the centre of gravity, 0, is in the centre 




MECHANICS. 323 

of the wall, and therefore D i = % D C. Now IDi is to be considered 
as a bent lever, the fulcrum of which is D, the weight of the wall acting 
in the direction of the centre of gravit}^ o, on the arm D i, and the press- 
ure of the water on the arm D I, or a force equal to that pressure thrust- 
ing in the direction Ir. 
Put P = pressure of the water, and W the weight of the wall. 

ThenP X DJ = PX^ = WX^,orP = 8 -|^. 

Note. When this equation holds, the wall or embankment will just he on the 

point of overturning ; but in order that the wall may have complete stability, this 
equation should give a larger value to P than its actual amount. 

The following formulas are for embankments of one foot in length ; for, 
if they have stability for that length, they will be stable for any other 
length. 

Let a represent depth of water and embankment, which are here supposed 

to be equal, b breadth of the embankment, S weight of water, and s that of the 

wall per cubic foot. 

a 2 
Then P = -^xlX§, also W = axbxlXs, each value being for 1 foot 

in length, which being substituted in the above equation, there will result 
a 2 Sbxabs 00 070 _ /3s „ 7S T 

-Q- S = ^ , or a 2 S = 3 6 2 5; 6w -~ = a, and aw sr — b ; 

which gives the breadth of an embankment or retaining wall that will 
just sustain the pressure of the water; the wall must therefore be made 
broader than shown by this equation, to give it due stabilitj'. 

Illustration. — The height of a wall, B C, equal to the depth of the water, is 12 
feet, and the respective weights of the water and the wall are 62.5 lbs. and 120 lbs. 
per cubic foot ; required the breadth of the wall, so that it may have complete stabil- 
ity to sustain the pressure of fresh water. 

3- = 1 VslO20 = 12X - 4166 = 5/ ^ 
the breadth that will just sustain the pressure of the water; therefore 1 foot should 
be added to this to give the wall complete stability ; hence 5 -J- 1 == 6, the required 
width of the wall. 

Illus. 2.— The width of a wall is 3 feet, and the weight of a cubic foot of it is 150 
lbs. ; required the height of the wall to resist the pressure of fresh water at the top. 

« = V¥= 3 /w = 3x2 - 683 = 8 - 049 ^- 

Illus. 3. — Required the thickness of a rectangular embankment or retaining wall, 
the height being 12 feet, and the weight of a cubic foot of its material 133.33 lbs., so 
that it may have just sufficient stability to retain its equilibrium against sea-water. 

12 X /i33^.= 12 s/w> = n V" 6 = 12X - 4 U "Z"*- 
When the Section is a Triangle — Fig. 8. 

Assume the Triangular embankment or wall, B C D, to 
sustain the pressure of the water, B C ef. 

Draw Dn bisecting B C in n; from the centre of press- 
ure, c, draw c I, perpendicular to B C, cutting D n in o, 
which is the centre of gravity of the triangular section 
of the wall ; also, draw o i D I respectively perpendicular 
™F f to D C, c I. Now I D i is to be considered as a bent lev- 
er, the fulcrum of which is D, the pressure of the water 
acting at I, and the weight of the wall in the direction of the centre of 
gravity, o, on D i. 

Put BC = o,DC = &, and the weights per cubic foot of the water and 
wall, S and s, as in the preceding cases. 




324 MECHANICS. 

Then cC = oi = lD = %a, and consequently T>i=z%DCz=%b; 
the weight of 1 foot in length of the wall == % abs, and the pressure at 
c of the same length of water = % a 2 S ; hence .QGGbx.&abs = .333 a 

/ S /2 s 

X .5 a 2 S ; whence aJ ~- = b, and &\/"a~ = «• 

Note. — The embankment, BCD, has equal resistance at any portion of its height 
for the corresponding depth of water. 

2 An embankment or retaining wall with a triangular section as above has great- 
er resistance than one with a rectangular section for equal heights, and like volumes 
and qualities of materials, in the proportion of 8 to 3. 

Illustration. — There is a triangular embankment of brick-work, weighing 117 
lbs. per cubic foot ; its depth is 14 feet ; required its width at the base to resist the 
pressure of fresh water standing at the surface. 

Hence the breadth of the base of the embankment must be at least 8 feet to insure 
perfect stability. 

When the Surface of the Water is below the Top of the Wall. 
Put d 0= depth of the water; then c = .bd 2 S, and W = .5abs, as be- 
fore : therefore. 6666x.5a65 = .333c?x.5cZ 2 S; whence d* — = b. 

V2as 

Illustration. — A triangular embankment is 12 feet in depth; the weight of the 
material is 130 lbs. per cubic foot ; required its width at the base to resist the press- 
ure of fresh water 10.5 feet deep. 

When the Wall has the Section of a Prismoid, or an Exterior Slope or 
Batter, as A D— Fig. 9. 

Assume the Prismoidal embankment, A B C D, to 
sustain the pressure of the water, B C ef. 

Draw A E perpendicular to D C ; put B C, as before 
= a, the top breadth AB = EC= &, and the bottom 
width, D E, of the sloping part, AED = c. 

Then the weights of the portions A C and A E D re- 
spectively for one foot in length are abs and .5 acs, 
these weights acting at the points n and i respectively. 

Now D n = D i + .5 E C = c -f .5 b, and D i = .666 D E = .666 c ; hence 
the sum of the moments of the embankment is a b s (c -f- .5 b) + .5 a c s x 
.666 c = .5 (b 2 -f 2 b c -f .666 c 2 ) a s, which must be equal to the pressure 
of the water. .-. .5 (b 2 + 2 b c + .666 c 2 )as = .333 a x .5 a 2 S, or (b 2 + 
2bc+ .666c 2 )s = .333a 2 S. 
Hence, when the depth, a, and the bottom width, b -4- c, are given, 
// 3(b+c) 2 s-a 2 S \ 

Illustration A trapezoidal embankment has a depth of 12 feet, and a bottom 

width of 6 feet ; required the top width, to resist the pressure of an equal depih of 
fresh water, the weight of the material being 100 lbs. per cubic foot. 

//3X62 X 100-122 X G2.5\ /1S0O A tif 

Consequently, 6 — 4.24 =± 1.76 feet, the top width. 

Note. — It frequently occurs that the face of an embankment has also a slope or 
batter ; in this case the section of the embankment is to be divided into two trian^ 
gles and a parallelogram, and the moments of the several parts added together, as in 
the last problem. 




MECHANICAL POWERS. 325 



MECHANICAL POWERS. 

Power is a compound of weight, or force and velocity : it can not 
be increased by mechanical means. 

The Powers are three in number— viz., Lever, Inclined Plane, and 
Pulley. 

Note. — The Wheel and Axle is a continuous or revolving lever, the Wedge a dou- 
ble inclined plane, and the Screw a revolving inclined plane. 

LEVER. 

Levers are straight, bent, curved, single, or compound. 

To Compute tlie Length of* a Lever, tlie Weight 
and Power being given. 

Rule. — Divide the weight by the power, and the quotient is the differ- 
ence of leverage, or the distance from the fulcrum at which the power 
supports the weight. 

W 

Or, — = p, W representing the weight, P the power, and p the distance of the power 

from the fulcrum. 
Example. — A weight of 1600 lbs. is to be raised by a power or force of 80 lbs. ; re- 
quired the length of the longest arm of the lever, the shortest being 1 foot. 

To Compute tlie Weight that can. be raised by a 
Lever, its Length, the Power, and. the Position 
of its Fulcrum being given. 

Rule. — Multiply the jJower by its distance from the fulcrum, and di- 
vide the product by the distance of the weight from the fulcrum. 

W 

Example.— What weight can be raised by a power of 375 lbs. suspended from the 
end of a lever 8 feet from the fulcrum, the distance of the weight from the fulcrum 
being 2 feet? 375x8 . * ■ 

— - — — 1500 lbs. 

2i 

To Compute the Position of the Fulcrum, the 
Weight and Power and the Length of the Lever 
being given. 

When the Fulcrum is between the Weight and the Power. 

Rule.— Divide the weight by the power, add 1 to the quotient, and di- 
vide the length by the sum thus obtained. 

Or, L-r- f — + 1 J = Z, I representing length of lever between the weight and fulcrum. 

Example.— A weight of 2460 lbs. is to be raised with a lever 7 feet long and a 
power of 300 lbs. ; at what part of the lever must the fulcrum be placed ? 
2450 
-^- = 8.2, "and 8.2 + 1 = 9.2. Then 84 (7x12) -4- 9.2 = 9.13 inches. 

When the Weight is between the Fulcrum and the Power. 

Rule.— Divide the length by the quotient of the weight, dividod by the 
power . w 

Or,L-£ = Z. 

Ee 






326 MECHANICAL POWERS. LEVER. 

To Compute the Length of Arm of the Lever to 
which tlie Weight is attached, the Weight, Pow- 
er, and. Length of Arm of the Lever to which the 
Power is applied being given. 

Rule. — Multiply the power by the length of the arm to which it is ap- 
plied, and divide the product by the weight. 

Or,— = *'. 

Example. — A weight of 1600 lbs. suspended from the fulcrum of a lever is sup- 
ported by a power of SO lbs. applied at the other end of the arm, 20 feet in length ; 
what is the length of the arm ? 

80x20 ■ ■ , t 

Note. — These rules apply equally when the fulcrum (or support) of 
the lever is between the weight and the power ;* when the fulcrum is at one 
extremity of the lever, and the power, or the weight, at the other ;f and 
whenihe arms of the lever are equally or unequally bent or curved. 

To Compute the Power required to Raise a given 
Weight, the Length of the Lever and the Posi- 
tion of the Fulcrum being given. 

Rule.— Multiply the weight to be raised by its distance from the ful- 
crum, and divide the product by the distance of the power from the ful- 
crum. ^ Wxw = p 

1 V 
Example— The length of a lever is 10 feet, the weight to be raised is 3000 lbs., 
and its distance from the fulcrum is 2 feet ; what is the power required ? 
3000x2 6000 „ Krt • 

To Compute the Length of Arm. of the Lever to 
■which the Power is applied, the Weight, Power, 
and Distance of the Fulcrum being given. 

Rule. — Multiply the weight by its distance from the fulcrum, and di- 
vide the product by the power. 

~ Wxio 
Or, — — =p. 

Example. — A weight of 400 lbs. suspended 15 inches from the fulcrum is sup< 
ported by a power of 5 J lbs. applied at the other ; what is the length of the arm ? 

400X15 . 

— — — = 120 inches. 
50 

Note. — When the arms of a lever are bent or curved, the distances 

taken from perpendiculars, drawn from the lines of direction of the 

weight and power, must be measured on a line running horizontally 

through the fulcrum. 

The General, Rule, therefore, for ascertaining the relation of Power 
to Weight in a lever, whether it be straight or curved, is, 

The power multiplied by its distance from the fulcrum is equal to I 
weight multiplied by its distance from the fulcrum. 

Or, P : W : ruj; p, or Fxp = Wxw; and 

l.^ = P. tS^W. 8. *£?=,. 4. **? = * 

p w P, r W 

* The pressure upon fulcrum is equal to the sum of the weight and the power. 

f The pressure upon fulcrum is equal to the difference of the weight and the potver. 



MECHANICAL POWERS. WHEEL AND AXLE. 327 

If several weights or powers act upon one or both ends of a lever, the 
condition of equilibrium is 

PX_p + P'X/>'-fP"X v'\ etc., = WXu>-f-W / Xw / ,etc. 
In a system of levers, either of similar, compound, or mixed kinds, the 
condition is VXyXp'Xp" _ 

wXw'Xw" 
Illustration— Let P = 1 lb., p and p' each 10 feet, p" 1 foot ; and if w and w' 
be each 1 foot, and w" 1 inch, then 
1X120X120X12 _ 172800 _ ±m , ^ .^ j ^ ^ support 1200 lbs. with levers of 

the lengths above given. 
Note. — The weights of the levers in the above formulae are not considered, the 
centre of gravity being assumed to be over the fulcrums. 

WHEEL AND AXLE. 

A Wheel and Axle is a revolving lever. 

The power, multiplied by the radius of the wheel, is equal to the 
weight, multiplied by the radius of the axle. 

Or, Px R = WXr. Or, PxV = Wx«,R and r representing the radii, and V and v 
the velocities of wheel and axle. 

As the radius of the wheel is to the radius of the axle, so is the ef- 
fect to the power. 

Or, Pv : r : : W : P. 

When a series of wheels and axles act upon each other, either by 
belts or teeth, the weight or velocity will be to the power or unity as 
the product of the radii, or circumferences of the wheels, to the prod- 
uct of the radii, or circumferences of the axles. 

Example. — If the radii of a series of wheels are 9, 6, 9, 10, and 12, and their pin- 
ions have each a radius of 6 inches, and the power applied is 10 lbs., what weight will 
it rake? 10x9x6x9x10x12 _ 

6XGX6X6X6 ; 

Or, if the 1st wheel make 10 revolutions, the last will make 75 in the same time. 

To Compute tlie Power of a Coni~bination ofWheels 
and. an Axle or Axles, as in Cranes, etc., etc. 

Rulic. — Divide the product of the driven teeth by the product of the 

drivers, and the quotient is their relative velocity, which, multiplied by 

the length of the winch and the power applied to it in lbs., and divided 

by the radius of the barrel, will give the weight that can be raised. 

v w P . 

Or, — — = W, w representing length of winch, r radius of barrel. 

Or, W r = v w P, P u power. 

Wr 

Or, =P, v " velocity, and W weight. 

vw 

Example A power of IS lbs. is applied to the winch of a crane, the length of it 

being 8 inches; the pinion having 6, the driving-wheel 72 teeth, and the barrel 6 
inches diameter. 

72 

— = 12, and 12xSxlS = 1728, which, -h 3, the radius of the barrel = 576 lbs. 

Ex. 2. — A weight of 94 tons is to be raised 360 feet in 15 minutes, by a power the 
velocity of which is 220 feet per minute ; what is the power required ? 

24x94 
360 -h 15 — Ufeet per minute. Hence *[ = 10.254 tons. 



328 MECHANICAL POWERS. BACK AND PINION. 

COMPOUND AXLE, OR CHINESE WINDLASS. 

The axle or drum of the windlass consists of two parts, the diameter of 
one being less than that of the other. 

The operation is thus : At a revolution of the axle or drum, a portion of 
the sustaining rope or chain equal to the circumference of the larger axle, 
is wound up, and at the same time a portion equal to the circumference of 
the lesser axle is unwound. The effect, therefore, is to wind up or shorten 
the rope or chain, by which a weight or stress is borne, by a length equal 
to the difference between the circumferences of the two axles. Conse- 
quently, half that portion of the rope or chain will be shortened by half 
the difference between the circumferences. 

To Compute the Elements of a Wheel and. Com- 
pound Axle, or Chinese Windlass— Fig. 1. 

Rule. — Multiply the power by radius of the wheel, arm, or bar to which 
" it is applied, and divide the product by half the differ- 
(1-) r ence of the radii of the axle, and the' quotient is the 

weight that can be sustained. 

Xir — r') 

PxR = "Wx X (r — r'), R representing radius of -wheel, etc., 
and r and r' radii of axle at its greatest and least diameter. 

Example. — What weight can he raised by a capstan, the ra- 
dius of its bar, a, being 5 feet, the power applied 50 lbs., and 
the radii, r r\ of the axle or drum 6 and 5 inches? 

— — : r^ s= — — = 6000 lbs. r= product of power and length 

% (b — o) .5 

of bar in inches -— % difference of radii of axle. 

WHEEL AND PINION COMBINATIONS, OR COMPLEX WHEEL- 
WORK. 

The power, multiplied by the product of the radii or circumferences, or 
number of teeth of the wheels, is equal to the weight, multiplied by the 
product of the radii or circumferences, or number of leaves of the pinions. 
Or, PxRXPw'XR", etc.=WXrXr'Xr", etc. 

Note. — The cogs on the face of the wheel are termed teeth, and tlrnse on the sur- 
face of the axle are termed leaves ; the axle itself in this case is termed a pinion. 

RACK AND PINION. 
To Compute the Po^^^eI* of* a Rack and Pinion. 

Rule. — Multiply the weight to be sustained by the quotient of the ra- 
dius of the pinion, divided by the radius of the crank, and the product is 
the power required. _ 

Or, Wx- = P. 

K 

When the Pinion on the Crank Axle communicates with a Wheel and 

Pinion. 

Rule. — Multiply the weight to be sustained by the quotient of the prod- 
uct of radii of the pinions, divided by the radii of the crank and the wheel, 
and the product is the power required. 






MECHANICAL POWERS. — INCLINED PLANE. 329 

Example. — The radii of the pinions of a jack-screw are each 1 inch ; of the crank 
and wheel 10 and 5 inches; what power will sustain a weight of 750 lbs. ? 



INCLINED PLANE. 



To Compute the Length of the Base, Height, or 
Length., when any Two of* them are given. 

When the Line of Direction of the Power or Traction is Parallel to the 
Face of the Plane. 
Proceed as in Mensuration or Trigonometry to determine the side 
of a right-angled triangle, any two of the three being given. 

To Compute the Power necessary to Support a 
"Weight on an Inclined. 3?lane, the Height and. 
Length heing given. 

Rulk. — Multiply the weight by the height of the plane, and divide the 
product by the length. 

Or, — - — — P, h and I representing the height -and length of the plane 

Example.— What is the power necessary to support 1000 lbs. on an inclined plane 
4 feet high and 6 feet in length ? 

12^ = 666.67 lis. 

To Compute the "Weight that may he Sustained "by 
a given IPovvrer on an Inclined IPlane, the Height 
and Length of* the 3?lane heing given. 

Rule. — Multiply the power by the length of the plane, and divide the 
product by the height. 

h 
Example.— What 13 the weight that can be sustained on an inclined plane 5 feet 
high and 7 feet in length by a power of 700 lbs. ? 

— ^— = 200 lbs. 
5 

Note. —In estimating the power required to overcome the resistance of a "body, be- 
ing drawn up or supported upon an inclined plane, and contrariwise, if the body is 
descending ; the weight of the body, in the proportion of the power of the plane (i. e., 
as its length to its height), must be added to the resistance if being drawn up or 
supported, or to the moment if descending. 

To Compute the Height or Length of an Inclined 
3?lane, the ^Weight and Power and one of the re- 
quired Elements being given. 

When the Height is required. 

Rule. — Multiply the power by the length, and divide the product by the 
weight. 

When the Length is required. 

Rule. — Multiply the weight by the height, and divide the product by 
the power. 

Or, — = *, and —£- = *■ 
Ee* 




330 MECHANICAL POWERS. INCLINED PLANE. 

Xo Compute the IPressnre on an Inclined Plane. 

Rule. — Multiply the weight by the length of the base of the plane, and 
divide the product' by the length of the face. 

Wx& 
Or, — - — = pressure, b representing length of base of plane. 

Example. — The weight on an inclined plane is 100 lbs., the base of the plane is 4 
jfeet, and the length of it 5 ; required the pressure on the plane. 

5 

When two Bodies on two Inclined Planes sustain each other, as by the 
Connection of a Cord over a Pulley, their Weights are directly as the 
Lengths of the Planes. 

Illustration If a weight of 50 lbs. upon an inclined plane, of 10 feet rise in 100, 

be sustained by a weight on another plane of 10 feet rise in 90 of an inclination, 
what is the weight of the latter ? 

Therefore 100 : 90 : i 50 : 45, the weight that on the shortest plane would sustain 
that on the largest. 

When a Body is Supported by two Planes, as Fig. 1 . 

The pressure upon them will be reciprocally as the 
sines of the inclinations of the planes. 
Thus the weight is as sin. A B D. 

The pressure on A B as sin. D B I. 
The pressure on B D as sin. A B H. 
Assume the angle A B D to be 90°, and D B I, 60 ; then the 
angle A B H will be 30° ; and as the sines of 90°, 60°, and 30° 
are respectively .1, .S66, and .5. if the weight = 100 lbs., then the pressures on AB 
and D B will be S6.6 and 50 lbs., the centre of gravity of the weight being in its 
centre. 

When the Line of Direction of the Power is parallel to the Base of the 

Plane. 

The power is to the weight as the height of the plane to the length of its 
base. 

Or, P : "W : : h : b, b representing the length of its base. 

HenC3 p =— ; w =— ' h =-w'' b =—- 

When the Line of Direction of the Power is neither parallel to the Face 
of the Plane nor to its Base, but in some other Direction, as P , Fig. 2. 

The power is to the weight as the sine of the angle of the plane's eleva- 
tion, to the cosine of the angle which the line of the power or traction de- 
scribes with the face of the plane. 

Thus, P' : W : : sin A : cos. P'co. 

(2.) Sin. A : cos. P' c o : : P' : W. 

8\ Cos. P' c o : sin. A : : W : P'. 

. P' Illustration. — A weight of 500 lbs. is required to be 
sustained on a plane, the angle of elevation of which, 
c AB, is 10° ; the line of direction of the power or trac- 
tion, P' e c, is 5° ; what is the sustaining power re- 
quired ? 

Cos. P' e c (5°) = .9962 : sin. A (10°) = .1737 : : 500 : 
87.1S lbs. 

Or, draw a line, Bs, perpendicular to the direction of the power's action 
from the end of the base line (at the back of the plane), and the intersec- 
tion of this line on the length, A c, will determine the length and height, 
n r, of the plane. 




MECHANICAL POWERS. — WEDGE. — SCREW. 331 

Illustration. — Of the last Example. 

By Trigonometry (page 296). A B, assumed to be 1, A r and n r are = .9S5 and 
.171. 

Hence — — '- — = 86.8 lbs = the product of the weight — the height of the plane 

-f- the length of it 

Note. — When the line of direction of the power is parallel to the plane, the power 
Lb least. 

WEDGE. 

A Wedge is a double inclined plane. 

To Compute the Power. 

1. When two Bodies or two Parts of a Bod// are Forced or Sustained in 

a Direction parallel to the Back of the Wedge. 

Rule. — Multiply the weight or resistance to be sustained by half the 
depth of the back of the wedge, and divide the product by the length of 
the wedge. 

Or, — '■ — =P, d representing the depth of the back, and I the length. 

Example. — The depth of the back of a double-faced wedge is 6 inches, and the 
length of it through the middle 10 inches ; what is the power necessary to sustain or 
overcome a resistance of 150 lbs. ? 

150X6--2 _45Q 

10 - 10 -*>"*• 

2. When one Body only is to be Forced or Sustained. 

Rule. — Multiply the weight or resistance to be sustained by the depth 
of the back of the wedge, and divide the product by the length of its base. 

Example.— What power, applied to the back of a wedge 6 inches deep, will raise a 
weight of 15,000 lbs., the wedge being 100 inches long on its base? 

15000x6 90000 nAA „ 

— . — — 900 lbs. 

100 100 

To Compute the Elements of a Wedge. 

1. £« W. 2. ™,£W. 

d-T-z d 

WWT2 =; % Wxd = L 1. Wxtt : 2 = r. 

Note. — As the power of the wedge in practice depends upon the split or rift in the 
wood to be cleft, or in the rise of the body to be raised, the above rules as regards 
the length of the wedge are only theoretical when a rift or rise exists. 

SCREW. 
A Screw is a revolving inclined plane. 

To Compnte the Length, and Height of the Plane 
of a Screw. 

As the screw is an inclined plane wound around a cylinder, the length 
of the plane is ascertained by adding the square of the circumference^ of 
the screw to the square of the distance between the threads, and taking 
the square root of the sum ; and the height or pitch of the screw is the 
distance between its consecutive threads. 



332 MECHANICAL POWERS. SCREW. 

To Compute tlie Power. 

Rile. — Multiply the weight or resistance, to be sustained by the pitch 

of the threads, and divide the product by the length of the plane. 

Wxp 
Or, — — = P, p representing the pitch. 

Example. — What is the power requisite to raise a weight of S000 lbs. by a screw 
of 12 inches circumference and 1 inch pitch? 

122 _i_ 12 — 145 an d ^145 — 12.0415?. Then 80 °° Xl = 604.36 lbs. 

-1*^.0410 

To Compute tlie "Weight. 

Bulk. — Proceed as above, substituting the power for the weighty and 
transposing the length of the plane and the pitch of the threads. 

p 

When the Diameters or Circumferences of the Screw or of the Point at 

ichich the Power is applied are given. 

Rule. — Ascertain the length of the plane from the diameter or circum- 
ference given, and proceed as before. 

Note. — When a lever is used to transmit the power, the circumference described 
by the power is not that due to the radius of the lever alone, but it is the path de- 
scribed in one revolution, i. e., the hypothenuse of the triangle (length of the plane), 
of which the circumference of the screw or lever is the buse and the pitch of the 
threads the height of it. Hence the diameter of a screw is not a necessary element 
in determining the weight it will support, when the point at which the power is ap- 
plied is given. 

WXP Pxrf 
The preceding formulae then become — - — == P. s#: W, d representing the 

distance described by the power. 

Example. — If a lever of 30 inches in length was added to the circumference of the 
screw in the preceding example. 

3 S19 

Then 12 -f- 3.1416 = 3.S19 — diameter of screw ; -^— -f 30 = 31.9095 = sum of 

radius of screw and length of lever ; and 01.9095x2x3. 1416 = 200. 493S = fAe cir- 
cumference described by the end of the lever — the base of the triangle. 

IIenceV200.493S2 + 12 = 200.5 = length of plane described by the power. 

8000x1 
Consequently, Q = 39.90 lbs. 

When a Screw and Lever is combined with a Wheel and Axle, etc. 

Rule. — Multiply the power by the product of the circumference de- 
scribed by it and the radius of the wheel, and divide this product by the 
product of the pitch of the screw and the radius of the axle of the wheel. 

P X c X 1 a. 
Or t = W, c representing the circumference described by the power, R and r 

the radii of the v:heel and its axle. 
Note. — As the screw applied to the wheel is an endless one, i. e., it revolves with. 
©at advancing, the circumference due to the radius of the lever or crank is the dis- 
tance described by the power. 

Example. — What is the power of a screw having a pitch of J^ths of an inch, driv- 
en by a lever 30 inches in length, with a force of 50 lbs. Jipplied at its extremity? 
S0x 2 X 3.1 416 = 1SS. 5 = circu mfcrence described by lever — base of triangle; Jgthi 
== .S75 =z height of triangle. 

Then "/1SS.5 2 -f- .S75~3 = 135.5 — length of plane described by power, 

50x \f 5 =mnAi b ,. 



MECHANICAL POWERS. PULLEY. 333 

Note.— If there is more than one thread to a screw the pitch must be increased at 
many time's as there are threads. 

Ex. 2 What weight can be raised with a power of 10 lbs. applied to a crank 32 

inches long, turning an endless screw of 3>£ inches diameter and 1 inch pitch, ap- 
plied to a wheel of 20 inches diameter, upon an axle of 5 inches ? 

32x2x3.1416 = 201 inches = circumference of 04 inches. 



10x201 X (20 -4-2) 10x2012 OA , A . _ , . ' / L.-K 

— — — — — = S040 := product of power and product of eircumfer* 

lX(5"^~2) 2.5 

ence described by it and the radius of the wheel -4- product of pitch of screw and 

radius of axle. 

When a Series of Wheels and Axles are in Connection with each other. 
The weight is to the power as the continued product of the radii of the 
wheels is to the continued product of the radii of the axles. 
W:P::Rw:rw. 
Or, r n : Rn : : P : W, n representing the number of wheels or axles. 
ExAMr-LE. — If a power of 150 lbs. is applied to a crank of 20 inches radius, turning 
an endless screw with a pitch of half an inch, geared to a wheel, the pinion of which 
is geared to another Avheel, and the pinion of the second wheel is geared to a third 
wheel, to the axle or barrel of which is suspended a weight ; it is required to know 
what weight can be sustained in that position, the diameter of the wheels being 18, 
and the pinions and the axle 2 inches. 

150X20X2X3.1416 •''. 

z=3<6S0 = power applied to face of first wheel. 

The diameters of wheels and pinions being 18 and 2, their radii are 9 and 1. 
Hence lXlXl :'9x9x9 : 376S0 : : 27468720 lbs. 

Differential Screw. 

When a hollow screw revolves upon one of less diameter and pitch (as 
designed by Mr. Hunter), the effect is the same as that of a single screw, 
in which the distance between the threads is equal to the difference of the 
distances between the threads of the two screws. 

Therefore the power, to the effect or weight sustained, is as the differ- 
ence between the distances of the threads of the two screws : to the cir- 
cumference described by the power. 

Illustration. — If the external screw has 20 threads, and the internal one 21 
threads in an inch pitch, and the power applied describes a line of 35 inches, the re- 

sultisas !*,- = -, or .00238. Hence -J*L = 14706. 



PULLEY. 

Pulleys are designated as Fixed and Movable, according as the 
cord is passed over a fixed or a movable pulley. A movable pulley is 
when the cord passes through a second pulley or block in suspension ; 
a single movable pulley is termed a runner; and a combination of pul- 
leys is termed a system of pulleys. 

To Compute the Power required, to Raise a given 
"Weight, the 3S"unVber of IParts of the Cord sup- 
porting the Lower Block: being given. 

When only one Cord or Rope is used. 

Rule. — Divide the weight to be raised by the number of parts of the 
cord supporting the lower or movable block. 

W 

Or, — = P. Or, raXP — W, n representing the number of parts of the cord sus- 
taining the lower block. 



334 



MECHANICAL POWERS. — PULLEY. 



Example.— What power is required to raise 600 lbs. when the lower block con- 
tains six sheaves and the end of the cord is fastened to the upper block, and what 
power when fastened to the lower block ? 

1. 77777, = 50 lbs. = weight -r- number of parts of rope sustaining lower block. 



6x2 

600 
6X2+1 



= 46.15 lbs. = weight -5- number of parts of rope sustaining lower block. 



To Complete the "Weight a given Power Avill Raise, 
the !N"nmber of Parts of the Cord, supporting tlie 

Lower Block being given. 

Rule. — Multiply the power by the number of parts of the cord support- 
ing the lower block. 

Or, PXn = W. 




(2.) 



To Compute tlie Number of Cords necessary to 
Sustain the Lower Block, the Weight and Pow- 
er being given. 

Rule.. — Divide the weight by the power, and the quotient is the number 
of parts of cord required. 

^ W 

Or,-=n. 

When more than one Cord or Rope is used. 

In a Spanish Burton, Fig. 1, where the ends 
of one cord, a P, are fastened to the support 
and the power, and the ends of the other, c o, 
to the lower and upper blocks, the weight is to 
the power as 4 to 1. 

In another, Fig. 2, where there are two cords, 
a and o, two movable pulleys, and one fixed 
pulley, with the ends of one rope fastened to 
the support and upper movable pulley, and the 
ends of the other fastened to the lower block 
and the power, the weight is to the power as 5 
to 1. 

In a System of Pulleys, Figs. 3 and 4, icith any Number of Cords, o o, 
the Ends being fastened to the Suj>po?-t. 





W 

^z=P; 2«XP = W; 



W 
P : 



: 2n, n representing the numbei 



of distinct cords. 
Example.— What weight will a power of 1 
lb. sustain in a system of four movable pul- 
leys and four cords ? 

1X2X2X2X2 = 16/6S. 

When fixed Pidleys, e e, are used in the 
place of Hooks, to Attach the Ends of 
the Rope to the Supjwrt — Fig. 4. 



^ = P; 3n X V-. 

3» 



:W; |, 



:3n. 



Example. — What weight will a power of 5 lbs. sustain with four 
movable and four fixed pulleys, and four cords ? 
5X3X3X3X3 = 405^5. 




CRANES, 335 

When the Ends of the Cord or the fixed Pulleys are fastened to the Weight, 
as by an Inversion of the last Figures, putting the Supports for the 
Weights, and contrariwise — Figs. 3 and 4. 

w w 

Fi S- 3 - -7r^-r, = J1 i <2»-l)XP = W; - = (2»-l). 
Fig. 4. _^_- = P; (3»-l)xP = W; ^ = (2*-l). 

Example.— What weight will a power of 1 lb. sustain in a system of two movable 
pulleys and two cords ? 

1X2X2 — 1 = 3 lbs. 

Ex. 2. — What weight will a power of 1 lb. sustain with a system of two movable 
and two fixed pulleys and two cords. 

1X3X3 — 1 = 8 lbs. 
And in the two examples preceding the last, 

1X2X2X2X2 = 10— 1 = 15/^5.; 5x3x3x3x3 = 4C5 — 1 = 404 Ms. 

When the Cords by which the Pulleys are sustained are not in a Vertical 
Direction — Fig. 5. 

e o, Fig. 5, is the vertical line through which the weight 
bears, and from o draw o r, o s parallel to D e and A e. 

The forces acting at e are represented by the lines e s, e r, 
and e o; and as the tension of every part of the cord is the 
same, and equal to the power P, the sides o s and o r of the 
parallelogram must be equal, and therefore the diagonal e o 
divides the angle r o s into two equal portions. Hence the 
weight will always fall into the position in which the two parts 
of the cord A e and e D will be equally inclined to the vertical 
line, and it will bear to the power the same ratio as e o to e s. 

Therefore W : P : : 2 cos. % e : 1, e representing the angle 
A e D. 

Or, 2 P X cos. % e = W. That is, twice the power, multiplied by the cosine of 
half the angle of the cord, at the point of suspension of the weight, is equal to the 
weight. 

Example. — What weight will be sustained by a power of 5 lbs., with an oblique 
movable pulley, Fig. 5, having an angle A e D of 30° ? 

5X2X.96533 = 9.6593 lbs. = twice the power X cos. 15°. 

When the Direction of the Cord is Irregular, the Weight not resting in the 
Centre of it. 

WxP = sin. ( < H-WXBm. a; FXdn - (g + M = W; WX Si " " = P, a zni b rep- 

sin. a sin. (a -f b) 

resenting the greater and lesser angles of the cord at e. 




CRANES. 

When the Post is Supported at both the Top and Foot. 

The usual form of a Crane is that of a right-angled triangle, the 
three sides being the post or upright, the jib or arm, and the stay or 
strut, which is the hypothenuse of the triangle. 

When the jib and the post are equal in length, and the stay is the diag- 
onal of a square, this form is theoretically the strongest, as the whole 
stress or weight is borne by the sta}-, tending to compress it in the direc- 
tion of its length; the stress upon it, compared to the weight supported, 
being as the diagonal to the side of the square, or as 1.4142 to 1. Conse- 
quently, if the weight borne by the crane is 1000 lbs., the thrust or corn* 
pression upon the stay will be "1414. 2 lbs., or as a e to c W, Fig. 1. 



336 



CRANES. 




The weight "VV is sustained by the rope or 
chain, and the tension is equal upon both parts 
of it ; that is, on the two sides of the square, 
ia and 6 W. Consequently the jib, ia, has no 
stress upon it, and serves "merely to retain the 
stay, a e. 

If the foot of the stay be set at rc, the thrust 

upon it, as compared with the weight, will be 

asawtoa to; and if the chain or rope from i to 

a is removed, and the weight is suspended from 

a, the tension on the jib will be as ia to a W. 

If the foot of the stay is raised to o, the thrust, as compared with the 

weight, will be as the line a o is to a W, and the tension on the jib will be 

as the line a r. 

B3- dividing the line representing the weight into equal parts, to repre- 
sent pounds or tons, and using it as a scale, the stress upon any other part 
may be measured upon the parallelogram. 

When the Post is Supported at the Foot onhj. 

If the post is wholly unsupported at top, and its foot is secured up to 
the line W, then the weight W, acting with the leverage, e W, will tend 
to rupture the post at e, with the same intensity or effect as if twice that 
weight was laid upon the middle of a beam equal to twice the length of 
e W, the point e being at the middle of the beam, which is assumed to be 
supported at both ends, the dimensions of the beam being alike to those 
of the post, and the depth being that of the line of rupture of the post. 

Or, the force exerted to rupture the post will be represented by the 
weight or stress, W, multiplied by 4 times the length of the lever, e W, 
divided by the depth or thickness of the post in the line of the stress, 
squared, and multiplied by the breadth of it and the Value of the material 
of which it is composed. 

The post of a crane is in the condition of half a beam supported at one 
end, the weight suspended from the other ; consequent^, it must be esti- 
mated as a beam of twice the length supported at both ends, the stress ap- 
plied in the middle. 

To Compute the Stress 011. tlie «Xi"b or Tension-rods, and 
on the Stay or Strut — Fig. 3. 

On the diagram of the crane, Fig. 2, mark off 
on the line of the chain, as a W, adistance, a b, 
representing the weight on the chain ; from the 
point b draw a line, c, parallel to the tension- 
rod or jib, a e, as the case may be, and where 
this intersects the stay or strut, draw a vertical 
line, c 0, extending to the jib or tension-rod, and 
the distances from a to the points b c and c, 
measured upon a scale of equal parts, will repre- 
sent the proportional strain. 

Thus, in the figure, the weight being 10 ton?, 
the stress on the stay or strut compressing, a c, 
will be 31 tons, and on the jib or tension-rods, a 0, 26 tons. 

Ity dividing the line representing the weight, as«Woraw, into equal 
parts, to represent tons or pounds, and using it as a scale, the stress upon 
any other part may be measured upon the described parallelogram. 

Thus, as the length of a W, compared to a e, is as 1 to 1.4142 : if a W is 
divided into 10 parts representing tons, a e would measure 14.142 parts 0* 
tons. 




CRANES. 



337 




In Fig. 3, the angle ab e and e b c being equal, the chain or rope is 
represented by ab c, and the weight by W ; the stress 
upon the stay b d, as compared with the weight, is as 
bd to ab or be. 

In practice, however, it is not prudent to consider the 
chain as supporting the stay ; but it is proper to disre- 
gard the chain or rope as forming part of the system, 
and the crane should be designed to support the load 
independent of it. It is also proper that the angles on 
each side of the diagonal staj r , in this case, should not 
be equal. If the side a & is formed of tension-rods of 
wrought iron, the point a should be depressed, so as to 
lengthen that side, and decrease the angle ab e; but 
if it be of timber, the point a should be raised, and the angle a b e increased. 

Fig. 4 shows the parts composing ( 5. ) 

a crane, arranged in the form of 

an equilateral triangle, in which 

the weight b d, the tension be, and 

the thrust a b are all equal to each 

other, the weight Wbeing suspend- 
ed from the point b. 

Fig. 5 shows a form of crane very 

generally used ; the angles are the 

same as in Fig. 3, and the weight 

suspended from it, being attached 

to the point d, is represented by the 

line b d. The tension, which is 
equal to the weight, is shown by the length of the line 
b c, and the thrust by the length of the line & a, meas- 
ured by a scale of equal parts, into which the line b d, 
representing the weight, is supposed to be divided. 

But if b e be the direction of the jib, then b g will show the tension, and bf the 
thrust {df being taken parallel to b e), both of them being now greater than before; 
the line b d representing the weight, and being the same in both cases. 




g 



// c 




To Compute tlie Stress upon tlie Stay of a Crane. 

Bulk. — Multiply the length of the stay in feet by the weight to be 
borne in pounds; "divide the product by the height of the jib from the 
point of bearing of the stay in feet, and the quotient will give the stress 
or thrust in pounds. 

Example. — The length of the stay of a crane is 28.284 feet, the height cf the post 
is 26.457 feet, and the weight to be borne is 22400 lbs. ; what is the stress ? 

28. 2S4X 22400 633561.6 



26.45T 



26.457 



= 23947 lbs. 



To Compute tlie Dimensions of tlie 3?ost of a Crane. 
When the Post is Supported at the Feet only. 

Bulk. — Multiply the weight or stress to be # borne in pounds by the 
length of the jib in feet measured upon a horizontal plane; divide the 
product by the Value of the material to be used, and the product, divided 
by the breadth in inches, will give the square of the depth, also in inches. 

Example. — The stress upon a crane is to be 22400 lbs., and the distance of it from 
the centre of the post is 20 feet ; what should be the dimension of the post if of 
American white oak ? 

Value of American white oak 50. Assumed breadth 12 inches. 

224 °^ X2 ° =± 8960, and ~ = 746.67. Then #746.67 = 27.32 inches. 
DO 12 

Ff 



338 



CENTRES OF GRAVITY. 



When the Post is Supported at both Ends. 

Rule. — Multiply the weight or stress to be borne in pounds by twice 
the length of the jib in feet measured upon a horizontal plane ; divide the 
product by the Value of the material to be used, and the product, divide*' 
by four times the breadth in inches, will give the square of the depth, als 
in inches. 

Example. — Take the same elements as in the preceding case. 

Assumed breadth 10 inches. 



Then 



22400x20x2 
50 = 



17Q20 
: 17920, and ^_ = 44S, and V44S - 21.166 inches. 



To Compute tlie Stress on tlie «Xi"b or Tension-rods, on 
the Stay of a Crane. 

On the diagram of the crane mark off on the line of the chain or rope a 
distance that represents the weight into a scale of equal parts, and by ap- 
plying this scale to the sides of the parallelogram representing the thrusts, 
the measure of each is obtained by inspection. 

Illustration. — The distance b d, Fig. 5, is divided into 10 parts representing tons, 
and the length of tlie sides, b c and a b, representing the thrust and tension, have 
respectively 10 and 14 parts ; consequently the stress on them is in their proportion 
in tons. 

CHAINS AND ROPES. 

Chains for cranes should be made of short oval links, and should not 
exceed* 1 inch in diameter. 

Ta"ble of Short-linked. Crane Chains and Ropes, 
showing the Dimensions and "W'eight of* each, 
and the Proof of the Chain in Tons. 



Diam. of 


Weight 


Proof 


Circumf. 

of 

Rope. 


Chains. 


Fathom. 


Strain. 


Ins. 


Lbs. 


Tons. 


Ins. 


X, 


6 


.to 


2.X 


^16 


8,5 


1.5 


8.# 


11. 


2.5 


4. 


V* 


14. 


3.5 


4.% 


X 


18. 


4.5 


5.X 


% 


24. 


5.25 


6.K 



Weight 




Weight 




Circumf. 


Weight 


of Rope 
pr. Fath. 


Chains. 


per 
Fathom. 


Strain. 


of 
Rope. 


of Rope 
pr. Fath. 


Lbs. 


Ins. 


Lbs. 


Tons. 


Ins. 


Lbs. 


1.5 


X 


28 


6.5 


7. 


10.5 


2.5 


H 


32 


7.75 


7.X 


12. 


3.75 


% 


36 


9.25 


8.}{ 


15. 


5. 


% 


44 


10.75 


9. 


17.5 


7. 


% 


50 


12.5 


9.X 


19.5 


8.7 


l. 


56 


14. 


10. 


22. 



The ropes of the sizes given are considered to be of equal strength with 
tlie chains, which, being short-linked, are made without studs. 
A crane chain will stretch under a proof of 15 tons, half an inch to a fathom. 



CENTRES OF GRAVITY. 

The Centre of Gravjcty of a body, or any system of bodies con- 
nected together, is the point about which, if suspended, all the pan 
will be in equilibrium. 

A body or system of bodies, suspended at a point out of the centre of 
gravity, will rest with its centre of gravity vertical under the point of 
suspension. 

A body or s}-stem of bodies, suspended at a point out of the centre of 
gravity, and successively suspended at two or more such points, the vert- 
ical line through these points of suspension will intersect each other at 
the centre of gravity of the body or bodies. 



CENTRES OF GRAVITY. 339 

The centre of gravity of a body is not always within the body itself. 

If the centres of gravity of two bodies, as B C, be connected by a line, 
the distances of B and C from the common centre of gravity, a, will be as 
the weights of the bodies. 

Thus, B : C : : C a : a B. 

LINE. 

Circular Arc. — = distance from the centre, r representing radius, c the chord, 

and I the length of the arc. 

SURFACES. 

Square, Rectangle, Rhombus, Rhomboid^ Gnomon, Cube, Regular Poly- 
gons, Circle, Sphere, Spheroids or Ellipsoids^ Spheroidal Zones, Cylinder, 
Circular Ring, Cylindrical Ring, Links, Helix, Plain Spiral, Spindles, all 
Regular Figures, and Middle Frustra of all Spheroids, Spindles, etc. The 
centre of gravity of the surfaces of these figures is in their geometrical 
centre. 

Triangles. — On a line drawn from any angle to the middle of the op- 
posite side, at % of the distance from the angle. 

Trapezium. — Draw the two diagonals, and ascertain the centres of grav- 
ity of each of the four triangles thus formed ; join each opposite pair of 
these centres, and it is at the intersection of the two lines. 

Trapezoid. — f J X - = distance from B on a line joining the middle of the 

two parallel sides B b, a representing the middle line. 

2 c r 
Sector of a Circle. — -r-y =: distance from the centre of the circle. 

4 r 
Semicircle.— = distance from the centre. 

o X O.1410 

Semi-semicircle 424 r = distance from both base and height and at their inter* 

section. 

c 3 
Segment of a Circle. — — — = distance from the centre, a representing area of 

segment. 

4 sin. J<> / r^ — r-> ^ 
Sector of a Circular Ring.— - X- — —r — X — t, = distance from centre of 

6 i— r* — ?v 

arcs, r and r' representing the radii. 

Hemisphere, Spherical Segment, and Spherical Zone at the centre of their 
heights. 

Circular Zone. — Ascertain the centres of gravity of the trapezoid and 
the segments comprising the zone; draw a line '(equally dividing the 
zone) perpendicular to the chords ; connect the two centres of the seg- 
ments b}- a line cutting the perpendicular to the chords ; then will the 
centre of gravity of the figure be on the perpendicular, toward the lesser 
chord, at such proportionate distance of the difference between the centres of 
gravity of the trapezoid and line connecting the centres of the segments as 
the area of the two segments bears to the area of the trapezoid. 

Prisms and Wedge. — When the end is a Parallelogram, in their geomeU 
rical centres ; when the end is a Triangle, Trapezium, etc., it is in the mid- 
dle of its length, at the same distance from the base as that of the triangle 
or trapezoid, of which it is a section. 

Prismoid. — At the same distance from its base as that of the trapezoid or 
trapezium, which is a section of it. 

Lune. — On a line connecting the centres of gravity of the two arcs at a 
point proportionate to the respective areas of the arcs. 



•340 CENTRES OF GRAVITY. 

Cycloid.— % of radius of generating circle = distance from the centre of 
the chord of the curve. 

Cone, Frustrum of a Cone, Pyramid, Frustrum of a Pyramid, and Un~ 
gula. — At the same distance from the base as in that of the triangle, par- 
allelogram, or semicircle, which is a right section of them. 

Spirals. — Plane, in its geometrical centre. Conical, at a distance from 
the base, % of the line joining the vertex and centre of gravity of the base. 

r 2 — r '2 

Frustrum of a Circular Spindle. — =e distance from the centre of the 

spindle, h representing the distance between the two bases. D the distance of the 
centre of the spindle from the centre of the circle, and z the generating arc, ex- 
pressed in units of the radius. 

_3 

Paraboloid of Revolution.-^- ^ + * 2)2 (3&2 ~ 2c2) + 2 e " = distance from 

( e 2 _|_ £2) a _ e a 

b 2 
vertex, a representing altitude, b radius of base, and e = — . 

2 a 

Any Plane Figure.— Divide it into triangles, and ascertain the centre 
of gravity of each ; connect two centres together, and ascertain their com- 
mon centre ; then connect this common centre and the centre of a third, 
and ascertain the common centre, and so on, connecting the last-ascer- 
tained common centre to another centre till the whole are included, and 
the last common centre will give the centre required. 

Parabola. — 2-5 of the height = distance from the base. 

Semi-spheroid or Ellipsoid and its Segment, and Segment of a Circular 
Spindle.— See HaswelVs Mensuration, pages 281-283. 

SOLIDS. 

Cube, Parallelopipedon, Hexahedron, Octahedron, Dodecahedron, Icosa- 
hedron, Cylinder, Sphere, Spherical Zone, Spheroids or Ellipsoids, Cylin- 
drical Ring, Links, Spindles, all Regular Bodies, and Middle Frustra of all 
Spheroids and Spindles, etc. The centre of gravity of these figures is in 
their geometrical centre. 

Tetrahedron.— Is the common centre of the centres of gravity of the triangles 
made by a section through the centre of each side of the figures. 

Cone and Pyramid.— % of the line joining the vertex and centre of gravity of the 
base = distance from the base. 

Frustrum of a Cone or Pyramid.— , ' ~ X 7 h = distance from centre 

(r + r,) 2 — r r, 4 J 

of lesser end, r and r, in a pyramid representing the sides. 

(vs\ 2 
r — J ■=- V "= distance from the centre, vs 

representing the versed sine, and v the volume of the segment. 
/8r-3A\ , 
I — — J X h = distance from the vertex, h representing the height. 

Hemisphere. — % r = distance from the centre. 

Spherical Sector.— % (r — % h) =■ distance from the centre. 

2r + Zh 

- = distance from the vertex. 

o 

Frustrum of a Sphere.— — — ——r — distance from the vertex of the frustrum. 

h > (0 a 2 h 2) 

Frustra of Spheroids.— Prolate. % —————— distance from centre ojsphfr 

roid, a representing semi-transverse diameter in a prolate frustrum, and the semi 
conjugate in an oblate frustrum 



: 



CENTRES OF GRAVITY. 341 

(d-f d') X (2a2— rf'2-f-rfs) 

jlny Frustrum. % — ^— _,,„ , „ . , ' — = distance from centre of sphe- 

6 a* — a z -\- (1 a -|- a* 
roid, d and d' representing the distances of the base and end of the segments from 
the centre of the spheroid. 
Semi- spheroids. — Prolate. X a.— Oblate. % a = distance from the centre. 

Segments of Spheroids. — Prolate. % - — -— . — Oblate. % _ . ,. = dis- 
- £> a -f- a k L a -\- d' 

tancefrom the centre of the spheroid, d and d' representing the distances of the base 
of the segments from the centre of the spheroid. 

Segment of an Elliptic Spindle at % of height from the vertex. 

Segment of a Circular Spindle and of a Parabolic Spindle. — See HaswelVs Men- 
suration, pages 192 and 199. 

Segment of a Hyperbolic Spindle, at % of the height from the vertex. 

Paraboloid of Revolution, at y z of the height from the vertex. 

Hyperboloid of Revolution. — ~|V Xh=z distance from the vertex, b represent' 

ing the diameter of the base. 

2 2 _|_ r 

Frustrum of Paraboloid of Revolution. — — — % h~ distance from the ver- 

r 2 -f- r 1 
tex, r and ri representing radii of base and diameter. 

Frustrum of Hyperboloid of Revolution.— % — «7~ 2 d ^ ±d'ri 4L2 = distance 

from centre of the base, a representing the semi-transverse axis, or distance from 
centre of the curve to vertex of figure; d and d' the distances from the centre of 
the curve to the centre of the lesser and greater diameter of the frustrum. 
Segment of Paraboloid of Revolution, at % of the height from the vertex. 

Segment of Hyperboloid of Revolution. 4* — r X h=. distance from the vertex. 

6 b -\- 4 h 

Of an Irregular Body of Rotation. 

Divide the figure into four or six equidistant divisions ; ascertain the 
volume of each, their moments with reference to the first horizontal 
plane or base, and then connect them thus : 

(A -f- 4 Aj, -]- 2 A 2 -J- 4 A 3 -f- A 4 ) — =: V, A A x , etc., representing the volume of the 

divisions and h the height of the body from the base ; 
;^ (0A + lx4A 1 + 2x2A 2 + 3x4A 3 + 4A 4 ) w ft « 

and A + 4A 1 + 2A, + 4A, + A 4 X- = distance of centre of gravity 

from base. 
Illustration.— A vessel generated by the rotation of a curve is divided into 4 
sections ; viz., A = 1, A l = 2.4, A 2 = 1.6, A 3 = 1, and A 4 == .15. 

Thrn ( ° Xl + 1X4x2 - 4 + 2x2xl - 6 + 3x4xl + 4x - 15 ) - 2 - 5 71 ' 5 Q(«Qn,«. 

Tll6n 1 + 4X2.4 + 2X1.6 + 4X1 + .15 X T = lTs= -" 5S mS " 

2 5 
and (1 + 4x2.4 -f- 2x1.6 + 4x1 + .15) X ^ = 5 - r83 cuhic inches, the volume. 

To Compute tlie Common Centre of Grravity of the En- 
gine, Boilers, etc., etc., or of any ISTixmlDer of "Weights in. 
a "Vessel. 

Rule. — Multipty the several weights on each side of £3 * bv their re- 
spective distances from it; divide the sum of the products by the sum of 
the weights, and the quotients will give the distances at which the sum 
of the weights on each side will produce the same effects as the several 
weights at their respective distances. 

* This is the symbol for Dead fiat, and it is the point in the length of a vessel where her frame has 
the greatest dimensions. 



342 



FRICTION. 



Reduce the weights to units, and add them together ; divide the sum 
of the distances by the sum of the units, and the product of the quotient 
and the respective units will give the distances of the contrary weights 
from the common centre of gravity. 

Example. — The weights and the mean distances from ^. of the engine, boilers, 
water in boilers, water-wheels, coal, coal-bunkers, spars and rigging, extra pieces, 
and engine stores in a steam vessel are as follows ; where is their common centre of 
gravity ? 



Engine, 
Boilers, 
Water, 
"Water-wheels, 



105 tons at 20 feet. 



60 
40 
20 



Forward. 

60 tons at 26 feet 
40 " " 26 » 
30 " " 4.33 " 



26 
26 
3.5 " 

= 1560 
= 1040 
= 130 



Coal, 140 tons at 4 feet. 

Bunkers, 20 - " " 3 * l 

Spars and rigging, 30 u "4.33" 
Extra pieces and stores, 5 " " 22. " 



130 



Aft. 

105 tons at 20 feet 



2730 



2730 
= 21 feet, and 



140 

20 

20 

__5 

290 

2900 _ 

290 " 



4 l 

3 « 
35 l 

22 ' 



2100 

560 

60 

70 

110 

2i)00 



10 feet. 



Then 1.3 + 2.9 = 4.2, 



130 and 290, when reduced, become respectively 1.3 and 2.9, 

and 2 '"t 10 = 7.3S09. 
4.2 

Hence 7.3809x1.3 = 9.5952 feet = the distance of 2.9 or 290 from the common 
centre, and 7.3809x2.9 — 21.4046 feet = the distance of 1.3 or 130/rom the common 
centre. 

Consequently 9.5952x290 = 21.4046x130, and 21.4046 — 21 — .40-46 feet = the 
distance aft of$Q of the common centre of gravity of the weights of the entire mass. 



FRICTION. 

Friction is termed sliding when surfaces move parallel with one an- 
other, as on a slide or over a pin ; and rolling when a body rotates 
upon the surface of some other, so that new parts of both surfaces are 
continually being brought in contact with each other. 

The force necessary to abrade the fibres of a body is termed the 
measure of the friction ; this is determined by ascertaining what por- 
tion of the weight of a moving body must be exerted to overcome the 
resistance arising from this cause : the fraction expressing the ratio is 
termed the Coefficient of the Friction. 

To Compute tlie Coefficient of the Friction of* T3od.ies. 

Place them upon a horizontal plane, attach a cord to them, and lead it 
in a direction parallel to the plane over a pulley, and suspend from it a 
scale in which weights are to be placed until the body moves. 

The weight that moves a bodj- is the numerator, and the weight of the 
body moved is the denominator of a fraction, which represents the coeffi- 
cient required. 

Illustration. — If by a pressure of 320 lb 3 , the friction amounts to SO lbs., the co- 

so 

efficient of friction in this case would be — - = .25. 

Hence, if the coefficient of friction of a wagon over a gravel road was .C5, and tho 
load S400 lbs., the power required to draw it would be 8400x.25z=2100 lbs. 



FRICTION 343 

Experiments and Investigations have adduced the following observa« 
tions and results: 

1. The amount of friction in surfaces of like material is very nearly pro- 
portioned to the pressure perpendicularly exerted on such surfaces. - 

2. With equal pressure and similar surfaces, friction increases as the di- 
mensions of the surfaces are increased. 

3. A regular velocity has no considerable influence on friction ; if the 
velocity is increased the friction is greater, but this depends on the sec- 
ondaiy or incidental causes, as the generation of heat and the resistance 
of the air. 

4. Similar substances excite a greater degree of friction than dissimi- 
lar. If the pressures are light, the hardest bodies excite the least friction. 

5. Friction is diminished by unguents. In the choice of unguents, 
those of a viscous nature are best adapted for rough or porous surfaces, as 
tar and tallow are suitable for the surfaces of woods, and oils best adapted 
for the surfaces of metals. 

6. A rolling motion produces much less friction than a sliding one. 

7. The friction of metals and woods varies with their hardness. 

8. Hard metals and woods have less friction than soft. 

9. That without unguents or lubrication, and within the limits of 33 lbs. 
pressure per square inch, the friction of hard metals upon each other ma}' 
be generally estimated at about one sixth the pressure. 

10. That within the limits of abrasion the friction of metals is nearly 
alike. 

11. That with greatly increased pressures friction increases in a very 
sensible ratio, being greatest with steel or cast iron, and least with brass 
or wrought iron. 

12. "With woods and metals, without lubrication, velocity has very lit- 
tle influence in augmenting the friction, except under peculiar circum- 
stances. 

13. When no unguent is interposed, the amount of the friction is, in ev- 
ery case, wholly independent of the extent of the surfaces of contact ; so 
that, the force with which two surfaces are pressed together being the 
same, their friction is the same, whatever may be the extent of their sur- 
faces of contact. 

14. The friction of abod} T sliding upon another will be the same, wheth- 
er the body moves upon its face or upon its edge. 

15. When the fibres of materials cross each other, friction is less thaa 
when they run in the same direction. 

16. Friction is greater between surfaces of the same character than be- 
tween those of different characters. 

17. That with hard substances, and within the limits of abrasion, fric- 
tion is as the pressure, without regard to surfaces, time, or velocity. 

18. The influence of the duration of contact (friction of rest) varies with 
the nature of the substances ; thus, with hard bodies resting upon each 
other, the effect reaches a maximum verv quickly ; with soft bodies, very 
slowly; with wood upon wood, the limit is attained in a few minutes; 
and with metal on wood, the greatest effect is not attained for some days, 



3U 



FRICTION. 



Experiments with 


TJnguents interposed.- [Morix.] 




Coef. of Friction. 




Materials and Surfaces in Contact. 


During 


After 


Unguents 




Motion. 


Quiesc. 




Beech upon oak, fibres parallel . . . 


.055 




Tallow. 


Brass upon brass .' 


.053 




Olive-oiL 


Brass upon cast iron 


.0S<3 


.106 


Tallow. 


Brass upon wrought iron 


.081 




Tallow. 


Cast iron upon brass 


.103 




Tallow. 


Cast iron upon brass 


.078 




Olive-oil. 


Cast iron upon cast iron 


.314 
.197 




Water. 


Cast iron upon cast iron 


Soap. 


Cast iron upon oak, fibres parallel 


.1S9 




Dry soap. 


Cast iron upon oak, " " 


.218 


.646 


Greased and saturated with water. 


Cast iron upon oak, u u 


.078 


.1 


Tallow. 


Cast iron upon wrought iron 


.103 


.1 


Tallow. 


Copper upon cast iron 


.072 


.103 


Tallow or olive-oil. 


Copper upon oak, fibres parallel. . 


.069 


.1 


Tallow. 


Elm upon cast iron 


.C66 

.07 


.142 


Tallow. 


Elm upon oak, fibres parallel 


Tallow. 


Oak upon cast iron 


.OS 
.136 




Tallow. 


Oak upon elm, fibres parallel 


Dry soap. 


Oak upon elm, u " 


.073 


.178 


Tallow. 


Oak upon oak, u " 


.164 


.44 


Dry soap. 


Oak upon oak, M u 


.075 


.164 


Tallow. 


Oak upon oak, fibres perpendicular 


.0S3 


.254 


Tallow. 


Oak upon oak, u u 


.25 




Water. 


Oak upon wrought iron 


09S 
.105 
.079 
.003 

.056 
.305 


.108 


Tallow. 


Steel upon cast iron 


Tallow. 


Steel upon cast iron 


Olive-oil. 


Steel upon wrought iron 


Tallow. 


Steel upon brass 


Tallow or olive -oil. 


Tanned ox-hide upon cast iron. . . 


Greased and saturated with water. 


Wrought iron upon brass 


.103 




Tallow. 


Wrought iron upon cast iron 


.103 




Tallow. 


Wrought iron upon cast iron 


.066 


.1 


Olive-oil. 


Wrought iron upon elm, fibres par. 


.078 




'Fallow. 


Wrought iron upon oak, " u 


.253 


.649 


Greased and saturated with water. 


Wrought iron upon oak, M " 


.214 




Dry soap. 


Wrought iron upon oak, u u 


.085 


.108 


Tallow. 


Wrought iron upon wrought iron . 


.082 




Tallow. 



Note — The extent of the surfaces bore such a relation to the pressure as to sepa. 
rate them from one another by an interposed stratum of the unguent. 

Deductions from the above Table, showing tlie Relative 
Values of some of tlie Sxibstances and Unguents to re- 
duce Friction. 



Substances. 




Relat. 
Value. 


Unguents 
used. 


Substances. 


Relat. 
Value. 


Unguents 
used. 


WOOD UPON WOOD. 






Wrought iron upon oak, 




Greased 


Beech upon oak, 


fibres 






fibres parallel 


.21 


and wet. 


parallel 




.96 


Tallow. 


METALS UPON METALS. 






Oak upon elm, fibr 


. par. 


.75 


Tallow. 


Brass upon brass 


.91 


Olive-oiL 


Oak upon elm, " 


u 


.73 


Tallow. 


Cast iron upon bras? . . . 


.69 , Olive-oiL 


Oak upon elm, u 


ii 


.74 


Tallow. 


Cast iron upon ca-t iron 


.83 Olive-oiL 


Oak upon oak, u 


u 


.21 


Water. 


Cast iron upon cast iron 


.64 


Water. 


METALS UPON WOOD. 






Cast iron upon cast iron 


.27 


Soap. 


Cast iron upon 


elm, 






Cast iron upon copper. . 


.8 


Olive-oil. 


fibres parallel . . 




.87 


Olive-oil. 


Steel upon brass 

Steel upon cast iron. . . . 


1. 


Olive-oil. 


Cast iron upon 


oak, 






.66 


Olive oil. 


fibres parallel . . 




.76 


Tallow. 


Steel upon wrought iron 


.57 


Olive-oiL 


Cast iron upon 


oak, 




Lard and 


Wrought iron upon cast 






fibres parallel . . 




.69 




iron 


.8 


Olive-oiL 


Catt iron upon 


oak, 






Wrought iron upon 






fibres parallel . . 




M 


u u 


wrought iron. ........ 


.75 


Olive-oiL 






FRICTION. 



345 



Relative Value of Unguents to reduce Friction. 



Unguents. 


Wood 
upon 
Wood. 


Wood 
upon 

Metals. 


Metals 
upon 

Metals. 


Unguents. 


Wood 
upon 
Wood. 


Wood 
upon 
Metals. 


Dry soap 


A 

.82 


.32 

.85 
.61 


.21 

.7 

.96 


Olive-oil 

Tallow 

Water 


1." 

.22 


1. 

.93 

.24 


Lard 


Lard and plumbago 



Metala 
upon 
Metals 



.18 



Tat>le of the Coefficients of Friction under Pressures in- 
creased, gradually up to the Limit of Abrasion. 

[G. Kennib.] 







Coefficient. 




Pressure 
per 










Wrought 


Wrought 


Steel 


Brass 


Square 


Iron upon 


Iron upon 


upon 


. upon 


Inch. 


Wrought 


Cast 


Cast 


Cast 




Iron. 


Iron. 


Iron . 


Iron. 


Lbs. 










32.5 


.14 


.174 


.166 


.157 


1S6 


.25 


.275 


.3 


.225 


224 


.271 


.292 


.333 


.219 


29S 


.297 


.32.) 


.344 


.211 


336 


.312 


.333 


.347 


.215 


390 


.376 


.363 


.353 


.205 







Coefficient. 




per 


Wrought Wrought 


Steel 


Brass 


Square 


Iron upon Iron upon 


upon 


upon 


Inch. 


Wrought Cast 


Cast 


Cast 




Iron. 


Iron. 


Iron. 


Iron. 


Lbs. 










485 


.403 


.366 


.353 


.221 


523 


.409 


.366 


.357 


.223 


600 




.367 


.£59 


.234 


672 




.376 


.403 


.233 


710 




.434 




.234 


820 




... 




.273 



liesxilts of Experiments upon Oils to determine their [Rel- 
ative Permanent Fluidity as exposed in Lubrication. 

[Naysmith.] 

Relative 
Fluidity. 



Description of Oil. 


Duration 

of 
Fluidity. 


Relative 
Fluidity. 


Description of Oil. 


Duration 
of 

Fluidity. 


Gallipoli 


9 
5 

7 


31.6 
17.3 
27.5 


Sperm, common .. 

Sperm, best 

Rape-seed 


9 


Lard 

Linseed 


7 
8 



100. 
80.1 
29. 



Coefficients of Friction. — Leather belts over wooden drums .47 of the 
pressure, and over turned cast-iron pulleys .28 of the pressure. 

Comparative Friction of Steam-engines, the Side Lever 
taken as the Standard of Comparison. 

Direct-action Engine, with rollers to slides, has a gain of .8 per cent., and 
with parallel motion a gain of 1.3 per cent. 

Vibrating Engine has a gain of 1.1 per cent. 

Direct-action Engine, with slides, has a loss of 1.8 per cent. 

Experiments upon different steam-engines have determined that the fric- 
tion, when the pressure on the piston is about 12 lbs. per square inch, does 
not exceed 1.5 lbs., or about one tenth of the power exerted. 

The friction of a double cylinder (50-inch diameter) direct-acting con- 
densing propeller engine is 1.25 lbs. per square inch of piston = 10.3 per 
cent, of the total power developed ; the friction of the load is .9 lbs. per 
square inch of piston = 7.5 per cent, of the total pressure ; and the fric- 
tion of the screw is 1.3 lbs. per square inch of piston £= 10.8 per cent, of 
the total power =i 28.6 per cent. 

The friction of a double cj-linder (70-inch diameter) inclined condens- 
ing water-wheel engine with its load is 15 per cent, of the total power de- 
veloped. 

The power required to work the air-pumps is 5 per cent., and to work 
the feed-pumps 1 per cent. 



346 



FRICTION. 



Results of Experiments \apon the Friction, of* Machinery. 

[Davison.] 

Steam-engine, vertical beam, one tenth its power; 190 feet horizontal, 
and 180 feet vertical shafting, with 34 bearings, having an area of 3300 
square inches, with 11 pair of spur and bevel -wheels ; 7.65 horses power. 

A set of three-throw pumps, 6 inches in diameter, delivering 5000 gal- 
lons per hour at an elevation of 165 feet ; 4.7 horses power, or about 13 per 
cent. 

Two pair iron rollers and an elevator, grinding and raising 320 bushels 
malt per hour; 8.5 horses power. 

An ale-mashing machine 800 bushels malt at a time ; 5.68 horses power. 

Ninety-five feet of Archimedes screw 15 inches in diameter, and an ele- 
vator conveying 320 bushels malt per hour to a height of 65 feet; 3.13 
horses power. 

Friction Clutch. — Driven by a leather belt 14 inches in width ; face of 
clutch 5 inches deep ; broke a cast-iron shaft 6.5 inches in diameter. 

Wood Bearings for 3?ropeller Sliaft. 

[Results of Experiments by Mr. Johx Pknx.] 



Bearings. 



S^lJcl'Ti- ofO^tio 



Babbit's metal on iron . 

Box on brass 

Box on iron 

Brass on brass 

Brass on iron 

Brass on iron , 

Brass on iron 

Cam-wood on brass 

Lignum-vitffi on brass . . 
Ugnuni-vita? on iron . . , 
Snake-wood on brass . . 



Lbs. 






1600 


S minutes. 


Rolled out sidewise. 


44S0 


5 minutes. 


Not cut. 


448 


SO minutes. 


No wear. 


443 


30 minutes. 


Little or no cutting. 


44S 


30 minutes. 


Little or no cutting. 


675 


1 hour. 


Abraded. 


4481 


— 


Set fast immediately. 


8007 


5 minutes. 


No wear. 


4000 


5 minutes. 


No wear. 


1250 


36 hours. 


No wear. 


400 > 


5 minutes. 


No wear. 



Marine Railway. — To draw 3000 tons upon greased slides a power of 
250 tons was necessary to move it, but when started 150 tons would draw it. 

Friction and Resistances (Screw Steamer). 

[By Vice-admiral C. R. Mooesom, R.X.] 

Moving friction of hull 07 

Moving friction of load 063 

Moving friction of rotation of blades of screw 09 

Slip of screw 171 

Resistance of hull .606 

1. 
Side Lever Steam-engine.— [J. V. Meebick.] 

Friction to work the air-pump 585 to .7 

Friction of weight of parts 5 " .5 

Friction of cylinder packing 15 " .3 

Friction of air-pump packing 046 " .092 

Friction of valves, parallel motion, resistance to air, etc. .169 " .178 

1.45 
1.45 + 1.85 



Hence 



= 1.65 lbs. per square inch. 



1.85 
If the journals are 



kept constantly lubricated, as with automaton lubrications, the friction 
of weight willbe reduced to .33, and the pressure will be reduced from 
1.65 — .33 to 1.32 pounds per square inch of piston to work the engine 
without load. The friction of the load, according as the journals are lu- 
bricated, the ends keyed up, etc., will range from 2 to 5 per cent. 



FRICTION. 



347 



Friction of Steam-engines in Pounds per Square Inch, of 
IPiston. — (Condensing".) 



Diameter of 


Oscillating 

and 

Trunk 


Beam and 


Direct-act- 
ing and 
Vertical. 


Diameter of 


Oscillating 

and 

Trunk 


Cylinder. 


Geared. 


Cylinder. 


10 


5 


6 


7 


50 


2.5 


15 


4 


5 


6 


60 


2.4 


20 


35 


4 


5 


70 


2.3 


25 


3 


3.6 


4.5 


SO 


2 


30 


3 


3.5 


4 


100 


1.6 


35 


2.6 


3 


3.5 


110 


1.5 



Beam and 
Geared. 


Direct-act- 
ing arid 
Vertical. 


2.T 
2.6 
2.5 
2.3 
2.2 
2 


3.3 
3 

2.7 
2.6 

2.5 
2.1 



Useful Effect of several Machines.— The useful effect or modulus of a 
machine is the fraction which expresses the value of the work compared 
with the power applied, which is expressed by unity. 



Bucket wheel 60 

Crab 80 

Endless screw 50 



Inclined chain pump 40 

Screw press 33 

Vertical chain pump 50 



Application of the preceding Results. 

Illustration. — A vessel, including the cradle, weighing 1000 tons, is to be drawn 
upon an inclined plane having a rise of 10 feet in 100 of its length ; what will be the 
resistance to be overcome, the cradle being supported on wrought-iron axles in cast- 
iron rollers, running on cast-iron rails ? 

— — = 100 tons '$= the power required to draw the vessel independent of friction 

Ratio of friction to pressure of wrought iron on cast, in an axle and its bearing, 
.075. Ratio of ditto of cast iron upon cast, say .005. 

Hence .075 -j- .005 = .08 of 1000 tons = 80 tons, which, added to the 100 tons be- 
fore deducted, gives ISO to?is, or the resistance to be overcome 

The power or effect lost by friction in axles and their bearing may be 
expressed by the formulas 

Wfd r _ 



230 



- = P,f representing the coefficient of the friction, d the diameter of the axle 
in inches, and r the number of revolutions per minute. 
Illustration. — The pressure on the piston of a steam-engine is 20.000 lbs., the 
number of revolutions 20, and the diameter of the driving shaft of wrought iron in 
a brass journal is 8 inches ; Avhat is the effect of the friction? 
20.000 X .0Tx8x20^ m93 ^ 



280 



Yv 



Hence — horses power, v representing circumference of shaft in feet X by 

revolutions per minute. 

The power or effect lost by friction in guides or slides may be expressed 
by the following formulae : 

*Wfsr 
(■Ov /^/s — 2\ ~ f) s representing the stroke of the cross-head, and I the length 

of the connecting rod in feet. 

On tlie Traction of Carriages, and tne Destructive Effects 
tliey prodnce on Roads.- [Experiments by M Mown.] 

The effects produced when a carriage is moved over a road are divided 
into two parts — the traction of the carriage, and its action upon the road. 

For woods, plasters, and hard bodies in general, the resistance to rolK 
ing is nearly, 

1. Proportional to the pressure. 2. Inversely proportional to the diam- 
eter of the bodv on rollers. 3. Greater as the breadth of the part in con- 
tact is diminished. 



348 



FRICTION. 



The relation or ratio between the load and the traction, upon a level 
road, is approximately given by the following formulae ; 

Vfs F 

For Carriages with. Two Wheels . — 



" W 



For Carriages ivith Four Wheels . 



2 ( Vfs ) 
txt' 



= — , U representing the constant 



multiplier in the following table./ the coefficient of friction, s the mean radius of 
the journals, r the radius of the wheels, t and V the radii of the fore and hind wheels, 
F the horizontal component of the traction, and W the total weight or pressure on 
the road 



Ta"ble of tlie ^Results of some Experiments -vvliich. were 
made at a ~W allying Pace. 

Weight in 
Pouuds. 



Artillery wagon . 



Cart without springs. 



Cart with springs. 

Carriages with six wheels) 
Two connected carr 
with six wheels 



six wheels) 
carriages, V 
►Is each . . j 



(Good sand and 
t dry 

/Hard gravel 
\ and diy 

(Ordinary sand 
\ and muddy. 

Sand, rutted 
and muddy 



15417 
10099 

(15713 

«J 9S12 

( 75G3 

352S 

11017 

CG15 
13230 



Traction 

m 
Pounds. 


Ratio of 

Traction to 

Load. 


393 
251 


1 to 3S.G 
1 to 40.2 


306 
20G 
151 


1 to 51.3 
1 to 47.7 
1 to 50.2 


87 
300 


1 to 40.S 
1 to 3G.3 


30G 
632 


1 to 21.6 
lto21 



An examination of the table furnishes the following results among 
others : 

On hard roads the traction is sensibly proportionate to the weights of 
the carriages, other elements being equal, and within certain limits the 
traction is independent of the number of wheels. 

Influence of the Diameter of Wheels. — The results of experiments show 
that on solid roads it may be admitted as a law that the traction is in- 
versely proportionate to the diameters of the wheels. 

Influence of the Width of Wheels. — Experiments made upon wheels of 
different breadths, having the same diameter, give the following results : 

1. On soft grounds, the resistance to rolling increases as the width of the 
felloe. 2. On hard ground and roads of Hie firmness, the resistance is very 
nearly independent of the width of the felloe. 



Influence of "Velocity. 



Carriage on a brass shaft. 



16-lb. carriage and piece . . . 



(Wet gravel) 
( and soft / 

JHard gravel,) 
\ even and dry/ 



C Good sand 
(Good sand 



::} 



Lbs. 
2300 



S-70 



7250 
7400 



Speed in 

Miles 
per Hour. 



3.13 
6.26 

2.S2 

3.4 

8.45 

2.77 
5.28 

S.05 



Trac- 
tion. 



Lbs. 
364 
370 

203 
203 
267 

313 
355 
406 



1 to 6.3 
lto 6.2 

1 to 40.8 
1 to 40.3 
1 to 31 

1 to22.S 
1 to 20.8 
1 to 13.3 



Wagon on six springs. . 

Wagon on six springs . . 

The results show that on soft grounds traction has no sensible aug> 
mentation with an increase of velocity, but that on solid and uneven-sur- 
faced roads it increases with an increase of velocity and in a greater de- 
gree, as the surface is uneven and the vehicle has less spring. 






or 


.25 of the load. 


or 


.125 


or 


.1 


or 


.04 


or 


.025 


or 


.033 



FRICTION. 349 

Experiments made with the carriage of a siege train on a solid gravel 
road and on a good sand road gave the following deductions: 1. That at 
a walk the traction on a good sand road is less than that on a good firm 
gravel road. 2. That at high speeds the traction on a good sand road in- 
creases very rapidly with the velocity. Thus a vehicle without springs, 
on a good sand road, gave a traction 2.64 times greater than with a simi- 
lar vehicle on the same road with springs. 

Friction of Roads. — According to Babbage and others, the friction of 
roads is as follows : A wagon with its load weighing 1000 lbs. requires a 
traction, 

In loose sand of 250 lbs., 

Fresh earth ** 140 u 

Common by-roads " 106 " 

Hard dry meadow u 40 " 

Dry high road 4% 25 u 

Macadamized road u 33 *f 

Bail-roads 6 lbs. to 3.5 " or -j ^059 « 

Upon well-paved roads the friction is J^st part of the loaa. 

Upon graveled roads the friction is J^ " 

Upon fresh earth the friction is ^ " 

Friction and. Higidity of Cordage. 

Experiments by Amonton and Coulomb, with an apparatus of Amon- 
ton's, furnish the following deductions: 

1. That the resistance caused by the stiffness of cords about the same 
or like pulleys varies directly as the suspended weight. 

2. That the resistance caused by the stiffness of cords increases not only 
in the direct proportion of the suspended weights, but also in the direct 
proportion of the diameter of the cords. 

Consequently, that the resistance to motion over the same or like pul- 
leys, arising from the stiffness of cords, is in the direct compound propor- 
tion of the suspended weight and the diameter of the cords. 

3. That the resistance to bending could be represented by an expression 
consisting of two terms, the one constant for each rope, and each sheave 
or drum ; the other proportional to the tension of the end of the rope 
which is being bent. 

4. That the resistance to bending varied inversely as the diameter of the 
sheave, or drum. 

5. That the complete resistance is represented by the expression -~ — 1 

S representing constant for each rope and sheave y and tvhich expresses the 
natural stiffness of the rope; T the tension of the rope which is being bent, 
which is expressed by CT; C being a constant for each rope and sheave ; 
and d the diameter of the sheave, including the diameter of the rope. 

6. That the stiffness of tarred ropes is sensibly greater than that of white 

ropes. 

Extending the results obtained by Coulomb, Morin furnishes the fol- 
lowing formula 

12 n 
For White Ropes : — (.00215 -f .00177 n -f .0012 W) — R. 

12 n 
For Tarred Ropes: — (.01054 -f- .0025 n -f .0014 W) = R, R representing the 

rigidity in pounds, n the number of yarns, d the diameter of the sheave in inches 
and the rope combined, and W the weight in pounds 

Go 



350 



FRICTION. 



Illustration. — What is the value of the stiffness or resistance of a diy white 
rope having a diameter of 60 yarns, which runs over a sheave 6 inches in diameter 
in the groove, with an attached weight of 1000 lbs. ? 

Assume the diameter for 60 yarns to be 1.2 ins. 

Then =q&£ (.00215 + .00177X60 + .0012X1000) = 100x (.00215 + .10620 +1.2) 

= 100X1.30S35 == 130.S35 lbs 



To Compute tlie Diameter of* a, Hope from tlie NnnVber 
of Yarns, and contrariwise. 

For White Ropes: ^(.©20739 n) — diam. in ins. For Tarred Ropes : V(-u2SS3 n) 
— diam, in ins. For White Ropes: d 2 -^-. 020739 = number of yarns For Tarred 
Ropes : d* -4- .02S83 = number of yarns. 

Example The number of yarns in a white rope is 60; what is its diameter? 

V-020739X60 = 1.116 inches Or, ^-^.-g = 60 yarns. 

The value of the natural stiffness of ropes increases as the square of 
the number of threads nearly, and the value of the stiffness proportional 
to the tension is directly as the number of threads, being a constant num- 
ber. Hence, having the rigidity for any number of threads, the rigidity 
for a greater or lesser number is readily ascertained. 

Wire Ropes. — Weisbach deduced from his experiments on wire ropes 
that their rigidity for diameters capable of supporting equal strains with 
hemp ropes is considerably less. 

Wire ropes, newly tarred or greased, have about 40 per cent, less rigidi- 
ty than untarred ropes. 

Friction of Axles. — With axles the friction of motion has alone been 
experimented upon. When the weight upon the axle and tlie radius of 
its journal is given, the mechanical effect of the friction may be readily de- 
termined. 

The mechanical effect absorbed hy, or of friction, increases with the 
pressure or weight upon the journal" of the axle and the number of revo- 
lutions. 

The friction of an axle is greater the deeper it lies in its bearing. 

Coefficients of Axle Friction.- [M Morin ] 



Hell metal upon bell metal , 

Cast iron upon bell metal 

Cast iron upon bell metal , 

Cast iron upon cast iron 

Cast iron upon lignum-vita; , 

Lignum-vitae upon cast iron , 

Lignum-vitae upon lignum-vitae . 
Wrought iron upon bell metal . . . 
Wrought iron upon cast iron 
Wrought iron upon lignum-vitae 



Condition of the Surfaces and Unguents. 




Greasy 
and wet- 
ted with 

Water. 


Oil, Tallow, or Lard. 


Vervsoft 


Dry and 

a little 
Greasy 






and pmi- 
fied Car- 
riage 
Grease. 




O 2 M 






.097 


.049 




.194 


.161 


.6:5 


,0-4 


.065 




.079 


.075 


.054 




.is5 




.1 

.110 


.092 
.07* 


.109 


.251 


.1S9 


.6-5 

.075 


.054 
.051 


.09 


.188 




.125 




... 



The friction of the journal of an axle which presses on one side only, as 
in a worn bearing, is less than when it presses at all points, the difference 
being about .005. 

If the journal of an axle revolves in a nave or eye, the radius of the 
friction path is the radius of the nave or eye. 



FRICTION. 351 

If the journal of an axle lies in a prismatic bearing, as in a triangle, 
etc., the friction is greater, as there is more pressure on, and consequent- 
ly greater friction in contact . in a triangular bearing it is about double 
that of a cylindrical bearing. 

To Compute tlie Mechanical Effect &£■ ITrictioii oix tlie 

Joviriial of an Axle. 
pt fWr 

— '— = F, p representing the ratio of the circumference of a circle to its diame- 
ter, t the number of revolutions, and r the radius of the journal in feet 
Example.— The weight of a wheel, with its axle or shaft resting on its journals, is 
3G0 pounds; the diameter of the journals 2 inches; and the number of revolutions 
30; what is the mechanical effect of the friction, the coefficient of it being .16? 

3.1416x30x.l6x3G0xr-=l2 452.4 • 

313 = -30" = 15 -° 8 lbS ' 

By the application of friction-wheels (rollers) the friction is much re- 
duced, and the mechanical effect then becomes, when the weights of the 
friction-wheels are disregarded, 

ptfWr r' 

~wr, — X = F, r' representing the radii of the axles of the friction* 

*° a' cos a~2 

wheels, a' the radii of the friction-wheels, and a the angle of the lines of di- 
icction between the axis oj the roller and the axis of the friction-wheels 

When a single Friction- wheel is used, —-—-- X /"W = F, and : = F', F' rep* 

00 J r'-^-a' 

resenting the mechanical effect. 

Example. — A wheel and its shaft, making 5 revolutions per minute, weighs 
30000 lbs. ; its diameter and that of its journals are 32 feet and 10 inches. The 
journals rest upon a friction-wheel, the radius of which is 5 times greater than its 
axle. 

1. What is the power at the circumference of the wheel necessary to overcome the 
friction? 2. What is the mechanical effect of the friction? 3. What is the reduc- 
tion of friction by the use of the friction-wheel? 

The coefficient of friction is assumed to be .075. 



32 2x12 

1st. — a ■ ' = 3S.4 times, i e , the circumference of the wheel = 3S.4 times that 
10 -r" 2 

of the axle. Hence -— = 58.59 lbs. = power required at circumference to 

overcome friction at axle. 

2d. -^ = 2. CIS feet = distance passed by friction. 

2 618x5 
Consequently, ' ±= .2181 feet = distance passed by friction at 5 revolutions 

in one second. Hence .2181x2250 (30000 X-075) = 490.725. 
3d. 1 -4- 5 = .2 = radius of friction-axle -4- by radius of friction -wheel. Hence 

490.725 
3S.4X.2 = 7.68 = friction referred to circumference of wheel, and — ^ — = 9S.145 

= mechanical effect by application of friction-wheel — a reduction of four fifths. 

Friction of Pivots.— Friction on Pivots is independent of their velocity, 
and increases in a greater degree than their pressures. 

Friction on Conical Bearings is greater than with like elements on plane 
surfaces. 

The figure of the point of a pivot, as to its acuteness, affects the friction : 
with great pressure the most advantageous angle for the figure ranges 
from 30 ' to 45° ; with less pressure it may be reduced to 10° and 12°. 

Relative "Value of Angles of P»ivots. 
60 1. 115° G6 | 45° 39 



352 



FRICTION. 



Relative Values of different ^Materials for vise as Pivots. 

Agate 83 I Granite 1 . I Tempered steel ... .44 

Glass 55 I Rock crystal 76 | 

The friction of a pivot approximates so near to that of sliding and axle 
friction, that the following coefficients are used: 

Coefficients of the Friction of ^Motion. 

Condition of the Surfaces and Unguents. 



Hemp cord,, etc ■■■■}$%£ 

Metal upon wool Mean . . . 

Sole-leather, smooth, upon wood 



or meta' . 



[ /Raw ... 
\D iy .... 



"Wood upon metal Mean . 

Wood upon wood 















TS 






gj 








g 










i 


o 




>> 


"3 


a 
> 


09 


"3 


>> 




a 




C 


- 


h 


C 


O 


.45 


.33 


.15 




.19 






.IS 


.31 


.07 


.09 


.09 


.20 


.13 


.54 


.36 


.16 




.20 






.34 


.31 


.14 




.14 






.42 


.24 


.06 


.07 


.OS 


.20 


.14 


.3C 


.25 




.07 


.07 


.15 


.12 



To Compute the Mechanical Effect absorbed, "by the Fric- 
tion of Pivots. 

It is necessary to have the coefficients of friction, and to know the mean 
space which the base of the point describes in a revolution. 

When the Section is that of a Parallelogram, ^ pfWr = F, r representing the 
radius of the base in feet 

Example. — A vertical shaft, having a pivot or journal at its base, 1 inch in diame- 
ter, with its superincumbent wheel, weighs 3G0 lbs., and makes 100 revolutions per 
minute ; what is the friction, and what the loss of mechanical effect? 

The coefficient of friction is assumed to be .1 : then .1x360 = 36 lbs. The space 
per revolution = f X3.14166X. 5 of J^ = 4.1SSx. 0416 = .1746 feet. 

Hence the mechanical effect per revolutions 36 X. 1746 = 6.2S56 lbs.; and, as the 

shaft makes 100 revolutions per minute, — — -X6.2S50 = 10.476 lbs. 

60 

When the Section is that of a Rmg having Sections of a Parallelogram, 

8 \r*-r'*) 
ExAMPLE.—The elements the same as in the preceding example, with the excep- 
tion that the radii of the ring are taken instead of the radius of the journals. Thus, 
the external and internal diameters of the ring are 1 inch and 5 inches. 
4 / 53 ojpv 1004 .1094 

§ x3 - i4jox (;l^5-> f T i 3 = 4 - ssx y of A= 4 - ss ><T«=- 2030 - 

Hence the mechanical effect per revolution = 36 X. 2036 = 7. 321' 6 lbs., and Yo^X 
S.G296 = 12.216 lbs. 

4 Wr 
When the Section is that of a Triangle, -- pf- =zF.a — the half of the an- 

J 3 sin. a 

gle of the vertex of the triangle. 

Example..— The elements the same as in the first example, with the addition of 
the angle a = 30°. 

-XS.UlGX-r^X^ = 4.1SSx^X.041G6 = .1746. 
3 sin. a ** .<* 

Hence the mechanical effect per revolution = 3GX.1740 = 6.2S6 lbs*, and l v %°-X 
6.286 = 10.476 lbs. 

Xomc. When the ends of pivots on vertical shafts are rounded, the friction is not 

diminished thereby. 



I 



FRICTION. 353 

Rolling Friction.— Rolling Friction increases with the pressure, and is 
inversely as the diameter of the rolling body. 

For rolling upon compressed wood, /= .019 to .031. 

When a Body is moved upon Rollers and the Power applied at the Base of the 

Body, if+f) — =F,/ and/' representing the coefficients of friction of the two 

surfaces upon which the rollers act. 

W 
When the Power is applied at the Circumference of the Roller, f — r= F. 

W 

When the Power is applied at the Axis of the Roller, f —rj^F. 

FRICTION OF MACHINERY AND TRAINS ON RAILWAYS. 

To Compixte the Resistance to a Train on a Railway. 

Let P represent the power requisite to draw a weight W, including its friction, on 
a plane with a rise of h feet m 100 ; R the power requisite to draw W on the plane 

exclusive of friction; and let F, the friction, be -— — ann part ofW. 

h W y 'i, > *'.' ui /AW . W\ 
By the formulae for inclined planes, R =^j7)(p and •" I loo "" "n / ' 

„ 100 nP _ hn+ 100XW 

Hence — — = W, and !-— = P. 

hn -+- 100 100 n 

Illustration. — If a train of 30 tons' weight is drawn on a level rail-road, what 
power is necessary to overcome the resistance or friction, it being S lbs. per ton ? 
2240 -r- 8 = 2S0 = hence the resistance = ^i_ of the weight. 

_ _ W 30X2240 
P = F = — = — — - — = 240 lbs. 
n 280 

2. — The grade of a railway is 2 feet in 100 feet ; what power is required to draw 

a load of 50 tons up the grade, the coefficient of friction being ^i^ , or n = 2S0 ? 

2X2SO-J-100X50X2240 73920000 „_,.„ 

Wb^2S0 =-2SUuO- = 2G4 ° ^ 

To Compute the Power, Speed., or Time of Running a 

Locomotive or Train upon a Railway. 

On a Horizontal Plane. — Let H represent the horses' 1 power, Sthe space in 
miles passed over in the time t in minutes, and W the weight in tons. 
L28*WXS H< 1.28 WXS _ II f _ w 

t ~ H T28W- S; H -' ; L28S- W * 

Illustration. — What is the required power of a locomotive to draw a train of 45 
tons at the rate of 5'J miles per hour ? 

1.28x45x50 2SS0 ; ■ . 

— = -^7 - = 4S horses. 

60 GO 

2.— In whit time will a locomotive of 50 horses' power, drawing a train of 1£5 
tons, run a distance of Si) miles? 

1.28X1 35X80 13S24 1 j & . r 

Kf . = — — — = 27G.24 minutes. 

50 50 

On an Inclined Plane. — Let h represent the rise or fall of the plane in 

eve?'?/ 100 feet. 

256(5±14ft)WxS 1000 ^XH _ 250 (5±14 ft ) WxS . 

1000 1 — *25G(5±14/i)W ' 100011 

1000<XH _.. IOOO^xH — 12S0WXS . 
r=W; feT—r^s = h. 



25G(5±14ft)S ' 35S4WXS 



* This constant represents a coefficient of friction of 1-280 of the weight of the train. 

Go* 



354 STABILITY. 

Illustration.— A train of 40 tons in weight ascends a road having a rise of 2 feet 
in 1UU at a speed of 15 miles per hour; what is the power of the engiue ? 

256X(5-f UX2)X40X15 506SS00 

— t — =. — : — i — -= S4.4b horses. 

1000X^0 0000 

Note. — When the train or load descends a plane, h must he taken negatively, or 
— , as gravity in this case assists the moving power; it also appears that, when — h 
r=5-14 of a foot, no moving power is required to draw the train. 

2. — A train of GO tons descends a road falling 3 inches in 100 feet with a speed 
of 50 miles per hour; what is the horses' power exerted hy the engine? 

Here h is negative, and 3 inches = .25 feet. 

25GX(5 — 14 25) X 50x00 1152000 



10^0x00 



- = 19.2 horses. 



Friction of Locomotives and. Railway Trains. 

Locomotive moving friction 15 lhs. per ton. 

Trains " " " " " 

Friction, developed in the Xjannching of Vessels. 

Experiments made by a committee of the Franklin Institute on the fric- 
tion of launching vessels gave, when the pressure or weight was from 2280 
to 35G0 per square foot, a coefficient of -^- = .0335. 



STABILITY. 

Stability, Strength, and Stiffness arc necessary to the permanence 
of a structure, under all the variations or distributions of the load or 
stress to which it may be subjected. 

Stability of a Fixed Body is the power of remaining in equilibrio with- 
out sensible deviation of position, notwithstanding the load or stress to 
which it may be submitted may have certain directions. 

Stability of a Floating Body. — A bod}* floating in a fluid is balanced, or 
at rest, when it displaces a volume of the fluid, the weight of which is 
equal to the weight of the floating body, and when the centre of gravity 
of the floating body and that of the volume, from which the fluid is dis- 
placed, are in the same vertical plane. 

AVhen a body in equilibrio is free to move, and is caused to deviate in a 
small degree from its position of equilibrium, if it does not tend to deviate 
further, or to recover its original position, its equilibrium is termed Indif- 
ferent ; when it tends to deviate further from its original position, its 
equilibrium is Unstable ; and when it tends to return to its original posi- 
tion, its equilibrium is termed Stable. 

A body in equilibrio may be stable for one direction of stress, and un- 
stable for another. 



STABILITY. 355 

/ Assume figure to represent the cross- 

M / *"*-^^ section of the hull of a vessel, G the 

r —~~~^r^^W w centre of gravity of the hull, w I the 

w -.., / . J] ""-H---, water-line, and c the centre of buoy- 

"~"/";c-— ., oj \ >*\ / / ancY °f the immersed section in the 

e s$?3^/ |~ = !~~- — 4-" ri J ~"" = ^ 7 - position of equilibrium. Conceive the 

/ c /--.-£ y ]""••• I vessel to be heeled or inclined over, so 

L_ Q fi'i / that ey becomes the water-line, and b 

./^^ / tne centre of buoyancy of the immersed 

— / section ; produce b M*, and the point M 

is the meta-centre* of the hull of the vessel. 

The Comparative Stability of different hulls or vessels is proportionate 
to the distance of G M for the same angles of heeling, or of the distance 
G b. The oscillations of the hull of a vessel may be resolved into a roll- 
ing about its longitudinal axis, pitching about its transverse axis and 
vertical pitching, consisting in rising and jinking below and above the 
position of equilibrium. 

If the transverse section of the hull of a vessel is such that, when the 
vessel heels, the level of the centre of gravity is not altered, then her roll- 
ing will be about a permanent longitudinal axis traversing her centre of 
gravity, and it will not be accompanied by any vertical oscillations or 
pitchings, and the moment of her inertia will be constant while she rolls. 
But if, when the vessel heels, the level of her centre of gravity is altered, 
then the axis about which she rolls becomes an instantaneous one, and the 
moment of her inertia will vary as she rolls ; her rolling must, then, nec- 
essarily be accompanied b} 7 vertical oscillations. 

Such oscillations tend to strain a vessel and her spars, and it is desira- 
ble, therefore, that the transverse section of her hull should be such that 
the centre of its gravity should not alter as she rolls, a condition which is 
always secured if all the water-lines, as w I and ef, are tangents to a com- 
mon sphere described about G; or, in other words, if the point of their in- 
tersections, o, with the vertical plane of the keel, is always equidistant 
from the centre of gravity of the hull. 

The Momentum of Stability of a floating bod}* is equal to the product of 
the weight of the fluid displaced, and the horizontal distances between the 
two centres of gravit} T of the body and of the displacement. 

To Determine the IVleasvire of tlie Stability- of tlie Htill of 
a Vessel or of a Floating Body, 

The measure of the stability of a floating body depends essentially upon 
the horizontal distance, G 6, of the meta-centre of the body from the cen- 
tre of gravity of the body ; and it is the product of the force of the water, 
or resistance to displacement of it, acting upward, and the distance of G &, 
or PxG&. If the distance, c M, is represented by r. and the angle of 
rolling, c M r, by M°, the measure of stability or S is determined by P r, 
sin. M° — S ; and this is therefore the greater, the greater the weight of 
the body, the greater the distance of the meta-centre from the centre of 
gravity of the body, and the greater the angle of inclination of this or of 
c M r. 

* Tli© mtta-eentre depends upon the position of the centre of huoyancy, for it is that point where 
ft vertical line drawn from the centre intersects a line passing through the centre of gruvity, of. the 
hull of the vessel perpendicular to the plane of the keel. 

The point of the meta-centre may be the same, or it may differ slightly for different angles of 
heeling. The angle of direction adopted to ascertain the position of the meta-centre should be the 
greatest which, under ordinary circumstances, is of probable occurrence ; in different vessels this an- 
glo ranges from 60° to 20°. 

If the meta-eentre is above the centre of gravity, the equilibrium is Stable ; if it coincides with it, 
the equilibrium is Indiffereut; and if it is below it, the equilibrium is Unstnblo. 



356 STABILITY. 

Illustration.— The weight of a floating body is 5515 lbs., the distance between 
its centre of gravity and the meta-centre is 11-32 feet, and the angle M is 20°. 
Hence S:= 5515x11. 32x. 34202 = 21352.24 lbs. 

To Compute tlie Elements of Stability of a Floating Body. 

A, s A.a A,a , — : — — c 

A« = A ( a; — a = s; - — Tz = g; i — : — T7 — ^> — r-±e sin. M = c; - — r- r = r; 
' ' A sin. MA sin. M 'A sin. M 

find sin. Mr = c, A representing area of immersed section; A, the section immersed 
by the careening of the body, as woe ; s the horizontal distance, cr, between the 
centres of buoyancy ; a the horizontal distance between the centre of gravity, n, of 
the areas immersed and emerged by careening ; g the distance, cM, below the cen- 
tre of buoyancy of the body, or of the water displaced and the meta-centre ; r the 
distance between the centre of gravity of the body and the meta-centre ; c the hori- 
zontal distance, Gd, between the centre of gravity of the body and of the line of dis- 
placement of it when careened, all in feet ; and H the angle of careening of the body. 
Note. — When the centre of gravity, G, is below that of the displacement, c, then e 
is -f- ; when it is above c, it is — ; and when it coincides with c, it is ; cr e is — 

g 
when — <[ 5 ; and a body will roll over when e sin. M = or > s. 

The other elements of the body above given, deduced by the formulae, etc., are, 
A = SO, A, = 21.5, a = 14.S, s = 3 7, r = 11. 32, g — 10.83, c = 3.ST, e = +.5, b ^ 
21.5, and P= 5515, b representing the breadth in feet, and P the weight or displace- 
ment of the body in lbs. 



Of the Hull of a Vessel— ( b ■ ± e)?, sin. M = S; d cos. % M = d\ d 

rpth of centre of gravity of displacement under water in equilibri 
h when out of equilibrium] all in feet. 

■= s > »km (!•-*) = ± e : p (x+^^i^ S! and P(s= 



representing depth of centre of gravity of displaceme7it under water in equilibrium, 
and d' the depth when out of equilibrium, all in feet. 

10 7t-ol3A ==g; ^^» ! "- J ' = ±g ^ p t^-f ^ - 

e sin. M) = S. 
Illustration — The displacement of a vessel is 10 000 000 lbs. ; area of immersed 
section 8Q0 square feet; vertical distance from centre of gravity of hull up to the 
centre of buoyancy or displacement, 1.9 feet ; and horizontal distance between the 
centres of gravity of the areas immersed and emerged, when careened to an angle 
of 9° 10' = 33.4 feet, the immersed area being 50 square feet. 

50 9 f QT^S 9 QQ 

Hence S = — x33.4 = 2.0S75/«f; ^ = 1^ = 13.1/^; r = -g| = 15 feet ; 

5">x33.4 /5')v QC 5 4- \ 

c = g ( . )Q + 1.9X.1593 = 2.30 feet; S = 10 000 000 (— ^— - -f 1.9 X. 1593 J = 

23 901 700 (hs. ; or S = 10 000 000 (2.0ST5 -f 1.9X.1593) — 23 901 700 lbs. 
2.— The length of a hull is 300, and breadth of beam 5 J feet. 

" eDCe S = (liroUo) +1-9X10000 000X.1593=23 905 896 Cta; g= r^^ 

J ' .1593 \10 000UOO / J 

Xo Coinpxite tlie Elements of tlie Power, etc., required to 
Careen a Body or Vessel. 

W/rrrPc. and AY / = F\ AT representing the iveight or power exerted, and I the d.is- 
tanre at which the weight or effort acts to careen the body taken from the centre of 
gravity of the displacement, perpendicular to the careening force in feet. 



Sin. M (h — n sin. M) -4- n sue. M — 5 = 7, and , 



1,3 3 j p 



10.7 to 13 A V G4. 125 LA ' 

ft representing the vertical height from centre of gravity of displacement to centre 
of the weight or effort to careen the body whtn it is in equilibrium, n the horizontal 
distance from the centre of the vessel to the cnitre of the weight or effort, L the 
length of the vessel, and m the meta-centre, all in fett. 

• The unit for the section of a parallelogram is 10.7 ; of a ■emicircle, IS ; and of a triangle, 12.8. 



STABILITY. 



357 



Illustration.— A weight is placed upon the deck of a vessel at a mean distance 
of 3.87 feet from the centre line of the hull ; the height at which it is placed is 10.82 
feet, and the other elements as in the first case g iven. 

Then fc= 10-82, n — 3.87. and .34202 (10.82 — 3.87X. 34202) -f-3, 67X1- 06418 — 3.7 
= .34202x9.5 + 4.12 — 3.7 = 3.67/ee*. 

To Compute tlie Measure of tlie careening Power of a 
Sailing Vessel. 

F/sin. w, cos. s = P, F representing area of sails, i square feet, /force of wind in 
lbs. per square foot, w angle of the wind to the sails, and s angle of sails to the 
plane of the vessel. 

To Compute the Measure of tlie Sailing Power ofaVesseh 

F f sin. it?, sin. s = P. 

To Compute the .Alteration of the Trim of a Vessel. 

distance in feet, w representing the weight or weights in tons moved through 

a mean distance, a ; and M = DXrr^r, D representing displacement in tons, Im dis^ 

tance of longitudinal meta-centre above the centre of gravity, and W I the length 
of the vessel at her load water-line. 

Illustration.— If a steamer has a displacement of 8625 tons, a length of vnter- 
line of 380 feet, and the height of her longitudinal meta-centre is 475 ien ; wh:it 
would be the alteration of her trim if 6 guns of 6 tons' weight were moved ait a dis- 
tance of 248 feet? 

Then 6> i|^! 48 = .Safest = 9.96 inches. 



iv a 



8625x|^ = 10781. 



380" 



10781 



Results of Experiments upon tlie Stability of Rectangu- 
lar Blochs of ATVood of Tiniform Length and Depth, Tout 
of different Breadths.— [W. Bland.] 

(Length 15, Depth 2, and Depression 1 inch.) 

| Kntios of Stability. 

Weight. 



With like 
Weights. 



By Squares of 
the Widths. 



Bv Cubes of 
the Width. 



Ins. 
3. 

4.5 

6. 

7.5 



Oz. 

24 
35 
45 
55 



1. 
3.5 
7. 
11. 



1. 

2.4 
3.7 



f. 

2.25 

4. 

6.25 



1. 

3.375 
8. 
15.625 



Hence it appears that rectangular and homogeneous bodies of a uni- 
form length, depth, weight, and immersion in a fluid, but of different 
widths, have stability for uniform depressions at their sides (heeling) 
nearly as the squares of their width ; and that, when the weights are direct- 
ly as their widths, their stability under like circumstances is nearly as the 
cubes of their width. 

Further experiments deduced the following results : 

1. That rectangular and homogeneous bodies of a uniform width, depth, 
and immersion in a fluid, but of different lengths, have stability for uni- 
form depressions at their sides nearly as their weight, and without refer- 
ence to their lengths, and that, when the weights are directly as their 
lengths, their stability under like circumstances is nearly directly as their 
lengths. 

2. That like bodies of a uniform width, length, an immersion of half 
their depth, but of different depths, have stability for uniform depressions 
at their sides nearly inversely as their depths, ana* that, when the weights 
are directly as the depths, their stability is inversely aa their depths. 



358 MODELS. 

Results ofExperiments upon the Stahilit^ and Speed, cf 
Models having Amidship Sections of different Forms, 
but of uniform Length., Width, and Weights.— (W. Bland.; 

(Immersion different, depending upon form of Section.) 



Form of Immersed Sect'on. 


Stability. 


Speed. 


Half-depth triangle ; the other half rectangle 


12. 
14. 

7. 

9. 


4. 
3 


Right-angled triangle* 


3 


Semicircle 


2. 



See page 611 for further result*. 



MODELS. 

The classes of forces to which Models are subjected are, 
1. To draw them asunder by tensile strain. 2. To break them by trans- 
verse strain. 3. To crush them by compression. 

The stress upon the side of a model is to the corresponding side of a 
structure as the cube of its corresponding magnitude. Thus, if the struc- 
ture is six times greater than the model, the stress upon it is as 6 3 to 1 = 
216 to 1. But the resistance of rupture increases only as the squares of 
the corresponding magnitudes, or as 6 2 to 1 = 36 to 1." A structure, there- 
fore, will bear as much less resistance than its model as its side is greater. 

To Compute the Dimensions of the Beam, etc., which a 
Structure can "bear. 

Rulk. — Divide the greatest weight which the beam, etc., in the model 
can bear by the greatest weight which it actually sustains, and the quo- 
tient, multiplied by the length of the beam, etc.", in the model, will give 
the length of the beam, etc., in the structure. 

Example. — A beam in a model is 7 inches in length, and PiiFtains a wright of 4 
pounds, but it is capable of bearing a weight of 26 lbs. ; what is the greatest length 
that the corresponding beam can be made in the structure? 
26-^ 4 = 6.5, and 6.5x7 = 45.5 inches. 

The resistance in a model to crushing increases directly as its dimen- 
sions ; but as strain increases as the cubes of the dimensions, a model is 
stronger than the structure, inversely as the squares of their comparative 
magnitude. 

Hence the greatest magnitude of a structure is ascertained by taking 
the square root of the quotient, as obtained by the preceding rule, instead 
of the quotient itself. 

Example. — If the greatest weight which a column in a model can sustain is £6 
lbs., and it actually bears 4 lbs. ; then, if the height of the column is IS inches, 
what will be the height of it in the structure? 

/26y 

: -^6.5 = 2.55, and 2.55xlS = 45.9 ins., height of the column in the structure. 



M- 



If, -when the length or height and the breadth are retained, and it is re- 
quired to give to tlie beam, etc., such a thickness or depth that it will not 
break in consequence of its increased dimensions, 

Then I yr) — v^>.5 = 2.55, which, X the square of the relative size of the model 
— the thickness required. 

If it were required to compute the resistance of a bridge from an exper- 
iment made with a model, 

n 2 W — — (n — 1) w = the load the bridge will bear in its centre. 
* Draught of water or immersion double that of the rectangle. 



HYDRAULICS. 359 

Suppose I the length of the platform of the model between the centres 
of its repose upon the piers to be 12 feet, its weight w 30 lbs., and the 
weight W it will just sustain at its centre 350 lbs. 

Let the comparative magnitudes, n, of the model and the bridge be as 
20, and the actual length of the bridge 240 feet ; what is the weight the 
bridge will sustain ? 

202X350-— x (20 — 1)X30 = 140000- 3800x30 = 20000 lbs. 



HYDRAULICS. 



Descending Fluids are actuated by the same laws as Falling Bodies. 

A Fluid will fall through 1 foot in % of a second, 4 feet in % of a sec- 
ond, and through 9 feet in % of a second, and so on. 

The Velocity of a fluid, flowing through an aperture in the side of a ves- 
sel, reservoir, or bulkhead, is the same that a heavy body would acquire 
by falling freely from a height equal to that between the surface of the 
fluid and the middle of the aperture. 

The Velocitv of a fluid flowing out of an aperture is as the square root 
of the height of the head of the fluid. The Theoretical velocity, therefore, 
in feet per second is as the square root of the product of the space fallen 
through in feet and 64.333 ; consequently, for 1 foot it is V 64.333 — 8.02 
feet. The Mean velocity, however, of a number of experiments gives 5.4 
feet, or .673. 

In short ajutages accurately rounded, and having the form of the con- 
tracted vein, or vena contracta, the coefficient of discharge = .974 that of 
the theoretical. 

Fluids subside to a natural level, or curve similar to the earth's con- 
vexity; the apparent level, or level taken by any instrument for that 
purpose, is onl) r a tangent to the earth's circumference; hence, in level- 
ing for canals, etc., the difference caused by the earth's curvature must be 
deducted from the apparent level to obtain the true level. 

Ded. notions from Experiments on the Discharge of Flnids 
from Reservoirs. 

1. That the quantities of a fluid discharged in equal times b}' the same 
apertures from the same head are nearly as the areas of the apertures. 

2. That the quantities of a fluid discharged in equal times by the same 
apertures, under different heads, are nearly as the square roots of the cor- 
responding heights of the fluid above the surface of the apertures. 

3. That the quantities of a fluid discharged during the same time bj T dif- 
ferent apertures, under different heights of the fluid, are to one anotlier in 
the compound ratio of the areas of the apertures and of the square roots 
of the heights of the fluid above the centre of the apertures. 

4. That, on account of the friction, small-lipped or thin orifices discharge 
pro portion all}" more fluid than those which are larger and of similar fig- 
ure, under the same height of fluid. 

5. That in consequence of a slight augmentation which the contraction 
of the fluid vein undergoes, in proportion as the height of a fluid increases, 
the flow is a little diminished. 

6. That circular apertures are the most effective, as they have less rub- 
bing or frictional surface for the same area. 

7. That the discharge of a fluid through a cylindrical horizontal tube, 
the diameter and length of which are equal to one another, is the same as 
through a simple aperture. 



360 HYDRAULICS. 

8. That if a cylindrical horizontal tube is of greater length than the di- 
ameter, the discharge of a fluid is much increased, and may be increased 
with advantage, up to a length of tube of four times the diameter of the 
aperture, 

9. That the discharge of a fluid by a vertical pipe is augmented, on the 
principle of the gravitation of falling bodies ; consequently, the greater 
the length of a pipe, the greater the discharge of the fluid. 

10. That the discharge of a fluid by an inclined pipe is greater than by 
a horizontal pipe of the same dimensions. 

11. That the discharge of a fluid is inversely as the square root of its 
density. 

12. That the velocity of a fluid line passing from a reservoir at any 
point is equal to the ordinate of a parabola, of which twice the action of 
gravity (2g) is the parameter, the distance of this point below the surface 
of the reservoir being the abscissa.* 

13. The volume of water discharged through an aperture from a pris- 
matic vessel which empties itself is only half of what it would have been 
during the time of emptying if the flow had taken place constantly un- 
der the same head and corresponding velocity as at the commencement 
of the discharge ; consequently, the time in which such a vessel empties 
itself is double the time in which all its fluid would have run out if the 
head had remained uniform. 

14. The mean velocity of a fluid flowing from a rectangular slit in the 
side of a reservoir is two thirds of that due to the velocity at the sill or 
lowest point, or it is that due to a point four ninths of the whole height 
from the surface of the reservoir. 

15. The velocity of efflux increases as the square root of the pressure on 
the surface of a fluid. 

16. In efflux under water, the difference of levels between the surfaces 
must be taken as the head of the flowing water. 

17. To attain the greatest mechanical effect, or vis viva, of water flowing 
through an opening, it should flow through a circular aperture in a thin 
plate in preference to a prismatic tube. 

18. The discharge of fluids through apertures slightly under water is 
nearly equal to the discharge in air. 

It is ascertained in practice that water issuing from a circular aperture 
in a thin plate contracts its section at a distance of half its diameter from 
the aperture to very nearly .8, the diameter of the aperture, so as to re- 
duce its area from i to about .617. f The velocity at this point is also as- 
certained to be about .974 times the theoretical velocity due to a body 
falling from a height equal to the head of water. The mean velocity in 
the aperture is therefore .974, which, x .617 = .6, the theoretical discharge ; 
and in this case .6 becomes the coefficient of discharge, which, if expressed 
generally by C, will give for the discharge itself 

a x /%g hXC =D, D representing the discharge per second, and a the area of the 
aperture. 
For Square Apertures it is .615, and for Rectangular .621. 

To Compute tlie Difference between True and. Apparent 

Level. 

it-i ,i f Feet, 1 »..■, ., f. 000000287^1 and the product is the 

When the \ ' I multiply the J I differe „ C e in inches 

distance \ Yards, Uquareofthe^ . 00000258c V. hm refraction is not 

* m [ Chains, J dlstance h r [ . 00125 J taken into account. 

* See D'Anbuisson, p 66 

f Bayer, .61 ; Bossut, .666: Ventnri, .637: Eytehvein, .64; Miobelotti, .64 The observed dis- 
charges* of water coincide nearer to the unit or Bayer than that of others. 



HYDRAULICS. 



361 



Notk. — If the distance is considerable, and the refraction must be considered, di- 
minish the distance by .0S33. 

Example.— What is the difference between the true and apparent level at a dis- 
tance of 18 chains, when the refraction is taken into account? 

1SX.0833 =-. 1.5, IS — 1.5 — 16.5, and 16.52X .00125 = .3403 inch. 

Deduction from Experiments on the Discharge ofWater 
~by Horizontal Conduit or Conducting JPipes.— [M. Bossut.] 

1. The less the diameter of the pipe, the less is the proportional dis- 
charge of the fluid. 

2. The .greater the length of conduit pipe, the greater the diminution of 
the discharge. 

3. The discharges made in equal times by horizontal pipes of different 
lengths, but of the same diameter, and under the same altitude of fluid, 
are to one another in the inverse ratio of the square roots of their lengths. 

4. In order to have a perceptible and continuous discharge of fluid, the 
altitude of it in a reservoir, above the plane of the conduit pipe, must not 
be less than .082 inches for every 100 feet of the length of the pipe. 

5. In the construction of hydraulic machines, it is not enough that el- 
bows and contractions be avoided, but also. any intermediate enlarge- 
ments, the injurious effects of which are proportionate, as in the following 
Table: 

Ttelative "Velocity of the Discharge of lihe Quantities of 
TTlxiid under lihe Heads in 3r*ipes having a different 
N"um"ber of Enlarged Ir*arts. 

Number of 
Parts. 



Relative 

Velocity. 


| Number of 1 Relative 
Parts. 1 Velocity. 


1. 


1 1 1. .741 



Number of 
Parts. 



Relative 
Velocitv. 



.509 



Number of 
Parts. 



Relative ' 
Velocity. 



.454 



Friction. 



The friction of a fluid gliding over a solid surface exerts a resistance to 
its motion which is proportional to the surfaces of contact and to the densi- 
ty of the fluid, and approximately to the square of the velocity of its mo- 
tion; that is, the resistance is approximately proportional to the weight 
of a prism of the fluid, the base of which is the surface of contact, and its 
height that due to its velocity. 

C Ws — = F, C representing the coefficient of friction, W the weight of a unit of 
volume of the fluid in pounds, and s the surface of contact in square feet 

The flowing of liquids through pipes or in natural channels is liable to 
be materially affected by friction. Liquids flow smoothly and with least 
retardation when the course is perfectly smooth and straight. 

Thus, if equal quantities of water were to be discharged through pipes 
of equal diameters and lengths, but of the following forms : 

and the time that the quantity discharged through 
the first is represented by 1 ; the time that will be re- 
quired to discharge an equal 
quantity through the second 
will be 1.11; and the time for 
the same quantity through the 
third, 1.55; and the Velocities 
of motion that would result un- 
der like heads would be as 1 for the first, .72 for the 
second, and .G4 for the third. 
When a fluid issues out of a circular aperture in s 
Hh 



It) 



(3.) 



(2.) 



362 HYDRAULICS. 

thin plate in the bottom or side of a reservoir, the issuing stream tends to 
converge to a point at the distance of about half its diameter outside the 
aperture, and this contraction of the stream reduces the area of its sec- 
tion from 1 to .644. 

When a fluid issues through a short tube, the vein is less contracted 
than in the preceding case, in the proportion of 16 to 13; and if it issues 
through an aperture which is alike to the frustrum of a cone, the base of 
which is the aperture, the height of the frustrum half the diameter of the 
aperture, and the area of the small end to the area of the large end as 10 
to 16, there will be no contraction of the vein. Hence this form of aper- 
ture will give the greatest attainable discharge of a fluid. 

The volume of a fluid that will flow out of a Vertical Rectangular Open^ 
ing or Slit, that reaches as high as the surface of the fluid, is .666 of that 
which would flow out of the same aperture if it were horizontal at the 
depth of the base. 

Discharges from. Compound or Divided. Reservoirs. 

The velocity in each may be considered as generated by the difference 
of the heights in the two contiguous reservoirs ; consequently, the square 
root of the difference will represent the velocities, which, if there are sev- 
eral apertures, must be inversely as their respective areas. 

Note.— When water flows into a vacuum, 32.166 feet must be added to the height 
of it; and when into a rarefied space only, the height due to the difference of the 
external and internal pressure only must be added. 

VELOCITY OF WATER OR OF FLUIDS. 

OoefricierLts of Discharge. 

The Coefficient of Discharge or Efflux is the product of the coefficients of 
Contraction and Velocity. 

The quantitj' of water or a fluid discharged in a given time from an 
aperture of a given area depends on the head, form of the aperture, and 
nature of the approaches. Two cases present themselves in practice : first, 
apertures in thin plates in the sides or bottom of a vessel or reservoir ; and, 
second, apertures at the ends of short or long tubes. If v represent the 
velocity per second acquired by falling from a height h, and g the veloci- 
ty acquired at the end of a second, then 

V%gh = v. 

The value of g varies slightly with the latitude and altitude of the place 
of observation, but may be assumed for most practical purposes equal to 
32.1G6 feet; then, for measures in feet, 



V 2 



2>2 = 64.333/i, and k=——~, h representing the height measured to the centre of the 

0-4. ooo 

opening. 

The head or heigkt, k, may be measured from the surface of the water to 
the centre of the aperture without practical error, for it has been proved 
by Mr. John Neville that for circular apertures, having their centre at 
the depth of their radius below the surface, and therefore the circumfer- 
ence touching the surface, the error can not exceed four per cent, in ex- 
cess of the true theoretical discharge, and that for depths exceeding three 
times the diameter the error is practically immaterial. For rectangulai 
apertures it is also shown that, when their upper side is at the surface of 
the water, as in notches, the extreme error can not exceed 4.17 per cent, 
in excess; and when the upper is three times the depth of the aperture 
below the surface, the excess is inappreciable. For notches, weirs, slits, 



HYDRAULICS. 363 

etc., however, it is usual to take the full depth for the head, when the above 
equation must be multiplied bj r two thirds to ascertain the discharge. 

Experiments show, 1. That the coefficient for similar apertures in thin 
plates, for small apertures and low velocities, is greater than for large 
apertures and high velocities, and that for elongated and small apertures 
it is greater than for apertures which have a regular form, and which ap- 
proximate to the circle. ' 

2. That it increases with the ratio of the aperture to the approaching 
channel when formed at the end of a short tube. 

3. That it increases when the aperture is at the side or bottom of a ves- 
sel, and when the contraction is partial. 

4. That it increases when the dimensions of the aperture and the head 
of water decrease. 

When the head, measured from the surface to the centre of an aperture, 
is the same, the discharge from them, whether horizontal or lateral, if of 
equal area, is practically the same. 

When the Discharge of a Fluid is under the Surface of another body of 
a like Fluid, the difference of the levels between the two surfaces must be 
taken as the head of the water or fluid. 

Or, V2g{k — h') — v. 
When the Outer Side of the opening of a discharging Vessel is pressed by a 
Force, the difference of the height of the head of the fluid and the quo- 
tient of the pressures on the two sides of the vessel, divided by the densi- 
ty of the fluid, must be taken as the heads of the fluid. 

Or, x /%g ( h — ^ ) = v, S representing the density of the fluid. 

Illustration. — Assume the head of water in the open reservoir of a steam hoiler 
is 12 feet above the water-line in the boiler, and the pressures of the atmosphere and 
the steam are respectively 14.7 and 19. T lbs. 

= 5.551 feet. 

When Water jloivs into a rarefied Space, as into the Condenser of a 
Steam-engine, and is either pressed upon or open to the Atmosphere, 'the 
height due to the mean pressure of the atmosphere within the condenser, 
added to the height of the water above the internal surface of it, must be 
taken as the head of the water. 

Or, V%g(h-)-ti)=v. 

Illustration. — Assume the head of water in the condenser of a steam-engine to 
be 3 feet, the vacuum gauge to indicate a column of mercury of 26.467 ins. (=13 
lbs.), and a column of water of 13 lbs. z= 29.9 feet. 

Then V*g (3 + 29.9) = "/64. 333x32.9 = V 2 H6.5C6 = Mfeet. 

Relative Velocity of the Discharge ofWater throxigh dif- 
ferent Apertures and. under like Heads. 

The velocity of motion that would result from the direct, unretard- 
ed action of the column of water which produces it, being a con- 
stant, or 1. 

The velocit}' through a cylindrical aperture in a thin plate 625 

Through a tube from two to three diameters in length, projecting 

outward 8125 

Through a tube of the same length, projecting inward 6812 

Through a conical tube of the form of the contracted vein 974 



364 HYDRAULICS. 

COEFFICIENTS FOR THE DISCHARGE OR EFFLUX OF WATER FOR 
VARIOUS OPENINGS AND APERTURES. 

Rectangular Weir. 
Height measured from the Surface of the Water to the Sill. 

Coefficients for the Discharge over Weirs. 

[From the Experiments o/James B Fbancis, Lowell, Mass , 1852.] 
Mean Head. , Length of Opening I Mean Discharge per Second, i Mean Coefficient. 

.6i to 1.55 feet. j 10 feet. J 32.9 cubic feet. .023 

Mean, .603; Small oblong openings as they approach the surface, .705 (Poncelet 
and Lesbros) ; Large square apertures as they approach the surface, .572 ; Opening, 
10 feet in length and 1 inch in depth, .591; 9 inches deep, .781 ; mean, .723 (Black- 
well). 

\_Deduced from the Experiments o/'Mr. Blackwell.] 

Heads in inches, measured from stiil water in the reservoir, 1 to 14. Thin plates, 
3 ftet long, .617; 10 feet long, .607. Planks 2 inches thick, square, 3 feet long, .571 ; 
C feet long, .563; 10 feet long, .547 ; 10 feet long, wing-boards, making an angle of 
C0°, .714. 

The principal causes for the variation in the coefficients derived from 
most experiments giving the discharge of water over weirs arises from, 

1. The depth being taken from only one part of the surface, for it has 
been proved that the heads on, at, ancl above a weir should be taken in or- 
der to determine the true discharge. 

2. The unequal widths of the crest, which, increasing the friction, re- 
duce the coefficients, particularly for smaller depths, very considerably.^ 

3. The nature of the approaches, including the ratio of the water-way in 
the channel above, to the water-way on the weir. 

When a weir extends from side to side of a channel, the contraction is 
less than when it forms a notch, or Poncelet weir, and the coefficient some- 
times rises as high as .667. When the weir or notch extends only one 
fourth, or a less portion of the width, the coefficient has been found to 
varv from .584 to .6. 

When the overfall is a thin plate, it discharges a greater proportionate 
quantity, when the stream is only 1 inch deep, than with greater depths, 
the vein contracting with the increased head. 

When the length of the weir is 10 feet, the coefficient is greatest with a 
depth of 5 inches ; and when wing-boards are added at an angle of about 
64°. the coefficient is greater than even when the head is less. 

When the heads remain the same, the coefficients decrease, at first more, 
and then less rapidly than the breadths of the weirs. 

Itectangxilar lS"otclies, or Vertical Apertures or Slits. 
Height measured from the Surface of the Head of the Water to the Sill 

Opening, 8 inches by 5 inches, mean .606 (Poncelet and Lesbros). 

The coefficient increases as the depth decreases, or as the ratio of the 
length of the notch to its depth increases. 

Opening, 18.4 inches by 1.8 and 6.75 inches, .648 to .63 (Du Buat). 

When the sides and under edge of a notch increase in thickness, so as 
to be converted into a short or small channel, open at the top, the coeffi- 
cient* reduce very considerablv, and to an extent beyond what the in- 
creased resistance from friction, particularly for small depths indicates. 
Poncelet and Lesbros found, for apertures 8x8 inches, that the addition 
of a horizontal shoot 21 inches long reduced the coefficient frcm .604 to 
.601, with a head of about 4 feet; but for a head of 4)4 inches the coefh- 



HYDRAULICS. 365 

cient fell from .572 to .483. For notches 8 inches wide, with the addition 
of a horizontal shoot 9 feet 10 inches long, the coefficient fell from .582 to 
.479 for a head of 8 inches, and from .622 to .34 for a head of 1 inch. 
Castel also found for a notch 8 inches wide, with the addition of a shoot 
8 inches long, inclined 4° 18', the coefficient for heads from 2 to 45 inches, 
to be .527 nearly. 

Triangular Notch. 

Professor Thomson deduced that for discharges from 2 to 10 cubic feet 
per minute in a thin plate the coefficient for a right-angled triangular 
notch was .617. 

Rectangular Openings or Sluices, or Horizontal Slits. 

Height measured from the Surface of the Water to the Centre of Pressure 
of the Opening. 

Opening, 1 inch by 1 inch. Head, 7 to 23 feet, .621) -*r;,>v» i^ff; 
u 3 t . u 3 .a |4 7 " 23 lt #614 f Micneiorci. 

" 1 " " 1 " " ' 1 " 5 " .637— Smeaton. 

2 " "2 " u 12 feet .617— Bossut. 

Equilateral triangle, 1 square inch, base down, .593) teA ,JV. 
" 1 " » base up, .585J- Eennle - 
Poncelet and Lesbros deduced that the coefficient of discharge increases 
with small and very oblong apertures as the}" approach the surface, and 
decreases with large and square apertures under like circumstances. 

The coefficients ranged, in square apertures' of 8 by 8 inches, under a 
head of 6 inches to rectangular apertures, 8 inches by .4 inches; under a 
head of 10 feet, from 572 to .745. 

In applying an open channel or canal to the exit of an opening in a res- 
ervoir, etc., the bottom and sides corresponding with the dimensions of the 
opening, Poncelet and Lesbros obtained the following coefficients : 

' Without channel, mean, .623 ; with channel, mean, .628. With a chan- 
nel, at a declination of .01, or 34', the coefficients were sensibly the same 
as when it was horizontal ; and when the declination was increased to .1, 
or 5° 44', the coefficients were increased. 
In a Thin Plate = .616 (Bossut) ; .61 (Michelotti). 
Sluice-boards or Convergent Sides. — When one side is inclined to the 
horizon, the coefficients for angles of 45° and 63° 30' are as .467 and .447. 

Circular Openings or Sluices. 

Height measured from the Surface of the Head of the Water to the Centre 
of the Opening. 

f Head of 6 inches; diameter of opening, 1. inch = .649 
Simple J Head of 1 foot ; diameter of opening, 6. inches — .642 
Aperture. | Head of 10 feet; diameter of opening, .5 u =.609 
[ Head of 10 feet ; diameter of opening, 6. u =.601 
Mean for 5 feet, and diameter of opening of 3 inches == .615 (Ponce- 
let and Lesbros). 

In a Thin Plate — 666 (Bossut) ; .631 (Venturi); .64 (Eytelwein). 

Contraction of section from 1 to .633, and reduction of velocity to .974 , 
hence . 633 x. 974 = .617 (Neville). 

1 to 3 inches in diameter, and heads from 6 to 23 feet, .614; and 3 to 6 
inches in diameter, with like heads, .62 (Michelotti). 

Cylindrical Ajutages, or Additional Tubes, give a greater discharge than 
apertures in a thin side, the head and area of the opening being the same ; 
but it is necessaiy that the flowing water should entirely fill the mouth 
of the ajutage. 

The mean coefficient, as deduced by Castel, Bossut, and Evtelwein, is .82. 

Hh* 



366 



HYDRAULICS. 



(1.) 



(2.) 




Sliort Tn"bes and Month-pieces. 

If an aperture be placed in the side of a vessel of from 1)^ to 2}^ diam. 
eters in thickness, it is converted thereby into a short tube, 
and the coefficient, instead of being reduced by the in- 
creased friction, is in- 
creased from the mean 
value up to about .815, 
when the bore is cylin- 
drical, as in Fig. 1 ; and 
when the junction is 
rounded, as in Fig. 2, to 

the form of the contracted vein, the coefficient increases 
to .974. 

In the conically di- 
vergent tube, Fig. 4, the 
coefficient of discharge is greater than 
for the same tube placed convergent, the 
fluid filling in both cases, and the smaller 
diameters,, or those at the same distance from the centres, o o, being used 
in the calculations. A tube, the angle of convergence, o, of which is 5° 
nearly, with a head of from 1 to 10 feet, the axial length of which is 3^ 
inches, small diameter 1 inch, and large diameter 1.3 inch, gives, when 
placed as at Fig. 3, .921 for the coefficient; but when placed as at Fig. 4, 
the coefficient increases up to .948. The coefficient of velocity is, howev- 
er, larger for Fig. 3 than for Fig. 4, and the discharging jet has greater 
amplitude in falling. If a prismatic tube project beyond the sides into a 
vessel, the coefficient will be reduced to .715 nearly. 

Cylindrical Prolongations or Ajutages. 





H?»ty,//M,//]M 



Length of pro- 
longation in 
Diameters of 
Aperture. 


Coefficient of 
Discharge 


Length of pro- 
longation in 
Diameters of 
Aperture. 


Coeffieiennt of 
Discharge. 


Length of pro- 
longation in 
Diameters of 
Aperture 


Coefficient of 
Discharge. 


1 and under 

2 to 3 


.62 

.82 


4 to 12 
25 ** 36 


.77 
.68 


37 to 48 
49 " 60 


.63 
.6 



The coefficients for prismatic tubes increase as the depths decrease, the 
same as for simple apertures. Bossut's experiments gave a mean of .807, 

Conically Convergent and. Divergent Tn"bes. 

The form of tube which gives the greatest discharge is that of a trun- 
cated cone, the lesser base being fitted to the reservoir (Fig. 4). Venturi 
concluded from his experiments that the tube of the greatest discharge 
has a length 9 times the diameter of the lesser opening (base), and a di- 
verging angle of 5° G' — the discharge being 2.5 greater than that through 
a thin plate, 1.9 times greater than through a short cylindrical tube, and 
1.46 greater than the theoretic discharge. 

D'Aubuisson and Castel's experiments give, for conically convergent 
tubes, Fig. 3, the following coefficients (Mr. Neville) : 



Coefficient of 
Velocity 



.85S 
.916 
.933 



Coefficient of 
Discharge. 



.853 
.920 
.931 



Converging 
Angle at 0. 



16° 
30° 
50° 



Coefficient of 
Velocity. 



.970 
.976 
.9S5 



Coefficient of 
Discharge. 



.937 
.S95 

.844 



Compound Month-pieces and .Ajntages. 
The following Table gives the coefficients of discharge for the different 
figures here given, and it will be found of great value, as the coefficients 
are calculated for the large as well as small diameters. 






HYDRAULICS. 



367 





The necessity for taking a | 
into, consideration the form of 
the junction of a pipe with a 
reservoir will be understood 
from the following results : 

Tatole. of Coefficients for IMoutli-pieces and. Sliort Tribes. 

[Calculated and reduced by Mr, NKViLLE,/Vo»t Vbktuki's Experiments.] 




Description of Aperture, Mouth piece, or Short Tube. 



.611 
.601 



.561 



.956 
.934 



.948 



1. An aperture IX ins. diameter in a thin plate .622 

2. Cylindrical tube IX ins. diameter, and 4% ins. long, Fig. 1 . 823 

3. Cylindrical tube, Fig. 2, having the junction rounded to 

the form of the contracted vein 

4. Short conical convergent mouth-piece, Fig. 3 

5. The like tube divergent, with the smaller diameter at the 
junction with the reservoir ; length 3% ins., lesser diam- 
eter 1 in., and greater diameter 1.3 ins 

6. The tube, Fig. 5, when ab—\% ins., o r=1.21 ins.,it?; = 

1.21 in., and ow = 2 ins., the cylindrical portion being 
shown by dotted lines 

7. The same tube when o u = 11 ins 

The same tube when o u = 23 ins 

8. The tube ab,or, s t, u v, Fig. 5, in which ab = st~ &t 

=s IX ins. from a to s \% ins., and 5 8 = 3 ins 

9. Tbe tube, Fig. 6, a b = \% ins 

10. The same tube, having the spaces aso and rtb between 

the mouth-piece and s £, and the cylindrical tube astb 

open to the influx of the fluid 

11. The double conical tube, ao STrb, Fig. 7, when a b = s t 
= 1>£ ins., or = 1.21 ins., ao = .92 in., and o 8=4.1 ins. 

12. The like tube when, as in Fig. 8, aor& = o6Tr, and a o s 

= 1.84 ins 

13. The like tube when st = 1.46 ins., and p 8 = 2.17 ins 

14. The like tube when st = 3 ins., and o 8 = 9>£ ins 

15. The like tube when os = 6^ ins., and st enlarged to 

1.92 ins 

16. The like tube when st=2^" ins., and o 8 = 12.% ins. ... 

17. A tube, Fig. 9, when os z=rt = 3 ins., or = st = 1.21 

ins., and the tube o btt the same as described in No. 11, 
viz., st = 1X ins., and s 8 = 4.1 ins 

18. The tube, Fig. 9, when st is enlarged to 1.97 ins., and ss 

to 7 ins., the other dimensions remaining as in No. 8 . . . 

19. When the junction of osrt with sst/, Fig. 5, is im- 

proved, the other parts remaining as described in No. 8 . 

Cylindrical Txibes or IPipes. 
The mean of various experiments with tubes of .5 to 3 inches in diam- 
eter, and with a head of fluid of from 3 to 20 feet, gives a coefficient of 
discharge of .813 : and as the mean for circular apertures in a thin plate 
is .63, it follows that, under otherwise similar circumstances and rela- 
.813 



Coefficients 

for the 

Diameter 

a b. 



Coefficients 

for the 

Diameter 

or. 

.974 ~ 



.6 

.507 

.5Jl 


.923 
.873 

.817 


.S23 
.804 


1.266 
1.237 


.785 


1.209 


.C2S 


1.428 


.823 
.823 
.911 


1.266 
1.266 
1.4 


1.02 
1.215 


1.569 
1.S55 


.805 


1.377 


.945 


1.454 


.S5 


1.309 



tions, 



.63 



: 1.29 times as much fluid flows through a cylindrical tube as 



through a like aperture in a thin plate. 



368 HYDRAULICS. 

These coefficients increase as the diameter of the aperture or tube is de- 
creased, and but slighth" with an increase of the velocity of efflux or 
height of the head of fluid. 

Thus, with tubes .65 of an inch in diameter, and with a head of from 9 
inches to 10 feet, the coefficient is from .946 to .957. 

Coefficients of* Discharge from. Short Cylindrical Tuoes 
Witli Square Junctions. 



Relation of the Length of 
the Tube to the Diameter. 


Coefficient of Dis- 
charge- 


Eelation of the Length of 
the Tube to the Diameter. 


Coefficient of Dis- 
charge. 


to \}i diameters 
2 « 
10 " 


.617 
.S14 
.75 


30 diameters 
50 " 
100 " 


.65 

.59 
.48 



DISCHARGE FROM RECTANGULAR WEIRS AND NOTCHES, AND VERTICAL 
APERTURES OR SLITS. 

A Notch is an opening, either vertical or oblique, in the side of a vessel, 
reservoir, etc., alike to a narrow and deep weir. 

Vertical Apertures or Slits are narrow notches, running to or near to the 
bottom of the vessel or reservoir. 

Note.— The mean velocity of a fluid issuing through a rectangular opening in the 
side of a vessel is % of that due to the velocity at the sill or lower edge of the open- 
ing, or it is that due to a point ^ of the whole height from the surface of the fluid. 

To Compute tlie Volume of Fluid, -which -will Flow out 
of any of the above Openings. 

Height measured from the Surface of the Head of the Water to the Sill of 
the Opening. 

B#L,rc. — Multiply the square root of the product of 64.333 and the height 
or whole depth of the fluid in feet by the area in feet, and by the coeffi- 
cient for the opening, and % of this product will give the discharge in cu- 
bic feet per second. 

V 

Or, ^/>/&v / 2#/iC=V, and ^ r=r- = t , t representing the time in seconds. 

3 §bhV2ghV 

Example. — The sill of a weir is 1 foot below the surface of the water, and its length 
is 10 feet ; what volume of water will it discharge in one second ? 

C = .G23, and V04.33x 1X10X1=80.2, and | 80.2X.623= 33.318 cubic feet. 
Note. — The mean* coefficient of discharge of weirs, the breadth of which is no 
more than the third part of the breadth of the stream, is ^ of 6 = .4; and for weirs 
which extend the whole width of the stream it is J .600 = .444. 

Rectangular Sluices and Horizontal Slits. 

Height measured from the Surface of the Head of the Water to the upper 
Side and to the Sill of the Opening. 

Rulk. — Multiply the square root of 64 333 and the breadth of the open- 
ing in feet by the coefficient for the opening, and by the difference of the 
products of the heights of the water and their square roots, and % of the 
whole product will give the discharge in cubic feet per second. 
Or, § b^/2g~<h\/h' — hy/h) C = V, h and h' representing the heights of the aperture 
and sill in feet. 
Examplk.— The sill of a rectangular sluice, 6 feet in width by 5 feet in height, is 9 
feet below the surface of the water ; what is the discharge in cubic feet per second? 
C = .C>25, f V~fg= 5.348, and h' =9-5 = 4; 5.348X6 (9>/9 -4 V '4)X.625 = 

32. 03>X 19 X. 625 = 381.04 cubic feet. __L 

* This includes the element of % in the formula. 



HYDRAULICS. 369 



DISCHARGE FROM CIRCULAR SLUICES, ETC. 

Height measured from the Surface of the Head of the Water to the Centre 
of the Opening. 

Rule.— Multiply the square root of the product of 64.833 and the depth 
of the centre of the opening from the surface of the water bj T the area of 
the opening in square feet, and this product by the coefficient for the 
opening, and the whole product will give the discharge in cubic feet per 
second. 

Or, \/2^A,aC=:V, a representing the area in square feet, and h the height of the 
surface of the fluid from the centre of the opening in feet 

Example. — The diameter of a circular sluice is 1 foot, and its centre is 1.5 feet 
below the surface of the water; what is the discharge in cubic feet per second? 

113 no 

Area of 1 foot — _-!_ — . TS54; C = .C4 ; V G4.ii33X 1.5 X.7S54 X .64 = 9.823 X 
144 
.7854X.64:= 4.938 cubic feet. 

When the Circumference reaches the Surface of the Water. 
V% g r, .9G04 aC = V,r representing the radius of the circle. 

Semicircular Sluices. 
When the Diameter is either Upward or Downward. 

l/2 g h a C = V, h representing the depth of the centre of gravity of the figure from 
the surface. 

Semicircular Weirs or !N"otclies. 

When the Diameter is Uppermost and Horizontal. 

V%gr .6103 aQ—Y. 

When the Diameter as above is at the Depth z, below the Surface, 

V 2g~z 1.188 aC = V. 

When the Circumference is Uppermost and Horizontal. 

V%gr .7324 a C = V, r representing radius of semicircle 

Example. — In what time will S00 cubic feet of water be discharged through a cir- 
cular opening of .025 square feet, the centre of which is 8 feet below the surface of 
the water ? 

C = 63. 

= *» = *» - = - S -^ = 2: 3 9.5S = 37 .&.; 19.0 sec. 

V"2ghX.025x.63 22.68X-025x.63 .35721 

DISCHARGE FROM TRAPEZOIDAL WEIRS OR NOTCHES. 

Height measured from the Surface of the Head of the Water to the Sill of 
the Opening. 

When either the Greater or Lesser Breadth is uppermost. 

15A \/2 g h (2 b -f- 3 b') C = V, b and b' representing the upper and lower breadths. 

Trapezoidal Sluices or Slits.* 
When the Greater Breadth is uppermost. 

5 5\ 

2 , — / 3. 3 2 H'3-A'j| 

- /2 g ( b'h"z — bh2 4--(b — b') — - r-JC — V, h and h' representing the depth 

\ 5 h — 11 ' 

from the surface to the upper and lower edges of the opening. 



For Triangular Sluices, etc., see Weisbnch, vol 1 , p 359, 



370 



HYDRAULICS. 



When the Lesser Breadth is uppermost. 

lLLUSTRATioN._The sill of a trapezoidal sluice 5 feet in height, the lesser breadth 
being uppermost, is 9 feet below the surface of the water, and the breadths of i ,2 
5.5 and 6.5 feet ; what is the discharge in cubic feet per second? DleadUls of Jt are 

C=.62, ? g V2g = 5.348, *'= 9 - 5 = 4; 5.34S (g.5 X V9 3 -5.5 X V* 3 -- (6.5- 

55) ^5^(x.62 = 5.348x(l75.5-44-|lx 2 i^ 2 ) x :G2=5.348X(131.5 
— 21.Dx.62 = 366.06 cubic feet. 

Triangular "Weirs and ^"otclies. 

Height measured from the Surface of the Head of the Water to Sill or Base 
of the Opening. 

When the Vertex is uppermost, %^2~g b <fd* C = V. 
When the Base is uppermost, xs^/2gb^/d 3 C = V. 
When a Notch is a Parallelogram, longest Diagonal being Vertical 

4. h / 5 5\ 

V29^Ah2-2dz)C=V. 

15 a y 

When Notch is a Right Angle, Base at Surface of Water, fy^/2gd^ = Y. 

When Vertex is at Surface of Water, % b dy/TgliC—Y, b representing breadth 
and d depth of opening in feet. 

The foregoing formulas* furnish an expression for the discharge from 
any rectilineal aperture whatever, as it can be divided into triangles, the 
discharge from each of which can be determined ; as an}- triangular aper- 
ture can be divided into two others by a line through one of the angles. 

Note. — For the greater number of apertures at any depth below the surface of the 
water, the product of the area, and the velocity due to the depth of the centre, or 
centre of gravity when it is practicable to obtain it, will give the discharge with 
sufficient accuracy. 

DISCHARGE FROM WEDGE- AND PYRAMIDAL-SHAPED VESSELS. 

Horizontal Triangular Prism — A Side uppermost. Paraboloid of Revolution — 
4 V 

Base uppermost. 5 X — t . 

3 Ga-flgk 
Note. — In this efflux the time is % greater than if the initial velocity remained 
uniform. 

6 V _, 

Quadrangular Pyramid— Base uppermost. ~X == — * 

C ay/2 g h 

Note. — As in this efflux the initial velocity of it decreases gradually to 0, the time 
of efflux is ^ greater than if the initial velocity remained uniform. 

Example.— In what time will a pond of water, the surface of which has an area 
of 765000 square feet, be discharged through a conduit 15 feet below its surface, 15 
inches in diameter, and 50 feet in length ? 

C = . 6. V = <C5fmQx15 — 5T37500 = volume of pond, a — 1 . 2272. 

4 5737500 22950000 ~. : .^ - . I' ' . .. 

-X-= := tr-zz-r* = — n „ — 334450 sec = 92 /?., 54 mm., 10 sec. 

3 .6X1.2272X31.0644 GS.62 

* For other formulas, see Neville's Treatise, p 51-58. 



HYDRAULICS. 371 

SPHERICAL AND PRISMOIDAL-SHAPED VESSELS. 

Sphere ' — ■ = t, 2 r being equal h. Hemisphere — Base uppermost : 

' 15Uay/2g 

— pr v ■ — t, r being equal h. Spherical Segment — Base uppermost : - 

l5Cay/2g 5 

Ah 
— t. Prismoid — Trilateral or Multilateral Pyramid — Bases uppermost: 

(3 q .}_ 8 Gi + 4/ Or Gx) = t, G and G' repfg the base and lower surface. 

15CaV 2 ^ 
Example. — A prismoidal reservoir filled with water is at top 5 feet in length by 3 
feet in breadth, and at a depth of 4 feet, at the point of insertion of a. short horizon- 
tal discharge, 1 inch in diameter by 3 inches in length ; it is 4 feet in length and 2 
feet in breadth ; what is the time required for the water in it to subside 2.5 feet? 
C — .815, a = . 00545, G = 15, G x = S. 

(3X 1 5 + 8XS + 4/15X8) 15x , 815x , V 0545x8 . 02 = 152.81SX^=1144.7 secoms, 
the time of emptying the whole vessel. 
Then 4 — 2.5= 1.5 ; hence G = 4. 375x2. 375 = 10. 39. 

(3X10.33 + 8x3 + 4\/l0.39x8) g *?{* 5 - 131.638x^1 ~ C03.9 seconds, and the 

.584 .584 

difference of these times gives the time in which the level of the water subsides 2.5 
feet ; viz., 1144.T — 603.9 = 540.8 seconds. 

DISCHARGE FROM IRREGULAR-SHAPED VESSELS, AS A POND, LAKE, ETC. 

To Compute tlie Time and. Volume d.isclia,rged.. 

Divide the whole mass of water into four or six strata of equal depths. 

™, * * c h — h± /A , 4 At , 2A 2 , 4A 3 , A 4 \ , ,, 

Then, for 4 Strata, 4** (— + — l + _i+— - 3 + _JL) = t, h, h', 

etc., representing the depths of the strata at A, A 2 , etc., commencing at the sur- 
face; Au A 2 , etc., being the areas of the first, second, etc., transverse sections of 

the pond, etc. ; and ~ 4 X A + 4 A L + 2 A 2 + 4 A 3 -f- A 4 = V. 

Example. — In what time will the depth of water in a lake subside 6 feet, the sur- 
faces of its strata having the following areas, the outline of the sluice being a semi- 
circle, 18 inches wide, 9 inches deep, and 60 feet in length? 

A = 20 feet (h ) depth of water = area of 600000 square feet. 
A 1 = 18.5 " (hO " u = .« 495000 " " 
A 2 = 17 " (h 2 ) » " = « 410000 « " 
A 3 = 15.5 " (A 3 ) " " = * 325000 u " 
A 4 = 14 " (hO " " = " 265000 " « 
a — area of 18 -H 2 = .8S3G square feet ; C = .537. 
Then 20 — 14 / 600000 4x495000 2x410000 4x3^5000 

12 X .537X.8S36 X S.02 X \ 4.472 + 4.301 + 4.123 ^ 3^937 *" 
265000X 6 

1TT7T = ak Pi-Ao x H94429 = 156938 sec. = 43 h.. 35 min., 38 sec. 
3.742 / 45.0048 ' ' 

And th e discharges r % X (600000 + 4x495000+2x410000+4x325000+265000) 
= 6 -r- 12X4965000 = 2482500 cubic feet. 

For 6 Strata, put 2 A 4 instead of A 4 , and 4A 3 and A 6 additional, and divide by 
18 instead of 12. 

Discharge through Pipes or Canals, when the Form and Dimensions of 
the Vessel of Efflux are not known. 

The volume discharged may be estimated by observing the heads of 
the water at equal intervals of: time. 



3 72 HYDRAULICS. 

Then VatWg f&hA) =V,/orl depth; CSfVS? (*&fa|/*i±VM 
= V for 2 *e}H&, and CatVT g ( V* + 4 •*, + » V», + 4V*, + V*. j =Y/b , 

4 depths. 

Note. — At the end of half the time of discharge, the head of water will be X of 
the whole height from the surface to the delivery. 

When discharged through Weirs or Notches. 

2 / — 

~ C b t V - g C\A 3 ~f~ 4 Vh 3 i + V^ 3 2) = V, 6 representing the breadth in feet. 

Example. — A prismatic reservoir 9 feet in depth is discharged through a notch 
2.222 feet wide, the surface subsiding 6.75 feet in 9S5 seconds; what is the volume 
discharged ? 

C = .6; fc 1 = 9— 6.75 = 2.25/«tf. 

? .6X2.222X935X8.02 (V^ + 4-/2T253 + \ffifi = 2221.6x(27 + 13.5+ 0) = 2221.5 

X40.5 = 89974. S cubic feet. 

When there is an Influx and Efflux. 

If a reservoir or vessel during an efflux from it has an influx into it, 
the determination of the time in which the surface of the water rises or 
falls a certain height becomes so complicated that an approximate de- 
termination is here alone essayed. 

A state of permanency or constant height occurs whenever the head of the water 

1 / I \2 

is increased or decreased by — I — — ) =&, I representing the influx in cubic feet per 
second. 
The time in which the variable head of water (x) increases by the volume V is 

At V 

given by the following formula : ; and the time in which it sinks the 

I — VaY^gx 
A V 

height, k, by — — 

CaY 2gx — I 
The time of efflux, in which the subsiding surface falls from A to A x , etc., and the 

head of water from h to h^ when k is represented by = y/ k, is — 

CaYtg 12CaY2g 

I A 4A t . 2A 2 4A 3 A 4 \_ 

\Y b — V* • Y^—y/k ■^/h 2 — y/k "^VAs- V* V^ — y/k)~ ' 

Example — In what time will the surface of the water in a pond, as in a previous 
example, fall 6 feet if there is an influx into it of 3.0444 cubic feet per second ? 
y/k = .8, C = .537. 

20 — 14 / 600000 4 X 495000 2 X 410000 4 X 325000 

12x.537x.S836x8.02 X \4.472 — .8 "*" 4.301 — .8 + 4.123 — .8~'~3.937 — .8 ~^ 
- X 1480201 = 194498 sec. = 54 h , 1 twin , 3S sec. 



3.742 — .8/ 45.064S 

Prismatic Vessels. — If the vessel has a uniform transverse section, A. 

Then = ( y/h — y^i + V^ X hyp. log.* (X— ZV_\ ) = t = the time in 

GaVVg^ W h i — Y k/ ' 

which the head of the water flows from htoh L . 

Example. — A reservoir has a surface of 500000 square feet, a depth of 20 feet; it 
is fed by a stream affording a supply of 3.0444 cubic feet per second, and the outlet 
has an area of .8S36 square feet ; in what time will it subside 6 ftet? 
y/k, as before = .8, C = .537. 

* For hyperbolic logarithm substitute the ordinary by multiplying by 2 303. 






HYDRAULICS. 373 

2X500000 /flA ; 4M , ft w ,__ /V20 — .8\__ rt _ 1000000. 



x (v , 20 _ V 14 + .8X .og. ($£$ x2 .S03) = ~x4.472-3.T42 



+ .8Xlog. (^^ =r g)x2.303 = 262784.5X(.73 4-.Sx.09621x2.3C3) = 262784.5x 
(.73 + .17726; = 238414 sec. = 6Q h., 13 min , 34 sec. 

To Compute the Fall iix a given. Time. 

This is determining the head h x at the end of that time, and it should be 
subtracted from the head h at the commencement of the discharge. Put 
into the preceding equation several values of h u until one is found to meet 
the condition. 

Illustration. — Take a prismatic pond having a surface of 3S750 square feet, a 
depth to the centre of the opening of the sluice of 10.5 feet, a supply oi 33.6 cubic 
feet, and a discharge of 40 cubic feet par second. 

V* = -S4. 

Putting these numerical values into the equation, and assuming different values 
for A 2 , a value which nearly satisfies the equation is 4. Consequently, 10.5 — 4 = 6.5 
feet, the fall. 

For a Weir or Notch. 

t = Tl [hyp. log. (yjW *)° + v™ •» (*"* = ^ TVI )\ ; 

/ I \2 

jt = [_ — J 3 5 arc (tang. = $r), the arc the tangent of which = y. 

According as A; is ^ A, and the influx of water, 1^ | C / V'tgW, there is a rise 
or fall of the fluid surface, the condition of permanency occurring when hi = A:. 

Example. —In what time will the water in a rectangular tank, 12 feet in length hy 
6 feet in breadth, rise from the sill of a notch, 6 inches broad, to 2 feet above it, when 
5 cubic feet of water flow into the tank per second? 

hi = 2, A = 0, A = 12X6 = 72, 1 = 5, b — .5, C = .6. 

k — ( 5 \| —3/3.1172 = 2.1338. 

Vf.6X.5x8.0-V 

_. 72 X 2.1338 / 2 -f- V2 x 2.1338 + 2.13C8 , 

Then __3 X 5 V iyP ' ^ (V2-V2.1338P " + ^ arC H* = 

vrgv» )) = 10 - 2423 x hyp - log - » - 3 - 4641 x arc ( tang - £® 

== 10. 2423 XLT. 96 1 — (3.461X arc, the tangent of which = .56497, or 29° 28' = 29.466, 
the length of which = .5143) = 1.781] = 10.2423 —7.961 — 1.781 = 10.2423X6.18 = 
63.297 sec. 

DISCHARGE OF WATER UNDER VARIABLE PRESSURES. 

To Compute tlie Time, tlie iRise and. Fall, and the Volume 

of "Water discharged tinder "Variable Pressures. 

— VZyoc = i>, x representing the variable head, A area of transverse horizontal sec- 
tion of vessel, and v theoretical velocity of efflux. 

Discharge from Reservoirs or Vessels not receiving any supply of Water. 

For prismatic vessels the general law applies, that twice as much would be dis- 
charged from like apertures if the vessels were kept full during the time which is re- 
quired for emptying them. 

Zky/h 2Ah 

To Compute the Time. -=r = _ = t. 

CaV2g D 

Example A rectangular cistern has a transverse horizontal section of 14 feet, a 

depth of water of 4 feet, and a circular opening in its bottom of 2 inches in diame- 

Ti 



374 HYDRAULICS. 

ter; in what time will it discharge its volume of water, the depth being maintained 
at 4 feet, and in what time when the supply to it is cut off and the cistern allowed to 
be emptied of its contents? 

h = 4feet, a = 22X.TS54 -H 144 = .021S, C = .6. 
V2g~kXaXC = 16.04X.0218X.6 = .209S cubic feet per second. 

2x14x4 
Hence — - -j- 2 = 2G6.9 seconds, the depth being maintained. 

Under a diminishing head of water the coefficient of efflux is increased ; hence, in 
the following case, it is taken at a mean of .CIS, and the volume discharged be- 
comes .2143. 

2x14x4 
Then — = 522.6 seconds, the vessel being emptied. 

.£l4o 

To Compute tlie Time and. Fall. 

The depression or subsiding of the surface of the water in a vessel, cor- 
responding to a given time of efflux, is h — h' = s, h representing the lesser 
depth, 

2A / GaV%g\ 
t.n. . /h>\ — * t«xt Q -.. 3Q itt (_/*. ty- — h' . 



2 A / 

/— (s/h — ^/h!) 35 1. Inversely, ( y/h - 



Example. — In what time will the water in the cistern, as given in the preceding 
example, subside 1.6 feet? 

A = 14, C = .6, a = .0218, i/2^= 8.02, A = 4, /V = 4 — 1.6 = 2.4. 

O y 1 A GO 

■ B X.»^x8.»^ ^-V2.4) = I ^- 9 X(2-1.55) = 120.1,ec. 

To Compute tlie Volume. 

At/ = V, y representing the extent of the fall, and V the volume of water dis' 
charged, as h — h'. 

Discharge from Vessels when the Reservoir of Supply is maintained at a 
uniform Height, 

2A-\A 

To Compute tlie Time. =r — t. 

CaVZg 
Example. — In what time will the level of the water in a receiving vessel having 
a section of 14 square feet attain the height of that in the supply, through a pipe 2 
inches in diameter, placed 4 feet below the level of the supply? 
n A<Q 2X14XV4 56 KOOO 

C = - 613 ' .-613 x.0218X8.02 ^O072^ 522 - 3geC 

When the Vessel of Supply has no Influx, and is not indefinitely great 
compared with the Receiving Vessel, 

r— - = t, A' representing section of receiving vessel, t the time in 

Ca(A+A')V2g 

which the two surfaces of water attain the same level ; and the time within which 

.* i isn r u. ... 2AA'(Vft — V/l') , 

the level falls from h to h' is j — = t. 

Ca(A-hA')V / 2 5 - 

Discharge from a Notch* in the Side of a Vessel when it has no Influx, 
3 A 



ilbxVtg 



r=r (—rr, 77 ) = t, b representing the breadth of the notch in feet. 

2s- W lt V V 



* When the notch extends to the bottom of the reservoir, etc., the time for the water to run out 
is indefinite, as h' = 0. 



HYDRAULICS. 375 

DISCHARGE FROM VESSELS IN MOTION. 

When the Vessel moves uniformly up or down. Vlgh — Y, g representing the 
accelerating force. 

When it Ascends with a retarded Motion. V%(g — p) h — Y,p representing the 
power applied. 

When it Descends with the same Retardation. v / 2 {g -f- p) h = V. 

Example A vessel containing water weighing 350 lbs. is drawn by a running 

weight of 450 lbs. over a roller ; what is its accelerating force, and what the velocity 
of its discharge '? 

450 — 850 100 ' . /T r— — ; 7 . -J..!' 

p = , g = g^- = .1-5 ; hence y 2x (§ + 8) g h = velocity of discharge. 

If the head of water is 4 feet, then V 2 X§X32.16x4 = IT. 01 feet. 



When the Vessel revolves. Vtgh+o2 = V, o representing the velocity of rota- 
tion of the aperture in feet per second. 

FLOW OF WATER IN RIVERS AND CANALS. 

Running Water. — Water flows either in a natural or artificial bed. In the first 
case it forms Streams, Brooks, and Rivers ; in the second, Drains, Cuts, and Canals. 

The Bed of a flowing water-course is formed of a bottom and the two banks or 
shores. 

The Transverse Section is a vertical plane at right angles to the course of the 
flowing water ; the Perimeter is the length of this section in its bed. 

The Longitudinal Section or Profile is a vertical plane in the course of the flow- 
ing water. 

The Slope or Declivity is the mean angle of inclination of the surface of the water 
to the horizon. 

The Fall is the vertical distance of the two extreme points of a defined length of 
the flowing course, measured upon a horizontal plane, and this fall serves to assign 
the angle for the defined length of the course. 

The Line of Current is the point when the flowing water attains its maximum ve- 
locity. * 

The Mid-channel i?, the .deepest point of the bed. The Velocity is greatest at the 
surface and in the middle of the current; and the surface of flowing water is highest 
in the current, and lowest at the banks or shore. 

A river, canal, etc., is in a state of permanency when an equal quantity of wtter 
flows through each of its transverse sections in an equal time, or when V, the prod- 
uct of the area of the section, and the mean velocity through the whole extent of the 
stream, is a constant number Hence, in the permanent motion of water, the mean 
velocities in two transverse sections are to each other inversely as the areas of these 
sections. 

To Compute tlie JJVIean Deptli of Flo-wing Water. 

Bulk. — Set off the breadth of the stream, etc., into any convenient num- 
ber of divisions ; ascertain the mean depths of these divisions; then di- 
vide their sum by the number of divisions, and the quotient is the mean 
depth. 

To Compute trie IVIean Area of* Flowing Water. 

Bulk 1. — Multiply the breadth or breadths of the stream, etc., by the 
mean depth or depths, and the product is the area. 

2. — Divide the volume flowing in cubic feet per second by the mean 
velocity in feet per second, and the qnotient is the area in square feet. 

To Compute tlie Volume of* Flowing Water. 

Bulk. — Multiply the area of the stream, etc., by the mean velocity of 
its flow in feet, ana* the product is the volume in cubic feet. 



376 HYDRAULICS. 

To Compute tlie IVLean Velocity of FloAviiig Water. 

Rulk. — Divide the velocity of the flow in feet per second by the area 
of the stream, etc., and the quotient will give the velocity in feet. 

From the experiments of Ximenes, Du Buat, and others, it is deduced that the 
coefficient of the mean velocity of a flowing stream is from .81 to .83 of the maxi- 
mum velocity or of that of the line of the current, and, contrariwise, the maximum 
velocity is 1.24 to 1.19 times that of the mean. 

The mean velocity at half depth of a stream has been ascertained to be as .915 to 
1, and at the bottom of it as .83 to 1, compared with the velocity at the surface. 

Thus, let v x , v 2 , v 3 , etc., be the surface velocities of a whole transverse profile of 
not a very variable depth: the corresponding velocities at a mean depth are .915 y, 
.915 i^, .915 v 2 , etc. ; hence the mean velocity in the whole profile, 

.915 — = v, n representing' the number of velocities put in the formula. 

Again, the velocity diminishes from the line of current toward the banks, and, to 
obtain the mean superficial velocity, 

2!+^±L»=.»]» v; hence, 
n 

To Compute tlie IVtean Velocity in. tlie -wliole ^Profile of a 
River, etc., 

.915X-915xt> = .8G7 C = .83 to .S4 per cent, of the maximum velocity, or of that 
of the line of current. 

Illustration. — In the line of current of a brook, the velocity of the flow of the 
water is 4 feet per second, and the depth 6 feet; what is the mean velocity of the 
flow, and what is its velocity at the bottom of the line? 

Assume C = .S2. Then 4X-82 = d.28 feet* 4X.83 = 3.32 /eeL 

The upper surface of flowing water is not exactly horizontal, as the water at its 
surface flows with different velocities with respect to each other, and consequently 
exert on each other different pressures. 

If v and v, are the velocities at the line of current and bank of a stream, the dif- 

V 2 Vl 2 

ference of the two levels is — ^ = h. 

IS 2 

52 _ .9x5 4.T5 
Illustration.— If v=z 5 feet, and v x .$v ; then ~— — = — .07S8/eeL 

A velocity of 7 to 8 ins. per second is necessary to prevent the deposit of slime and 
the#growth of grass, and 15 ins. is necessary to prevent the deposit of sand. 

The maximum velocity of water in a canal should depend on the character of the 
bed of the channel. 

Thus, the Mean Velocity should not exceed, 



Second. 

Over a slimy bed ... 8 ins. 
Over common clay. . 6 u 
Over river sand 1 ft. 



Second. 

Over small gravel 1 ft. 

Over large shingle ... 3 u 
Over broken stones . . 4 u 



Second. 

Over stones 6 ft. 

Over rocks 10 u 



To Compute tlie "Velocity of* tlie Flow or Discharge of 
Water in Canals, Streams, 3?ipes, etc. 

1. When the Volume discharged per Minute is given in Cubic Feet, and 

the Area of the Canal, etc., in Square Feet. 
Rulk. — Divide the volume by the area, and the quotient, divided by 60, 
will give the velocity in feet per second. 

2. When the Volume is given in Cubic Feet, and the Area in Square Inches. 
Rule. — Divide the volume by the area ; multiply the quotient by 144, 

and divide the product by 60. 

3. When the Volume is given in Cubic Inches, and the Area in Sq. Inches. 
Rulk. — Divide the volume by the area, and again by 12 and by 60. 



HYDRAULICS. 377 

Examine. — The flow of water through a drain of 20 ins. area is 25 cubic feet per 
minute ; what is the velocity of the flow? 

25 

-Xl44-f-60 = 3/ee*. 

To Compute tlie Flow or Volume of tlie Discharge. 

1. When the Area is given in Square Feet. 

Rule. — Multiply the area of the flow by its velocity in feet per second, 
and the product, multiplied by 60, will give the volume in cubic feet. 

2. When the Area is given in Square Inches. 

Rule. — Multiply the area by its velocity, and again by GO, and divide 
the product by 144. 

To Compute tlie Height of tlie Head of Flowing "W ater. 
1. When the Volume and the Area of tlie Flow are given in Feet. 

Rule. — Divide the volume in feet per second by product of the area, 
and % coefficient for opening and square of the quotient, divided by 64. 3o, 
will give the height in feet. 

Example. — Assume volume 26G.4S cubic feet, area 40 square feet, and C = .623. 

Then ( 266 ' 48 )V64.33 = 2 -gIJg=4/^. 
\40xf.t)23y C4.33 

Note 1. — The velocities and discharges here deduced are theoretical, the actual 
results depending upon the coefficient of efflux used. The mean velocity, however, 
as before given, page 359, may be taken at y/2 g .673 — 5.4 feet instead of 8.02 feet. 

2. — As a rule, with large bodies, as ships, etc., their floating velocity is somewhat 
greater than that of the flow of the water, not only because in floating they descend 
an inclined plane, formed by the surface of the water, but because they are but 
slightly affected by the irregular intimate motion of the water: the variation for 
small bodies is so slight that it may be neglected. 

Illustration, of tlie preceding Rules. 

The breadth of a stream is set off into three divisions— viz., 3.1, 5.4, and 4.3 feet; 
the mean depths of the sections of these divisions are 2.5, 4.5, and 3 ftet, and their 
mean velocities are 2.9, 3.T, and 3.2 feet per second ; what is the volume of the wa- 
ter flowing, and what the mean velocity ? 

Hence 3.1x2.5x2.9 + 5.4x4.5x3.1 + 4.3x3x3.2 =^153.6G5 cubic feet; andS.lX 

153 665 
2.5+5.4x4.5+ 4.3X3 — ^S5sq.feet, area of the sections ; and '' ( — 3.419 ft. 

CANAL LOCKS. 

When a fluid passes from one level or reservoir to another, through an aperture 
covered by the fluid in the latter, tlie effective bead on each point of the aperture, 
and consequently the head due to the velocity of the efflux at each instant, is the dif- 
ference of the levels of the two reservoirs at that instant. 

Hence C aV^gh' = V per second, h' representing the diJJ'crcnce of the levels. 

To Compute tlie Time of Filling and. Discharging a Single 

T^ocli. 

When the Aperture or Sluice in the Upper Gate is. entirely under Water, 
and above the Lounr Level, 

— time of filling up to centre of sluice, h representing the height of the 

CaV'Agh 

centre of the sluice in the upper gate from the surface of the canal or reservoir, and 

h' the height of the centre of the sluice in the upper gate from the lower surface, or 



378 HYDItAULICS. 

the water in the lock or river, all in feet; and = time of filling the re- 

CaViyh 

maining space, where a gradual diminution of the head of water occurs. 

« U' + 2A)A , I'-' . , . . 

Consequently, = time of filling a single lock. 

UaVtgh 

When the Aperture or Sluice in the Lower Gate is entirely under Water, 
and above the Lower Level. 

• == time of emptying or discharging it, a' representing area of sluice, h 

Ca'V '_(/ 
height of upper surface of canal from centre of sluice, and h' height of centre of 
sluice from lower surface. 

Example. — The mean dimensions of a lock are 200 feet in length by 24 iu breadth ; 
the height of the centre of the aperture of the sluice from the upper and lower sur- 
faces is 5 feet; the breadth of both upper and lower sluices is 2.5 feet; the height 
of the upp.-r is 4 feet, and of the lower — entirely under water — 5 feet ; required the 
times of filling and discharging. 

h — ^h' — b, A = 200x24 = 4S00, C = .615. a = 4x2.5= 10, a' = 5x2.5=12.5. 

• • = , = 217. C2 seconds = time of filling the lock up to the 

.6i5xl0xi/2yfc U«-«8G 

, , , • , 2X4S00X5 4S000 

centre of the sluice ; and ■ = = 435.23 seconds = time of 

.615X10X^2(7/* H0.2t>6 
filling the remaining space, or the lock above the centre of sluice, and 217.624-435.23 
= 652. S2 seconds, the whole time. 

_ (54-2x5)x4S00 72000 rKO QK ' ..„. 

Or, - — = = C52.85 seconds = time of filling. 

.615x10X1/2 5- A H0.266 

2x4S00i/5-f 5 3035S.0S ' , J . , 

• ~f= „ = 492.39 seconds = time of discharging. 

.615x12.5X1/2*/ 0L0d4 

When the Aperture or Sluice in the Upper Gate is entirely under Water, 
and below the Lower Level, 



— — = time of filling the lock. 

CaV2g 

When the Aperture or Sluice in the Lower Gate is in part above the Surface 
of the Lower Level and in part below it. 

■ 2 A (h + h') 

C bV27j [dVh + h' — - + d'Vh-j-h'J = time of discharging, d and d' represent- 
ing the distances of the part of the aperture above and below the surface of the lower 
water, b the breadth of the aperture, and h and h' as before. 

Example Assume the sluice in the preceding example to be 1 foot above the 

lower levil of the water, or that of the lower canal; what is the time of the dis- 
charge of the lock ? d = 1, d l = 4. 

2X4S00 (5 + 5) 900,"0 _ 96000 

.615x2.5 X S.02[lXl/5+5-ri-2)+4XV'5^ 
= 494.9 seconds = time of discharging. 

Double Lock. 

A double lock is not a duplication of a single lock in its operation, for in the lower 
chamber the supply of water is from the upper one, having no influx, instead of a 
uniform Mipply flowing directly from the surface level of the canal or feeder. 

The operation, therefore, of a double lock is complex, the addition to the formula 
for a single lock bsing that of the discharging of the water in the upper lock to fill 
the lower, the he-id of water gradually decreasing in the chamber, which is closed 
from the upper reach during the discharge into the lower. 



HYDRAULICS. 379 

To Compute the Time of* Filling the Lower Lock. 

When the Sluice in the Middle Gate is wholly below the Level of the Lower 

Lock. 

2AA\A ., i T , 

= = t, A' representing area of the lower or receiving lock, and h the 

QaV2g (A + AO 

initial height or difference of the levels of the surfaces of the water. 

Example A double lock has the following dimensions — viz., area of upper 

chamber 2300 square feet, and of lower 2200 square feet; height of surface of water 
in upper chamber from surface of water in lower chamber 13 feet, and area of sluice 
13.5 square feet; what is the time of filling the lower chamber, or that in which the 
level of the water is the same in both chambers ? 

2X2300x2200x^/13 _-,^ Q 
U - M .55X13.5X8.02 (2300 + 2200) ~ U °' J S6C ' 
Note. — This is also the time of emptying of the upper lock or chamber. 

When the Sluice in the Middle Gate is wholly above the Level of the Lower 

LjOcIc. 

2 A K'-\/h' 

±= time of filling up to centre of sluice, h' representing the distance 

(JaVtg (A-f-A0 

from the lower level to the centre of the sluice ; and ■ = time of fill- 

Cav / 2 i /(A-|-A / ) 
ing the remaining space. 

OVERFALL WEIRS. 

Weirs are designated as Perfect when their sill is above the surface of 
the natural or down stream, and as Lmperfect or Submerged when their 
sill is below that surface. 

To Compute the Volume of Water discharged over a 

Weir. 
2 r 3 3"| 

- C b V 2 g \(h -\- k)2 — kzj == V, h representing head or depth of water over sill, b 

breadth of the weir, and k height due to the velocity of the water as it flows to the 

weir == — . 
2<gr 

This formula, however, is not directly applicable to the determination of the dis- 
charge, because k, or the height due to the velocity, is dependent upon V, or the 
volume. When, therefore, k can not be determined by observation, it will answer to 

2 , 

put -CbV'lgh — V. 

To Compute the Depth of tlie Flo-w over a Sill or Saddle 
that "will Discharge a given. Volume of Water. 



\2UbV2tr I 



'bV2g 

When the back-water is raised considerably, say 2 feet, the velocity of the water 
approaching the weir (k) may be neglected. 

Then a -j- h — ( nH ffl = x,a representing the original depth of the stream 

v -2C6v / 2r,/ 
or of the back-water below the weir, h the depth of water over the si/','., h' the height 
to which the surface of the natural stream has been raised by the dam, and x the 
height of the dam. 

Hence W -j- # — h -j- x, and a-\- h' — h=x. 

Illustration.— A stream 30 feet wide and 3 feet deep discharges 310 cubic feet 
of water per second. It is required to raise it at this point 4.5 feet by the aid of a 
dam ; what should be the height of it ? 



380 HYDRAULICS. 

Note. — As the height of water to be raised is considerable, the simple formula 
may be used. 

a = 3, h' = 4.5, V = 310, & = 30, C = .75, V2g = SM. 

3 + 4 .5 _ (■■ i X °'V n V? = T.5 - fef = T.5 -VStF = 5.G2/^. 
T \2X.T5X 30xS. 0_7 \b6u.iy v J 

If it were required to raise the water 1.5 feet, the dam would not be re- 
quired to be raised above the level of the natural stream, and hence the 
weir would be submerged. 

Applying then the following formula in this case, which is that of a 
submerged weir, 

Putting *=|5± {^T0 = -° 155 tSu) 2 = • 0155X5 ' 272 = M *«\ 

Assuming C in thii case = .S. 

310 2(1.532)2 — (.0817)2 , %<1QO 2 1 9S<>— .0233 

Then ===- — = r.-RoT — 1 - 2s3 — "> TTwa 

.Sx30xv / (J4.33o(1.5-|-.US17) 3 V 1 -^ » L ?f« 

= 1.2S3 — 1.041 = .24:2 feet. 

Hence, as a 4- A' = A -J- x, h = A' -f- a — a?, and h = 1.5 -f-. 242 = 1.742, ar = 3-4- 
1.5 — 1.742 = 2.753 /ee*. 

SLUICE WEIRS OR SLUICES. 

The discharge of water by Sluices occurs under three forms — viz., 
Unimpeded, Impeded, or Partly Impeded. 

To Compute the Discharge when. "Unimpeded. 

C db^/2 gh = V, d representing the depth of the opening. 

To Compxite the Discharge -when Impeded. 

Cd6V2./A = V, h representing the difference of level between the supply and the 
back-water. 

To Compute the Discharge when it is partly- Impeded. 

C bV2g ldy/h — a -f- d'^/h\ = V, d' representing the depth or height of the back- 
water above the upper edge of the sluice. 

Illustration. — Dimensions of a sluice are 18 feet in breadth by .5 in dtpth; 
height of opening above surface of water is .7 feet, and difference between levels of 
supply and surface of water is 2 feet; what is the discharge per second? 

.6X1SXS 02 fo\/2 — -f .5^2) = 80.12x1 COG = 139.11 cubic feet. 
FLOW OF WATER IN REDS. 

The flow of water in beds is either Uniform or Variable. It is uniform 
when the mean velocity at all transverse sections is the same, and conse- 
quently when the areas of the sections are equal ; it is variable when the 
mean velocities, and therefore the areas of the sections, vary. 

To Compute the I^all of the Flow. 

F — X- = A,F representing the coefficient of friction, I the length of the flow, p 
A 2 (j 
the perimeter of the sides and bottom of the river, and h the fall in feet. 



HYDRAULICS. 



381 



To Compute tlie Velocity of tlie ITlow. 

-. r, A representing area of vertical section. 



Vr 



A 



2ghz 



FX lp 

Tat>le of tlie Coefficients of Friction of tlie Flow of^V"ate^^ 
in Beds, as in Rivers, Canals, Streams, etc. 

IN FEET PER SECOND. 



Velocity 


Coefficient. 


Velocity. 


Coefficient 


Velocity 


Coefficient. 


Velocity 


Coefficient. 


.3 


.DOS 15 


.7 


.00773 


1.5 


.00759 


5 


.00745 


.4 


.0011)7 


.8 


.00769 


2 


.00752 


8 


.00744 


.5 


.00": 85 


.9 


.00766 


2.5 


.00751 


10 


.00743 


.6 


.00778 J 


1. 


.00763 


3 


.00749 


12 


.00742 



By experiments of Du Buat and others, reduced by Eytelwein, F^ .007505 for a 
velocity of 1.5 feet, and v == 92.35 for measures in feet. 

Illustration 1. — A canal 2600 feet in length has breadths of 3 and 7 feet, a 
depth of 3 feet, with a flow of 40 cubic feet per second ; what is its fall ? 

„ = 3 + 2 y (LEljV^lO.*; A = i±I><- 3 = : 1 5; e = « = 2 .«a 



15 



Hence . 007565* X 



260DX10.2 2.062 



■AMI feet 



15 2g 

2. — A canal 5S00 feet in length has breadths of 4 and 12 feet, a depth of 5 feet, 
and a fall of 3 feet ; what is the velocity and volume of the flow ? 



Hence 



. /- 

V .o 



^^4 + 2 v /( 1 -?- ¥ -i)V5 2 = lG.S; A: 



12-f-4x5_ 



:40. 



40 



= 3.23 feet, and 40 x 3.23 = 129.2 cubic feet. 



.007565 X5S00X 16.8 

Forms of Transverse Sections. 

The resistance or friction which the bed of a stream, etc., opposes to 
the flow of water, in consequence of its adhesion or viscosity, increases 
With the surface of contact between the bed and the water, and therefore 
with the perimeter of the water profile, or of that portion of the transverse 
section which comprises the bed. 

The friction of the flow of water in a bed is inversely as the area of it. 

That the friction of a flowing stream may be the least practicable of at- 
tainment, its transverse section, omitting any part of its surface in con- 
tact with the air, must have that form in which the perimeter for a given 
area is a minimum, or the area for a given perimeter a maximum. 

Of all regular figures, that which has the greatest number of sides has 
for the same area the least perimeter; hence, for inclosed conduits, the 
nearer its transverse profile approaches to a regular figure, the less the 
coefficient of its friction ; consequent^, the circle has the profile which 
presents the minimum of friction. 

Th trapezoidal and rectangular sections are those generally given to 
canals, cuts, etc. 

The order of the regular figures applicable to Canals, Cuts, etc., in their 
least resistance to the flow of water, is as follows : 

^ 1. Semicircles. 2. Half a Decagon. 3. Half a Hexagon, or a Trape- 
zium. 4. Half a Square. 

When a canal is cut in earth or sand and not walled up, the slope of its 
sides should not exceed 45°. 



* See Note on Table of Coefficients, above. 



382 



HYDRAULICS. 



To Compute tlie Form or Profile corresponding to a given 
Slope, as a b e. 



/ A sin. L_a A . 

' \/ — — = a, and d, cotang. 

V -I — cos.Z_a d ° 



senting the depth, and b the breadth at the bottom, 
b Example. — What dimensions must be given to the 

transverse section of a canal when its banks are to have 
a slope of 40°, and which is to convey a volume of 75 cubic feet of water with a 
mean velocity of 3 feet ? 

75 
A — — = 25 square feet. 

1 25 sin. 40° /25X. 64279 /16.0698 'J >« 

V 2 - cos. 40° = V 2^70004 = V T^T = 3 - 6 ° 9 ^ d ^ 



25 
3.G0J 



— 3.609X cotang. ^_ 40° =6.927 — 3.009x1.1917 = 2.626 feet breadth at bottom. 

The slope a c, or cut of the bank ±= 3.609 X cotang. 40° = 4.301. Hence 4.301x2 

2d 
-f 2.626 = 11.22S feet, the breadth at the top., and b -j- - == perimeter of the 

water profile ; then 2.626 + 2 ^f.;^ 9 = 13.S55/ee*, and \ = 1M5? = .5542, *Ae ra- 

.042iy , A 25 

too determining the friction. 

ITatole of the Profiles which correspond, to different An- 
gles of Slope. 

Dimensions of Transume Profile 



Angle of 
Slope. 



90° 
60° 
45° 
40° 

36° 52' 
S5° 
30° 

26° 34' 
Semicircle 



Relative 
Slope. 



0. 

.577 
1. 

1.192 
1.333 
1.402 
1.732 
2. 



Depth. 
d 



.707VA 
.76 VA 
.74 VA 
.T22/A 

.707VA 
.697V A 
.GG4v/A 
.63GVA 



Lower Breadth. 
6 



1.414/ A 
.8T7yA 
.613^/A 
.525v/A 
.47VA 
.439^A 
.35GVA 
.3 VA 



Slope. 



0. 
.439VA 
.74 VA 
.86 VA 

.C43VA 

.995v/A 

1.15 VA 

1.272 V A 



Upper Breadth 



1.414 V /A 
1.755VA 
2.092VA 
2.243/A 
2.537VA 
2 43 VA 
2.656 v/A 
2.S44VA 
1.596 V A 



Quotient 
of 

t 
A 



2.8^8 
VA 

2.632 
VA 

2.704 

VA 
2.771 

VA 
2.81S 

VA 
2.87 
VA' 
3.012 

VA 
3.144 

VA 
2 507 

VA 



''"he Angle of Slope = angle a be. Relative Slope = length of a e to be. 
n c= length of slope cut off the banks. 

From this Table it appears that the quotient £ is least for the semicircle, and 

greatest for the trapezium of 26° 34'. 

Illustration.— What dimensions must be given to a profile having an area of 40 
square feet, and a slope of its banks of 35° ? 



HYDRAULICS. 383 

By the preceding Table, .697V40 — 4.408 feet depth; .439V40 = 2.776 feet, lower 
breadth; .995v/40 = 6.292 feet, slope ; 2. 43V40 = 15.367 feet, upper breadth; and 
2 S7 

- ' __ = .4533, the ratio determining the friction. 
"/40 

To Compute tlie Transverse Section, -when tlie Volume 
and. Fall are given. 

/mlV 2 \%- / mlV 2 \% 

.026S ( r = A, and ( F r- ) = A, m representing the unit in the precede 

\ h ) \ 2gh/ 

ing Table. 

Example A trench for a length of 3650 feet, with a fall of 1 foot, is to discharge 

12 cubic feet of water per second ; what dimensions are to be given to the transverse 
profile, it being of a serai-hexagonal figure ? 

.0268 ( 2 - G32X8 f° Xl2 y = 7.665 square feet ; » = ^= 1^/eet. 

(2 6*P V^650vl22\2 
• 00T5S ' »»,,.* < P = ?- 6 ? square feet. 
ZX 32. loo XI / 

Therefore the depth = .76^/ A — 2 104 feet, the lower breadth =.S77-/A = 2. 428 
feet, and the upper breadth — 2x2.42S = 4.S56/eeJ. 

"Variable IVIotion. 

The variable motion of water in beds of rivers or streams may be re- 
duced to the rules of uniform motion when the resistance of friction for 
an observed length of the river can be taken as constant. 

To Compute the Volume of "Water flowing throxigli a 

River. 

V2(/h 

— V, A and A^ representing the areas of the 

, / -1 X \ F lp ( l \ 1 \ 
V A2, A2" 1 " Ai-f-A \A2iAy 

upper and lower transverse sections of the flow. 
Example. — A stream having a mean perimeter of water profile of 40 feet for a 
length of 300 feet has a fall of 9.6 inches ; the area of its upper section is 70 feet, and 
of its lower 60 square feet ; what is the volume of its discharge ? 



/T0 + 60X 9 - 6 
^ 92.35V 4llYm , 



To obtain F for the velocity due to this ca*e, 92.35 V — — — — — = 8.59 feet, 

4UX«>0J 
the coefficient for which, see preceding Table =..00744. 



7' 



M * 8X ^ 7.174 



i , ^ 300x40/1 , iv = V^m^i = 85s ' 7 cublc feet ; the 



7jL_i_ + .00744 *?^(±+±^ 
V 702 602 ^ 70 + 60 \70 2 T 602/ 



35S.7 
mean velocity of which = — 5.52 feet, the exact coefficient for which is .00745. 

2 

Friction in 3?ipes and Sewers. 

The Resistance of Friction in the flow of water through pipes, etc., of a 
uniform diameter is independent of the pressure, and increases directly as 
the length, very nearly as the square of the velocity of the flow, and in- 
versely as the diameter of the pipe. 

With ^wooden pipes the friction is 1.75 times greater than in metallic. 

The time occupied in the flowing of an equal quantity of water through 
Pipes or Sewers of equal lengths, and with equal heads, is proportionally 
as follows : In a Right Line as 90, in a True Curve as 100, and in a Eight 
Angle as 140. 



384 



HYDRAULICS. 



When Pipes branch off from Mains, or when they are deflected at right 
angles, the radius of the curvature should be proportionate to their diam- 
eter. Thus, 





Ins. 


Ins. 


Diameter 

Radius 


2 to 3 
18 


3 to4 
20 



Ins. 


Ins. 


Ins. 


6 
30 


8 
42 


10 
60 



Discharge of Water in. Pipes or Sewers for any- Lengtli 
and Head, and. for Diameters from 1 Incli to lO Feet. 

[Bhardmobk.] 
IN CUBIC FEET PEE MINUTE. 



Diameter. 
Ft. Ins. 


Tabular No. 1 


Diameter. 
Ft. Ins. 


Tabular No. 


1 Diameter. 
| Ft. Ins. 


Tabular No. 


1 


4.71 


1.7 


7433. 


3.7 


57265. 


IX 


8.48 


1.8 


8449. 


3.8 


$643. 


1% 


13.02 


1.9 


9544. 


3.9 


64156. 


1% 


19.15 


1.1 


10722. 


3.1 


677S2. 


2 


26. G9 


1.11 


119S3. 


3.11 


71526. 


2X 


46. 6T 


2. 


13328. 


4. 


75392. 


3 


73.5 


2.1 


14758. 


4.1 


793S0. 


3X 


108.14 


2.2 


16278. 


4.2 


83492. 


4 


151.02 


2.3 


178S9. 


4.3 


87730. 


4^ 


,194.84 


2.4 


19592. 


4.6 


101207. 


5 


203.87 


2.5 


21390. 


4.9 


115354. 


6 


416.54 


2.6 


23232. 


5. 


131703. 


7 


612.32 


2.7 


25270. 


5.3 


148791. 


8 


854.99 


2.8 


27358. 


5.6 


167139. 


9 


1147.6 


29 


29547. 


5.9 


1867S6. 


10 


1493.5 


2.10 


31S34. 


6. 


207754. 


11 


1S94.9 


2.11 


34228. 


6.6 


2537S1. 


1. 


2356. 


3. 


36725. 


7. 


305437. 


1.1 


2S76.7 


3.1 


39329. 


7.6 


362935. 


1.2 


3463.3 


3.2 


4204^. 


8. 


42(3481. 


13 


4115.9 


3.3 


44S63. 


8.6 


496275. 


1.4 


4836.9 


3.4 


47794. 


9. 


57250S. 


1.5 


5G2S.5 


35 


50835. 


9.6 


655369. 


1.6 


6493.1 


36 


539C5. 


10. 


745038. 



Note. — This Table is applicable to Sewers and Drains by taking the 
same proportion of the tabular numbers that the cross-section of the wa- 
ter in the sewer or drain bears to the area of the whole area of the sewer 
or drain. 

The formula upon which this Table is constructed is, 
2356 y/d> 

= V,d representing diameter, and h height of fall of the water in feet. 



■J: 



APPLICATION OF THE TABLE. 



To Compute the Volume of Fluid discharged, the Length 
of tlie 3?ipe or Sewer, tlie Height or Fall, and tlie Diam- 
eter being given. 

Rule. — Divide the tabular number, opposite to the diameter of the tube, 
by the square root of the rate of inclination, and the quotient will give the 
volume required. 

Example.— A pipe has a diameter of 9 inches, and a length of 4750 feet ; what i 
its discharge per minute under a head of 17.5 feet? 



/4750 _ 

VTT5- 



V211A — 16.47, and tabular number for 9 ins. = 1147.61. 



Then l-i-^-69.67 cubic feet per minute. 
16.m7 



HYDRAULICS. 385 

To Compute the Diameter, the Length, Fall, and. Dis- 
charge "being given. 

Rule. — Multiply the discharge by the square root of the ratio of incli- 
nation ; take the nearest corresponding number in the Table, and oppo- 
site to it is the diameter required. 

Example. — Take the demerits of the preceding case. 



/'475U _ 



69.67X W -ry-r = 1147. Gl, and opposite to this is 9 inches. 

To Compute the Head, the Length, the Discharge, and 
the Diameter "being given. 

Rulk. — Divide the tabular number for the diameter by the discharge, 
square the quotient, and divide the length of the pipe by it ; the quotient 
will give the head necessary to force the given volume of water through 
the pipe in one minute. 

Example Take the elements of the preceding cases. 

1 4t^ — 16.47 ; 16.472 = 271 .4 ; 4750 -4- 271.4 = 17.5 feet. 

OJ.Ol 

To Compute the whole Head necessary to furnish the 
requisite Discharge. 
See Formula and Illustration, page 3S6. 

To Compute the Velocity, the Volume and the Diameter 
alone Deing given. 

Rule. — Divide the volume when in feet by the area in feet, and the 
quotient, divided by 60, will give the velocitj^ in feet per second. 
Example. — Take the elements of the preceding case. 

J? 9 ' 6 !., -*- 60 =2.65 feet. 
.752 x. 7854 J 

When the Volume is not given.* 

Rulk. — Multiply the square root of the product of the height of the 
pipe by the diameter in feet, divided b}^ the length in feet, by 50, and the 
product will give the velocity in feet per second. 

Examine Take the elements of the preceding case. 



■/■■ 



"w 5 *" ="•*•<■ 



To Compute the Volume of Water discharged from a 

, IPipe.f 

h lV> 
39.27 \/ —j- = V in cubic feet per second. 

Illustration.— The diameter of a pipe is 1 foot, the head of the flow 9, and the 
length of the pipe 9000 feet ; what is the volume of the discharge? 



39.27 X V IbTu = 39 * 2Tx V-001 = 1.242 cubic feet. 

•iamet 
tnd th< 

y {^if x {= dinfeet - 



To Compute the Diameter of the !Pipe, the Volume of the 
Flow, the Head, and the Length of the Pipe being given. 



Illustration.— Take the elements of the preceding case. 



^ 



1.242\2 D000 5 7 

^||) x±f=V .oo\xum = V i = ifoot. 



* Beardmore. t Ibid. 

Kk 



1386 



HYDRAULICS. 



To Compute the Inclination, of a IPipe, the Volume of the 
Flow, the Diameter and. Length of the IPipe "being given. 



\39.27/ d*~~ V 



Illustration. — Take the elements of the preceding case. 

7^242X2^1 _ 001xl __ >001 _ ratio j height t0 i en gth. 



FRICTION OF WATER IN PIPES.— [Wkisbach] 
To Compute the Head necessary to overcome the Fric- 
tion of the IPipe. 

(. 01444- : I X— X — — h', h' representing the head to overcome the friction of 

\ l y/v J d 5A 

the flow in the pipe in feet, I the length of pipe in feet, d internal diameter of 
pipe in inches, and v velocity of the water in feet per second. 

Illustration.— The length of a conduit-pipe is 1000 feet, its diameter 3 inches, 
and the required velocity of its discharge 4 feet per second ; what is the required 
head of water to overcome the friction of the flow in the pipe ? 

/ 0144 + ^^) X 1 -^X^ = .02313X333.333X2.C63=: 22 S&feet. 

The head here deduced is the height necessary to overcome the friction 
of the water in the pipe alone. 

The whole or entire head or fall includes, in addition to the above, the 
height between the surface of the suppty and the centre of the opening of 
the pipe at its upper end. Consequently, it is the whole height or vertical 
distance between the supply and the centre of the outlet. 

To Compute the whole Head, or the Height from the Sur- 
face of the Supply to the Centre of the Discharge. 

I 7)2 

1.5 is taken as a mean, and is the coefficient of friction for the interior orifice, or 
that of the upper portion of the pipe. 

Illustration. — Take the elements of the preceding case. 

Note. — In the preceding formula I was taken in feet, as the multiplier of 12 for 
inches was canceled by taking 5.4 for 2#, hut in the above formula it is necessary 
to restore this multiplier. 

For facilitating the calculation, the following Table of the coefficient of 
resistance is introduced : 

Coefficients of Friction of Water in Pipes at different 
"Velocities. 

(A Reduction of the following Formula.) 



Ft. Ins. 


c. 


Ft. Ins. 


c. 


4 


.0443 


3.4 


.0239 


8 


.0356 


3.S 


.0234 


1. 


.0317 


4. 


.0231 


1.4 


.0294 


4.4 


.0227 


1.8 


.0278 


4.8 


.0224 


2. 


.0266 


5. 


.0221 


2.4 


.0257 


5.4 


.0219 


2.8 


.025 


5.8 


.0217 


3. 


.0244 


6. 


.0215 



Ft. Ins. 


c. 


Ft. Ins.' 


c. 


6.4 


.0213 


11. 


.0196 


6.8 


.0211 


11.6 


.0195 


7. 


.0209 


12. 


.0194 


7.4 


.0208 


12.6 


.0193 


7.8 


.0206 


13. 


.0191 


8. 


.0205 


13.6 


.019 


8.6 


.0204 


14. 


.0189 


9. 


.0202 


15. 


.0188 


10. 


.0199 


16. 


.0187 



Illustration. — The coefficient due to a velocity of 4 feet per second is .0231. 



HYDRAULICS. 



387 



Thus, by the Formula (.0144+ '— — J == .0231 ; and by the preceding Table a ve- 
locity of 4 feet per second — .0231 for its coefficient. 
Hence for (.0144-f- * — - — ) read, when practicable to do so, C, a coefficient. 

Table shovtdng the Velocity ofWater flowing from IPipes 
and Sewers, as computed "by the Formulae of Beard- 
more and 'Weishach. 











(Beakdmork ) 


(Wkisbach.) 




Diam- 






Discharge 




V2jh 


Actual 


eter of 


Head. 


Length. 


per 
Minute. 


MJ\xd = v. 


— = v. 


Pipe. 


V cx ^+ 15 


Velocity. 


Ins. 


Feet. 


Feet. 


Cubic Feet. 


Feet per Second 


Feet per Second. 


Feet. 


1 


1 


100 


.471 


1.44 


C= .0294 1.32 


1.43 


1 


9 


100 


1.41 


4.33 


C = .0227 4.49 


4.31 


1 


9 


18t)0 


.333 


1.02 


C fca .0317 .92 


1.02 


1 


25 


225 


1.57 


4.81 


C = .0224 5.09 


4.79 


2 


25 


225 


8.89 


6. Si 


C= .0211 7.32 


6.79 


2 


25 


5250 


1.84 


1.41 


C=.0294 1.32 


1.4 


3 


25.6 


1000 


11.78 


4. 


C = .0231 4.19 


4. 


4 


4 


144 


25.17 


4.8 


C=.0224 4.8 


4.8 


4 


36 


300 


52.25 


10. 


C— .0199 10.92 


9.99 


6 


4 


144 


69.42 


5.9 


C=.0217 5.76 


5.89 


6 


8 


144 


9S.24 


8.33 


C=.0204 8.35 


8.33 


6 


50 


1000 


93.2 


7.91 


C= .02(5 8.65 


7.91 


12 


1 


1000 


74.5 


1.58 


C= .0294 1.44 


1.59 


12 


9 


441 


336.5 


7.15 


C--.0209 7.05 


7.14 


12 


9 


9000 


74.5 


1.5S 


C=.0209 1.75 


1.59 


12 


64 


1600 


471.2 


10. 


C= .0199 11.11 


9.99 


24 


3 


300 


1332.8 


7.07 


C = .0209 6.44 


7.07 


24 


9 


900 


1332.8 


7.07 


C — .0209 7.2S 


7.97 


24 


9 


9000 


421.5 


2.24 


C=.026 2.21 


2.24 


24 


70 


13720 


95.2 


5.05 


C=.0221 5.43 


5.05 


36 


6 


600 


3672.5 


8.66 


C= .0204 8.32 


8.66 


S6 


6 


15000 


734.5 


1.73 


C=.0356 1.47 


1.73 


36 


9 


2025 


2448.3 


5.76 


C=.0217 5.99 


5.77 


36 


TO 


43750 


1469. 


3.46 


C= .0239 3.57 


3.46 



(Beardmore), I — representing length of pipe, and d its diameter in feet. 
( Weisbach), / = representing length of pipe, and d its diameter in inches. 
Note. — For values of C and the constant of 1.5, see page 386. 
The preceding cases are selected with a view to give the range of or- 
dinary operations, and to show the application of the Formulas to high 
and low heads, large and small diameters, and extremes of length. 

To Compute tlie Distance a Jet of "Water -will "be project- 
ed from a "Vessel throngh an Opening in its Side. 
B C is equal to twice the square root of A oXo B. 
If s is 4 times as deep helovv A as a is, s will discharge 
twice the quantity of water that will flow from a in the 
same time, as 2 is the -\/ of As and 1 is the y/ of A a. 

Note. — The water will spout the farthest when o is 
equidistant from A and B ; and if the vessel is raised 
above a plane, B must be taken upon the plane. 
^ " The quantities of water passing through equal aper- 

tures in the same time are as the square roots of their depths from the surface. 

Rule. — Multiply the square root of the product of the distance of the 
opening from the surface of the water, and its height from the plane upon 
which the water flows, in feet by 2, and the product will give the dis- 
tance in feet. 




388 HYDRAULICS. 

Example. — A vessel 20 feet deep is raised 5 feet above a plane ; how far will a jet 
reach that is 5 feet from the bottom of the vessel ? 

20 - 5x5 + 5 = 150, and • v / 1 50x2 = 24.495/eef. 
The velocity of a jet of water flowing from a cylindrical tube is determ- 
ined to be .82 of that due to the height of the reservoir. Hence the vol- 
ume of the discharge through a cylindrical opening = .82 ay/"lgh. 

Jets d'Eau. 

That a jet may ascend to the greatest practicable height, the communi- 
cation with the supply should be perfectly free. 

Short tubes shaped alike to the contracted fluid vein, and conically con- 
vergent pipes, are those which give the greatest velocities of efflux. Hence, 
to attain the greatest effect, as in fire-engines, long and slightly conically 
convergent tubes or pipes should be applied. 

In order to diminish the resistance of the descending water, a jet must 
be directed with a slight inclination from the vertical. 

The effect of the combined causes which diminish the height of a jet 
from that due to the elevation of its supply can only be determined by 
experiments. Great jets rise higher than small ones." 

With cylindrical tubes, the velocity being reduced in the ratio of 1 to 
.82, and as the heights of the jets are as the squares of these coefficients 
or ratios, or as 1 to .67, the height of a jet through a cylindrical tube is % 
that of the head of the water from which it flows. 

With conical tubes, the velocity being from .55 to .95, the heights of the 
jets are as the squares of the coefficients 1 and .9 (a mean), or as 1 to .81, 
which is equal to j% that of the head of the water from which it flows. 
Hence the relative values of Cylindrical and Conical tubes are as .67to .81. 

To Compute tlie "Vertical Keiglxt of a Stream projected. 
from tlie IPipe of a Fire-engine or Pump. 

Note. — In Five-engines, the difference between the actual discharge and that as 
computed by the capacity and stroke of the cylinder, as ascertained by Mr. Lamed, 
1S53, is IS per cent. 

Rulic. — Ascertain the velocity of the stream by computing the quanti- 
ty of water running or forced through the opening in a second; then, by 
fjule in Gravitation, page 397, ascertain the height to which the stream 
would be elevated if wholly unobstructed, which multiply by a coefficient 
for the particular case. 

Example. — If a fire-engine actually discharges 14 cubic feet of water through a 
pipe % inch in diameter in one minute, how high will the water be projected, the 
pipe being directed vertically? 

14x1728 -J- .4417 area of pipe, -f- 12 inches in a foot, -j- 60 seconds = 76.07 feet ve- 
locity ; and as the velocity of a stream of water from a vessel is but y B that due to 

3 
its head, then 7G.07Xr — 114.1 feet. 

Then, by Rule, page 397, 114.1 -r- S.02 = 14. 22, and 14.222 — 202.21/ee*. 

According to the elements furnished by observation, the mean coeffi- 
cient in this case would be .57 ; hence 114.1 x .57 = 65.04 feet. With 40 
feet of hose, coefficient was .5. 

In great heights and with small apertures, the coefficient should be re- 
duced. In consequence of the varying elements and conditions of opera- 
tion of fire-engines, it is difficult to assign a coefficient for them. 

A steam fire-engine of the Portland Company, discharging a stream 1% 
ins. in diameter, through 100 feet 2% in. hose, gave a theoretical head, 
computed from the actual discharge, of 225 feet, and the stream vertically 
projected was 200 feet ; hence the coefficient in this case was .88. 



HYDRAULICS. 



389 



Table of the !P 


roportional Rise of Water in Rivers 


, occa- 




sioned t>y the Erection, of Piers, etc. 




Velocity 








of Cur- 
rent in 


Amount of Obstruction compared with Area of Section of the R 


iver. 


Second. 


.1 


.2 


.3 


•4 


.5 


.6 


.7 


.8 


.9 




Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


1 


.0157 


.0377 


.06U3 


.1192 


.2012 


.3521 


.6:8 


1.609 


6.639 


2 


.0277 


.0665 


.1231 


.2102 


.3548 


.6208 


1.196 


2.83S 


11.71 


3 


.0477 


.1144 


.2118 


.3618 


.6107 


1.069 


2.05S 


4.885 


20.15 


4 


.076 


.1822 


.3372 


.5759 


.9719 


1.701 


3.276 


7.775 


32.07 


5 


.1165 


.2793 


.5168 


.S782 


1.49 


2.607 


5.02 


11.92 


49 15 


6 


.1558 


.3736 


.6912 


1.181 


1.993 


3.487 


6.715 


15.94 


C5.75 


7 


.2078 


.4983 


.9221 


1.575 


2.658 


4.651 


8.958 


2L.26 


87.71 


8 


.2678 


.6423 


1.188 


2.03 


3.426 


5.995 


11.54 


27.4 


113. 


9 


.3359 


.8054 


1.49 


2.557 


4.296 


7.517 


14.48 


34.36 


141.7 


10 


.4119 


.9817 


1.827 


3.122 


5.268 


9.219 


1 17.75 


42.14 


173.8 



Table of tlie Velocities ofFlow necessary to clear Circular 
Drains or Sewers. 



Diam. 


Velocity 


Ins. 


per Min. 


4 


240 


6 


220 


7 


220 


8 


220 


10 


210 



lin 
lin 
lin 
lin 



1 in 119 



Diam. 


Velocity 


Ins 


per Min. 


12 


190 


15 


ISO 


18 


180 


21 


180 


24 


180 



Gradial. 



1 in 175 
1 in 244 
1 in 294 
1 in 343 
1 in 392 



| Diam. 


Velocity 


Ins. 


per Min. 
ISO 


30 


36 


ISO 


42 


180 


48 


180 


54 


180 



Gradial. 

1 in 490 
1 in 5S8 
1 in 6S6 
1 in 784 
1 in 882 



Friction of Water upon a Plane Surface. 

By the experiments of Beaufoy, it was ascertained that the friction in- 
creased very nearly as the square of the velocity, and that a surface of 50 
square feet, at a velocity of 6 feet per second, presented a resistance of 6 lbs. 

50 
Hence — = 8.33 square feet = 1 Jb. resistance at a velocity of 6 feet ; and, 
6 

consequently, — — = .12 lbs. resistance per sq. foot at the same velocity. 
o.oo 

For a velocity of 10 feet, 6 2 : 10 2 : : .12 : .333 lbs. 

To Compute tlie Horses' Power necessary to Raise Wa- 
ter to any given Elevation. 

Rule. — Multiply the weight of the column of the water by its velocity 
in feet per minute, and divide the product by 33000. 

Example. — It is required to raise 1000 gallons of fresh water per minute to an 
elevation of 140 feet, through a cast-iron pipe 560 feet in length ; what is the re- 
quired diameter of the pipe, and what the power ? 

1000 galls, fresh water = 1000x231 (page 23) = 231000 cubic inches, and - OQ = 

li ^8 
133.68 cubic feet per minute. 

Then, by Formula, page 385, 133. 68 X /— = 267.36 ; and opposite to the near- 

V 140 
est tab. number, page 384, is 5-f- inch = diam. of pipe. 
Hence 133. 6SX 62. 5x140-^-33. 000 = 35. 44 horses 9 power. 

Volume of Water per Acre of Area. 1 

A depth of 1 inch = 3630 cubic feet. 
Kk* 



390 AEROSTATICS. 



MISCELLANEOUS ILLUSTRATIONS. 

1. What is the head required to discharge 66.63 cubic feet of water per second 
over a weir 10 feet in breadth by 4 in depth ? 

C = .623 
/ 66.43 \2 

(5 ■) ^-2 5 r = 16.0PH-64 33 = 4/^. 

Vf.629 X 10 X 4/ 

2. If the external height of fresh water, at 60° above the injection opening in the 
condenser of a steam-engine, is 3 feet, and the indicated vacuum at 23 inches, the 
velocity of the water flowing into the condenser is determined thus : 

v = V% g (h-{-h'), h' representing the height of a column of water equivalent to the 
pressure of the atmosphere within the condenser 

A column of 2.03593 inches of mercury == 1 lb. pressure per square inch; 1 inch of 
mercury = .490774 lbs 

Assuming the mean pressure of the atmosphere =14.723 lbs. per square inch, the 
height of a column of fresh water equivalent thereto = 33.95/<?e£. 

Then, if 1 inch — .490774 lbs., 23 inches = 11.2S7S lbs. ; and if 14.723 lbs. = 03.95 
feet; 11. 2S73 Z£s. = 26.013 feet. 

Hence v = a/2 g (3 + 26.013) — 43.203 feet, less the retardation due to the coeffi- 
cient of efflux. 

3. If a stream of water has a mean velocity of 2.25 feet per ?econd at a breadth 
of 560 feet, and a mean depth of 9 feet, Avhat will be its mean velocity when it has a 
breadth of 320 feet, and a mean depth of 7.5 feet? 

560X9X2.25 11340 . 'U. . : 

= = 4. i 25 feet. 

320x7.5 2400 J 



AEROSTATICS. 

One cubic foot of Atmospheric Air at the surface of the earth, when 
the barometer is at 30 inches, and at a temperature of 34°, weighs 
527.04 grains = .0752914 lbs. avoirdupois, being 829.43 times lighter 
than water. 

Specific gravity compared with water, at G2.4491 == .0012056. 

The mean weight of a column of air a foot square, and of an alti- 
tude equal to the height of the atmosphere, is equal to 2120.14 lbs- 
avoirdupois, being equal to the support of 33.95 feet of water. 

13.282 cubic feet of air weigh a pound avoirdupois. 

It consists, by volume, of oxygen 21, and nitrogen 79 parts; and in 
10000 parts there are 4.9 parts of carbonic acid gas. By weight, it 
consists of 77 parts of oxygen, and 23 of nitrogen. 

The rate of expansion of Air, and all other Elastic Fluids for all 
temperatures, is uniform. From 32° to 212° they expand from 1000 
to 1376, equal to ? ^-g = .002088 for each degree of their bulk for every 
degree of heat. From 212° to 680° they expand from 1376 to 2322 
= .002 for each degree of heat. 



AEROSTATICS. 



391 



The Standard pound is computed with a mercurial barometer at 30 
inches; hence, as a cubic inch of mercury at 60° weighs .490774 lbs., 
the pressure of the atmosphere at a temperature of 60° = 14.72322 
lbs. per square inch. 

The Elasticity of air is inversely as the space it occupies, and di- 
rectly as its density. 

When the altitude of the air is taken in arithmetical proportion, its 
Rarity will be in geometric proportion. Thus, at 7 miles above the 
surface of the earth, the air is 4 times rarer or lighter than at the 
earth's surface ; at 14 miles, 16 times; at 21 miles, 64 times, and so on. 

At the temperature of 32°, the mean velocity of sound is 1092.5 feet 
per second. It is increased or diminished half a foot for each degree 
of temperature above or below 32°. 

The velocity of sound in water is estimated at 4900 feet per second. 

The motions of air and all gases, by the force of gravity, are precise- 
ly alike to those of fluids. 

The head or altitude of the atmosphere at the ordinary density is 
equal to a column of mercury 30 inches in height, divided by the spe- 
cific gravity of air compared with mercury. 

Hence 30 ins. =2.5 feet, which, divided by .00008878, the specific 
gravity of air compared with mercury = 28160 feet — 53.3 miles. 

The theoretical velocity, therefore, with which air will flow into a 
vacuum, if wholly unobstructed, is \/2g h — \ 347. 4: feet per second. In 
operation, however, it is 1347.4 X .707 = 952.61 feet. 

The coefficients for the efflux of air through openings are as follows •• 

Circular aperture in a thin plate 65 to .7 

Cylindrical ajutage . . 92 

Conical ajutage 93 

To Compute the "Velocity of Sound through .A.ir. 

1092.5s yi-|-.uu2>'b;8 (6 — 61) — v in feet per second, t representing temperature 
of air in degrees. 
Example. — The flash of a cannon from a vessel was observed 13 seconds before the 
report was heard ; the temperature of the air was 60° ; what was the distance to the 

vessel ? 

1032.5 X 13 a/ 1 + .002088 (60° — 32) z=z 14C23 feet, or 2.77 miles. 

Height of the Barometer at different Levels above tlie 
Surface of the Earth. 



Feet. 

"ioocT 

2000 
3000 



Inches 



28.91 
27.86 
26.85 



Feet. 



4000 
5000 
lmile 



Inches. 

~25.87 
24.93 
24.67 



Miles. 


Inches. 


Miles. 


Inches 


2 
3 
4 


20.29 
16.68 
13.72 


5 
10 
15 


11.28 
4.24 
1.6 



Measurement of Heights "by a Barometer. 

Approximate Rule. — For a mean temperature of 55°, x required 
difference in height in feet, h the height of the mercury at the lower station, 
and h! the height of the mercury at the upper station in feet. 

55000X == x. Add ^J^ of this result for each degree which the mean temper- 

ature of the air at the two stations exceeds 55°, and deduct as much for each degree 
below 55°. 



392 



AEROSTATICS. 



To Compute >*^hat Degree of Rarefaction may "be effected 
ixx a Vessel. 

Let the quantity of air in the vessel, tube, and pump be represented by 
1, and the proportion of the capacity of the pump to the vessel and tube 
by .83 ; consequently, it contains }£ of the air in the united apparatus. 

Upon the first stroke of the piston this fourth will be expelled, and % 
of the original quantity will remain ; % of this will be expelled upon the 
second stroke, which is equal to X& °f tne original quantity ; and, conse- 
quent!}', there remains in the apparatus % of the original quantity. Pro- 
ceeding in this manner, the following Table is deduced : 



No. of Strokes. 


Air expelled at each Stroke. 


Air remaining in the Vessel. 


1 


w**u 


% = % 


2 


3 3 

16 4x4 


9 _ 3X 3 
16 ~ 4 x 4 


3 


9 3x3 


27 3x3x3 




64 ~" 4 x 4 x 4 


64 ~~ 4x4x4 



And so on, continually multiplying the air expelled at the preceding 
stroke by 3, and dividing it by 4 ; and the air remaining after each stroke 
is ascertained by multiplying the air remaining after the preceding stroke 
by 3, and dividing it by 4. 



Distances at -which different Sounds are -Aju.di"ble. 



Feet. 



Miles. 

.087 



A full human voice speaking in the open air, calm .... 460 
In an observable breeze, a powerful human voice with 

the wind can be heard 15840 

Report of a musket 16000 

Drum 10560 

Music, strong brass band 15840 

Cannonading, very heavy 575000 

In the Arctic, conversation has Vjeen maintained over water a distance 
of 6696 feet. 



3 

3.02 
2 
3 
90 



Miles 


Feet 


Pressure 


per 


per 


on a Sq. 


Hour. 


Minute. 


Ft in Lbs. 


1 


88 


.005 


2 


170 


.02 \ 

.045/ 


3 


264 


4 


352 


.08 


5 


440 


:i:5) 


6 


528 


.18 V 


8 


T114 


.32 ) 


10 


8S0 


.5 


15 


1320 


i:iZ5 


20 


1760 


2. 



"Velocity and Force of "Wind. 



Description of the 
Wind. 



Barely observable. 

Just perceptible. 

Light breeze. 

Gentle, pleasant 
wind. 

Fresh hreeze. 
Bri~k blow. 
Stiff breeze. 



Miles 


Feet 


Pressure 1 


per 


per 


on a Square 


Hour. 


Minute 


Foot in Lbs. 


25 


2200 


3.125 


30 


2640 


4.5 \ 

6.125/ 


35 


30S0 


40 


35?0 


8. 


45 


of 60 


10.125 


50 


4400 


12.5 


60 


5280 


18. 


80 


7040 


32. 


100 


88)0 


50. 



Description of the 
Wind. 



Very brisk. . 

High wind. 

Very high wind. 

Gale. 

Storm. 

Great storm. 

Hurricane. 

Tornado. 



To Compute the Capacity and Diameter of a Balloon, see 
Page 162. 

For Expansion of Air, see Practical Mechanics' 1 Journal, vol. viii., 1st series, 
pages 231-252. 

For Treatise on Aerometry, of D'Aubuisson de Voissons, see Journal of Franklin 
Institute, pages 124, 1S6, 234, 313, vol. 39. 



AEROSTATICS. 



393 



Table for foretelling tlie Weather throngh tlie Lunations 
of tile ]VIoon. — [Dr. Hkkschell and Adam Clakke ] 

This Table and the accompanying remarks are the result of manj^ears' 
actual observation, and will, by inspection, show the observer what kind 
of weather will most probably follow the entrance of the moon into any 
of its Quarters. 

If the New Moon, the First Quar- 
ter, the Full Moon, or the Last 
Quarter enters 



Between midnight and 2) 
A.M. j 

Between 2 and 4 A.M. 
Between 4 and 6 A.M. 
Between 6 and 8 A.M. 

Between 8 and 10 A. M. 

Between 10 and 12 A.M. 
At 12 o'clock M. and 2) 
P.M. j 

Between 2 and 4 P.M. 
Between 4 and 6 P.M. 

Between 6 and 8 P.M. 



Fair. 

Cold, with frequ't showers 

Rain. 

Wind and rain. 

Changeable. 

Frequent showers. 

Very rainy. 

Changeable. 
Fair. 

Fair, if wind N.W. ; rainy, 
if S. orS.E. 



(Hard frost, unless the wind 
( is S. or E. 

Snowy and stormy. 

Rain. 

Stormy. 
(Cold rain, if the wind be 
\ "W> ; snow if E. 

Cold, and high wind. 

Snow or rain. 

Fair and mild. 

Fair. 
( Fair and frosty, if the wind 
1 is N. or W. 
( Rain or snow, if S. or S. E. 

Ditto. 

Fair and frosty. 



Between 8 and 10 P.M. Ditto. 
Between 10 and midnight. Fair. 

Obskrvations. — 1. The nearer the time of the moon's change, first quar- 
ter, full, and last quarter are to midniyht, the fairer will be the weather 
during the seven following days. The range for this is from 10 at night 
till 2 next morning. 

2. The nearer to mid-day, or noon, the phases of the moon happen, the 
more foul or wet weather may be expected during the following seven 
days. The range for this calculation is from 10 in the forenoon till 2 in 
the afternoon. 

These observations refer principally to the summer, though they affect 
spring and autumn nearly in the same ratio. 

3. The moon's change, first quarter, full, and last quarter, entering dur- 
ing six of the afternoon hours, i. e., from 4 to 10, may be followed by fair 
weather; but this is mostly dependent on the wind, as noted in the Table. 

4. Though the weather, from a variety of irregular causes, is more un- 
certain in the latter part of autumn, the whole of winter, and the begin- 
ning of spring, yet, as a general rule, the above observations will appl} T to 
those periods also. 

Classification, of Clonds. 

1. Cirrus — Like to a feather. 2. Cirro-cumulus — Small round 
clouds. 3. Cirro-stratus — The concave or undulated stratus. 4. Cu- 
mulus — When in conical, round clusters. 5. Cumulo-stratus — The 
two latter mixed. 6. Nimbus — A cumulus spreading out in arms, and 
precipitating rain beneath k. 7. Stratus — A level sheet. 
Note. — The Cirrus is the most elevated. 

Classification, of Lightning. 
1. Striped or Zigzag — Developed with great rapidity. 2. Sheet — 
Covering a large surface. 3. Globular — When the electric fluid ap- 
pears condensed, and it is developed at a comparatively lower veloci- 
ty. 4. Phosphoric — When the flash appears to rest upon the edges 
of the clouds. 



^94 DYNAMICS. 



DYNAMICS. 

Dynamics is the investigation of the laws of Motion of Solid Bod- 
ies, or of Matter, Force, Velocity, Space, and Time. 

The Mass of a body is the quantity of matter of which it is com- 
posed. 

Force is divided into Motive, Accelerative, or Retardative. 

Motive Force, or the Momentum of a body, is the product of its mass 
and its velocity, and is its quantity of motion. This force can, there- 
fore, be ascertained and compared in any number of bodies when these 
two quantities are known.* 

Accelerative or Retardative Force is that which respects the velocity 
of the motion only, accelerating or retarding it ; and it is denoted by 
the quotient of the motive force, divided by the mass or weight of the 
body. Thus, if a body of 5 lbs. is impelled by a force of 40 lbs., the 
accelerating force is 8 lbs. ; but if a force of 40 lbs. act upon a body 
of 10 lbs., the accelerating force is then only 4 lbs., or half the former, 
anl will produce only half the velocity. 

With equal masses, the velocities are proportional to their forces. 

With equal forces, the velocities are inversely as the masses. 

With equal velocities, the forces are proportional to the masses. 

Motion. — The succession of positions which a body in its motion 
progressively occupies forms a line which is termed the trajectory, or 
path of the moving body. 

A motion is Uniform when equal spaces are described by it in equal 
times, and Variable when this equality does not occur. When the 
spaces described in equal times increase continuously with the time, a 
variable motion is termed accelerated, and when the spaces decrease, 
retarded ; and when equal spaces are described within certain inter- 
vals only, the motion is termed Periodic, and the intervals periods. 
Uniform motion is illustrated in the progressive motion of the hands 
of a watch ; Variable motion in the progressive velocity of falling and 
upwardly projected bodies ; and Periodic motion by the oscillation of 
a pendulum or the strokes of a piston of a steam-engine. 

Let body, force, velocity, space, and time be represented by bfv s t, 
gravity by g, and momentum or quantity of motion by m ; this being the 
effect produced by a body in motion. 

If two or more bodies, etc., are compared, two or more corresponding 
letters, as B, /;, b', V, v, v ', etc., are employed. 

Uniform Motion. — The space described by a body moving uniform- 
ly is represented by the product of the velocity into the time. 

W T ith momenta, m varies as b v. 

Illustration.— TVo bodie?, one of 20, the other of 10 lbs., are impelled by the 
same momentum, say 60. They move uniformly, the first for 8 seconds, the second 
for 6 ; what are the spaces described by both ? 

ThenTV = 3xS = 24 = S, and tv = 6x6 = 36 = 5, the spaces respectively. 

* It is compared, because it is not referable to any standard, as a ton, pound, etc. Thus, suppose 
a cannon-ball weighing 15 lbs, projected with a velocity of 1500 feet per second, strike a resisting 
body, its momentum, according to the above rule, would be 15X1500 =22500; not pounds, for weight 
is a pressure with which it can not be compared. 



DYNAMICS. 395 

Uniform Variable Motion. — The space described b}' a bodj' having uni- 
form variable motion is represented by the sum or difference of the veloci- 
ty, and the product of the acceleration and the time, according as the mo- 
tion is accelerated or retarded. 

Illustration. — A sphere rolling down an inclined plane with an initial velocity 
of 25 feet, acquires in its course an additional velocity at each second of time of 5 
feet ; what will be its velocity after 3 seconds ? 

25 + 5X3;= 40 /ee*. 

2 A locomotive having an initial velocity of 30 feet per second is so retarded 

that in each second it loses 4 feet ; what is its velocity after 6 seconds ? 
30 — 4x6 = 6/otf. 

Motion. Uniformly Accelerated. 

In this motion, the velocity acquired at the end of any time whatever is equal to 
the product of the accelerating force into the time, and the space described is equal 
to the product of half the accelerating force into the square of the time, or half the 
product of the velocity and the time of acquiring the velocity. 

The spaces described in successive seconds of time are as the odd numbers, 1, 3, 5 
7, 9, etc. 

Gravity is a constant force, and its effect upon a body falling freely in a vertical 
line is represented by g. and the motion of such body is uniformly accelerated. 

The following theorems are applicable to all cases of motion uniformly accelerated 
by any constant force, F : 

v* 2 s 



= %tv=.%g-Ft* = 



2#F* v gF V%gF' 



2 — — JL— I 

v-g~F-\J^ 



t s v hJ gt gt* 2gs 

When gravit}' acts alone, as when a bod}- falls in a vertical line, F is 
omitted. Thus, 

.<, .„ & v 2 s 12s 

2g . g v V g 

. 2 s v 2 s v* 

v = gt=z — =W2gs £• = - — — == — . 

6 t v s * t P 2s 

If, instead of a heavy body falling freely, it be projected vertically up- 
ward or downward with a given velocity, v, then s = tv^%gt 2 ] an ex- 
pression in which — must be taken when the projection is upward, and 
-j- when it is downward. 

Illustration — If a body in 10 seconds hajs acquired a velocity by uniformly ac- 
celerated motion of 2C feet, what is the accelerating force, and what the space de- 
scribed, in that time ? 

2 6 

2G -r- 10 = 2. 6 — accelerating force ; ~ x 102 — 130 f eet — the space described. 

2. A body moving with an acceleration of 15.625 feet describes in 1.5 seconds a 

15.625X (1.5)2 
space = = IT. 578 feet. 

3. A body propelled with an initial velocity = S feet, and with an acceleration = 5 

72 
feet, describes in 7 seconds a space = 3x7 +5x-~ = 14S.5 feet 

4. A body which in 180 seconds changes its velqcity from 2.5 to 7.5 feet, traverses 
in this time a distance of — —— -xl$0 = 900 feet. 

5. A body which rolls up an inclined plane with an initial velocity of 40 feet, by 

which it suffers a retardation of 8 feet per second, ascends only — = 5 secorids, and 

8 

402 

— - = 100 feet in height, then rolls back, and returns, after 10 seconds, with a ve- 
locity o f 40 fee t, to its initial point : and after 12 seconds arrives at a distance of 40 

Xl2 — 4x1^2 =. £6 feet below the point, assuming the plane to be extended backward. 



396 



GRAVITATION. 



GRAVITATION. 

Gravity is an attraction common to all material substances, and 
they are effected by it in exact proportion to their mass. 

This attraction is termed terrestrial gravity, and the force with which 
any body is drawn toward the centre of the earth is termed the weight 
of that body. 

The force of gravity differs a little at different latitudes : the law of the 
variation, however, is not accurately ascertained ; but the following theo- 
rems represent it very nearly : 

{g (1 — .002S37 cos. 2 lat.), "i g representing the force of gravity at lati~ 

g (1 -\- .002337), at the poles, > tude 45°, and g s the force at the other 
g (1 — .002837), at the equator, ) places 

In bodies descending freely by their own weight, their velocities are as 
the times of their descent, and the spaces passed through as the square of 
the times. 

The Times, then, being 1, 2, 3. 4, etc.. the Velocities will be 1, 2, 3, 4, etc. 
The Spaces passed through will be as the square of the velocities acquired 
at the end of those times, as 1, 4, 9, 16, etc. ; and the spaces for each time 
as 1, 3, 5, 7, 9, etc. 

A body falling freely will descend through 16.0833 feet in the first sec- 
ond of time, and will" then have acquired a velocity which will cany it 
through 32.166 feet in the next second. 

The velocity acquired at any period is equal to twice the mean velocity 
during that period. 

The motion of a falling body being uniformly accelerated by gravity, 
the motion of a body projected vertically upward is uniformly retarded in 
the same manner. (See Note at end ofsubject, p. 401.) 

A body projected perpendicularly upward with a velocity equal to 
that which it would have acquired by falling from any height, will as- 
cend to the same height before it loses its velocity. 

Table exlii"biting tlie Relation, of Time, Space, and. "Ve- 
locities. 



Seconds from 
the begin- 
ning- of the 
Descent 


Velocity acquired 

at the End of that 

Time. 


Squares of 
the Time. 


Space fallen 

through in that 

Time. 


Spaces for 
this Time 


Space fallen 

through in the 

last Second of 

the Fall. 


1 


32.166 


1 


16.083 


1 


16.08 


2 


64.333 


4 


64.333 


3 


48.25 


3 


96.5 


9 


144.75 


5 


80.41 


4 


128.665 


16 


257.33 


7 


112.58 


5 


160.832 


25 


402.08 


9 


144.75 


6 


193. 


36 


579. 


11 


176.91 


7 


225.166 


49 


788.08 


13 


209.08 


8 


257.333 


6-4 


1029.33 


15 


241.25 


9 


289.5 


81 


1302.75 


17 


273.42 


10 


321.666 


100 


1608.33 


19 


305.58 



and in the same manner the Table may be continued to anv extent. 

Note. —In considering the action of gravitation on bodies not far distant from the 
surface of the earth, it is assumed, without sensible error, that the directions in which 
it acts are parallel, or perpendicular to the horizontal plane. 

A distance of one mile only produces a deviation from parallelism less than one 
minute, or the 60th part of a degree. 



GRAVITATION. 397 

To Compute the Time which a Body v^ill "be in falling 
through a given Space. 

Rule. — Divide the space in feet by 16.083, and the square root of the 
quotient will give the required time in seconds. 
Example.— How long will a body be in falling through 402.08 feet of space? 
\/402.08-M6.0S3 =. 5 seconds. 

To Compute the Time which a Body will "be in falling, 
the "Velocity per Second "being given. 

Rule. — Divide the given velocity by 32,166, and the quotient is the 
time. 

Example. — How long must a body be in falling to acquire a velocity of 800 feet 
peV second ? 800 -r- 32.160 = 24.87 seconds. 

Ex. 2.— Compute the time of generating a velocity of 193 feet per second, and the 
whole space descended. 

193-^- 32.166 = 6 seconds; 62xl6.083 = 579/e*rf. 

To Compute the Velocity a Body ^vill acquire by falling 
from any given Height. 

Rule. — Multiply the space in feet by 64.333, and the square root of the 
product will give the velocity acquired in feet per second. 
Example. — Required the velocity a body acquires in descending through 579 feet. 
A/579 X 64. 333 = 193 feet. 
As the velocity acquired at any period is equal to twice the mean velocity during 
that period. 

Ex. 2. — If a ball fall through 2816 feet in 12 seconds, with what velocity will it 
strike* ^ 

' 2316 -M2 — 193, mean velocity, which X 2 = 3S6 feet — velocity. 

To Compute the Velocity a Falling Body -will acquire in 
any given Time. 

Rule. — Multipty the time in seconds by 32.166, and the product will 
give the velocity in feet per second. 
Example. — What is the velocity acquired by a falling body in 6 seconds? 
32.166X6 = 192.996/ee*. 

To Compute the Space fallen through, the Velocity "being 

given. 

Rule. — Divide the velocity by 8.02, and the square of the quotient will 
give the distance fallen through to acquire that velocity. 

Example. — If the velocity of a cannon-ball is 579 feet per second, from what 
height must a body fall to acquire the same velocity ? 

579 -=-8.02 =72.2 and 72.22 — 5212.S4/ee*. 

To Compute the Space throxigh which a Body will fall in 
any given Time. 

Rule. — Multiply the square of the time in seconds by 16.083, and it 
will give the space in feet. 
Example. — Required the space fallen through in 5 seconds. 
52 = 25, and 25X16.0S3 = 402.08 feet. 
The distance fallen through in feet is very nearly equal to the square of the time 
in fourths of a second. 

Ex. 2.— A bullet being dropped from the spire of a church waa 4 seconds in reach- 
ing the ground ; what was the height of the spire ? 

4X4 = 16, and 162 = 256/eef. 
By Rule, 4x4x16.0833 = 257.33 feet. 

Ll 



398 GRAVITATION. 

Ex. 3. — What is the depth of a well, a bullet being 2 seconds in reaching the bot- 
tom? 2x4 = S, andS2 = W/ccf. 

By Rule, 2x2xl6.0S33 = 64.33/ee*. 

By Inversion In what time will a bullet fall through 256 feet? 

V256 = 16, and 16 H- 4 = 4 sec. 

Let s represent the space described by any falling body, t the time, v the velocity 
acquired in feet per second, and x the space in feet which the body falls in the ttt> 
second. ~ 

Then v — %V 16. 063 s, or 32.166*, or ~ ; x = 32. 166 (* — %). 

tv v 2 Is v 2 s 

, = 16.083 («, or ¥ , or gj-; t= J^® «*g^g, or-. 

Ascending bodies are retarded in the same ratio that descending bodies are accel- 
erated. Hence a body projected upward is ascending for one half of the time it is in 
motion, and descending the other half. 

To Compute tlie Space moved tliroiagli "by a Body pro- 
jected. Upward or Downward -witli a given Velocity. 

If projected Up ward. 
Eulk. — From the product of the given velocity and the time in seconds 
subtract the product of 32.166, and half the square of the time, and the 
remainder will give the space in feet. 

Or, *2 X 16.0S3 — vxi = s 
Example. — If a body be projected upward with a velocity of 30 feet par second, 
through what space will it ascend before it begins to return? 

30 -^-32.166 =.9326= the time to acquire this velocity. 
Then 30X.9326 = 2T.9S = product of velocity of projection and the time. 

32.166X 1 - r— = 13.98= product of 32.166, and half the square of the time. 

Hence 27.98 — 13.93 = 14 feet. 

Ex. 2 — If a body be projected upward with a velocity of 96.5 feet per second, it is 
reiuired to ascertain the point of the body at the end of 10 seconds. 
96.5-^-32.163 = 3 seconds, the time to acquire this velocity, and 3 2 Xl6.0S3 = 144.75, 
the height the body reached with its initial velocity. 

Then 10 — 3 = 7 seconds left for the body to fall in. 

Hence, by Rule (page 397), 7*xl6.0S3 = 7SS.07, and 7SS.07 — 144.75 = 643.32 /<?«* 
= the distance below the point of projection. 

Or, 102x16.033 = 1603.3 feet, the space fallen through under the effect of gravity, 
and '.'6.5x10 — 9(jb feet, the space if gravity did not act. Hence 1608.3 —905 = 
613.S feet. J * 

Ex. 3. — Tf a shot dischargod from a gun return to the earth in 12 seconds, how 
high did it ascend? 
The shot is half the time in ascending. 

62xl6.0S3 = 579/ee* = product of the square of the time and 16.033. 

If projected Downward. 
Rule. — Proceed as before, and the sum of the products will give the 
space in feet. 

Or, £2 X 16.0S3 + ?;X* = s. 
Example.— Ifgg body be projected downward with a velocity of 96.5 feet per sec- 
ond, through wli >pace must it descend to acquire a velocity of 193 feet per second? 
96.fr 32.166 = 3 seconds, the time to acquire this velocity. 
193 - )e 32.166 = 6 seconds, the time to acquire this velocity. 
Hence 6 — 3 = 1 seconds, the time of the body falling. 
Then 96.5x3 =W89. 5= product of velocity of projection and the time. 

: 2 : 
Therefore 289.5 + 144.75 = 434.25/ee*. 



GRAVITATION. 399 

Promiscuous Examples. 

1. A ball is 1 minute in falling, how far will it fall in the last second? 

Space fallen through = square of the time, and 1 minute =. 60 seconds. 
602x16.083 = 57898 feet for 60 seconds, 
592 x 16. 0S3 = 55984 " " 59 " 
1914 1 " 

2. Compute the time of generating a velocity of 193 feet per second, and the whole 
space descended. 

193-^32.166 = 6 seconds; 62xl6.0S3 = 579 /<?<??. 

8. If a hall fall through 2316 feet in 12 seconds, with what velocity will it strike ? 
2316 -^- 12 = 193X2 = 380 feet. 

IMotion and. Gravitation of Bodies on Inclined Planes. 

The space which a body describes upon an inclined plane, when descend- 
ing the plane by the force of gravity, is to the space it would freely fall in 
the same time as the height of the plane is to its length ; and the spaces 
being the same, the times will be inversely in this proportion. 

If a body descend in a curve, it suffers no loss of velocity. 

If two bodies begin to descend from rest, and from the same point, the 
one upon an inclined plane, and the other falling freely, their velocities 
at all equal heights below the surface will be equal. 

Example. — What distance will a body roll down an inclined plane 300 feet long 
and 25 feet high in one second by the force of gravity alone ? 
As 300 : 25 : : 16.0S3 : 1.34025/etf. 

Hence, if the proportion of the height to the length of the above plane is reduced 
from 25 to 300 to 25 to 600, the time required for the body to fall 1.34025 feet would 
be determined as follows : 

As 25 : 600 : : 1.34025 : 32.166 = 16.083x2 == twice the time required for one half 
the proportion of height to length. 

Or, as ^ : ^ : : 1.34025 : 32.165, as above. 
25 25 

Notes. — The times of descending different planes of the same height 
are to one another as the lengths of the planes. 

A body acquires the same velocity in descending any inclined plane as 
by falling freely through a distance equal to the height of the plane. 

When bodies move down inclined planes, the accelerating force is expressed by 
-; the quotient of the height 4- the length of the plane; or, what is equivalent there- 
to, the sine of the inclination of the plane, i e sin. i. 

The Formulae to determine the several elements are: 

1 « • • v2 1, 

1. s = -gt 2 sin. i=; i — : — Z tv ! 

2 & 2^-sin.i 2 4. v = Y3pgtsm.i. 

■ * ' 2 s 

2. v = gtsm. i— ^(2gs sin.z) = — ; 1 V 2 
t 5. s=z\ r t=p-gt*sm. i= - 



a (=v (_^)=:^i 

\g sin. i) v 



2s \ 2s 2 2g sin. i 1 



^representing the space fallen through in feet, v the velocity in feet per second, t the 
time in seconds, g 32. 166 feet, and V the velocity in feet per second of the body 
when projected. 

Illustration.— An inclined plane having a height of one half its length, the 
space fallen through in any time would be one half of that which it would fall freely. 

All of the preceding elements are required, the time assumed to be 5 seconds, and 
the velocity with which a body is projected upward being 96.5 feet, and downward 
16.0S3 feet. 

The velocity which a body rolling down such a plane would acquire in 5 seconds 
is 80.410 feet. 



400 GKAVITATION. 

2. i; — 32.16GX5X.5 = 80.416 — V(2X32.1C6X201.04X.5)_ 80.416= 2X201 '° 4 - 

80.416. 

2X201.04 



// 2X201.04 __ S 



80.416 



= 5. 



4. v = 16.083 -f 32. 166 X 5 X. 5 = 96.499 = velocity acquired at the end of 5 seconds 
when projected downward with a velocity of 16.083 feet per second. 

5. 5 = 96.5X5 -— x52x.5 = 2S1.4625/e<?£ = the space through which the 

a 

body will be projected upward in 5 seconds. 

Or, 1fi ft' g — ^ = 289.5/(?ei = the space through which the body will be project- 
ed upward before its motion is lost. 

What time will it take for a ball to roll 38 feet down an inclined plane, 
the angle t =? 12° 20', and what velocity will it attain at 38 feet from the 
starting-point? 

"VA VftXll6 = M "» ,i ' ; » = " SiD - < = 32.106X3.S3X.2136 
= 22.SS feet per second. 
Retailed Motion. — Bodies projected vertically will obtain inversely the 
same velocity as when descending, as the same'force acts upon them, and 
causes retarded motion when they ascend, and accelerated when they de- 
scend. 
The Formulae to determine the several elements are : 

32.166 <2 32.166*2 

1. s=zYt = tv-\-- 



2 



o T7 oQirr* * 32.166* . V 2A / 2 h y/h 

2. v=V-32.1Mt = 7 — , 4. ^ = 32166- y^V 32U66 = 4Xr 



,_ y-v _ v _ n 

' l ~ 32.166" 32.166 V 32. 



V2 2h » , 32.166^2 v*' V2 

- 5. h — - 



S.1662 32.166' 2 2 64.083' 

V representing any time less than t, and h height in feet to which the body will 
ascend. 

Illustration. — An ascending ball starts with a velocity of 135 feet per second; 
with what velocity will it strike an object 60 feet above ? 

^-x/sS 'sIFc"- 47 seconds > until ltstrikes - 

Then u = 135 — 32.166x.47 = 119.88 /e^joer seco7id. 

If a cannon-ball is projected at an angle to the horizon, there are two forces act- 
ing on the ball at the same time— viz., the force of gunpowder, which propels the 
ball uniformly in a right line, and the force of gravity, which causes the ball to grav- 
itate at an accelerated motion ; these two motions (uniform aud accelerated) cause 
the ball to move in a curved line (Parabola). 

V = «**\/& 5 P = tS : * = 243T8 1 sin. i> co,^; 

V representing velocity of the ball, W weight of the ball in pounds, s the greatest 
height of ball over horizontal line, t the time of fight, v pounds of powder in the 
charge, b the horizontal range, and x angle with the horizon. 

Illustration A cannon loaded to give a ball a velocity of 900 feet per second, 

the angle i — 45° ; what is the horizontal range, and what the time t f 
90^2 X sin, 46° X cob. 40* 900»X-5 f fc 

32.166 32.166 J 

Note. — As the distance b will be greatest when the angle 1 = 45°, the product of 
sine and cosine is greatest for that angle. Sin. 45° X cos. 45 == .5. 



ANIMAL STRENGTH. 401 

To Compute tlie Velocity- of a Falling Stream of Water 
per Second at tlie End. ofanj'- given Time, the perpen- 
dicular Distance "being given. 

Example. — What is the distance a stream of water will descend oa an inclined 
plane 10 feet high, and 100 feet long at the base, in 5 seeonds ? 

5 2 X 16. 083 =402. 08 feet = the space a body will freely fall in this time. 

Then, as 100 : 10 : : 402.08 : 40.21 feet = the proportionate velocity on a plane of 
these dimensions to the velocity when falling freely. 



Tlie various Formulas here given are for Bodies Pro- 
jected Upwards or Falling Freely, in Vacuo. 

When, however, the weight of a body is great compared with its volume^ and the 
velocity of it is low, deductions given are sufficiently accurate for ordinary purposes. 



ANIMAL STRENGTH. 



The mean effect of the power of a man v unaided by a machine, working 
to the best practicable advantage, is the raising of 70 lbs. 1 foot high in a 
second, for 10 hours in a day. 

Two men, working at a windlass at right angles to each other, can raise 
70 lbs. more easily than one man can 30 lbs. 

The result of observation upon animal power furnishes the following as the maxi- 
mum daily effect : 

1. When the effect produced varied from % to .2 of that which could be produced 
without velocity during a brief interval. 

2. When the velocity varied from % to % for a man, and from .08 to .06G for a horse, 
of the velocity which they were capable for a brief interval, and not producing any 
effort. 

3. When the duration of the daily work varied from ^ to ^ for a brief interval, 
during which the work could be constantly sustained without prejudice to the health 
of the man or the animals; the time not extending beyond 18 hours per day, how- 
ever limited may be the daily task, so long as it represents a constant attendance in 
the shop. 

By Mr. Field's experiments in 1838, the maximum power of a strong man, exerted 
for 2>£ minutes z= 18000 lbs. raised one foot in a minute. 

A man of ordinary strength exerts a force of 30 lbs. for 10 hours in a day, with a 
velocity of 2^ feet in a second == 4500 lbs. raised one foot in a minute = .2 of the 
work of a horse. 

A man can travel, without a load, on level ground, during 8>£ hours a day, at the 
rate of 3.7 miles an hour, or 31^ miles a day. He can carry 111 lbs. 11 miles in a 
day. Daily allowance of water for a man, 1 gallon for all purposes ; and he requires 
from 220 to 240 cubic feet of air per hour. 

A porter going short distances, and returning unloaded, can carry 135 lbs. 7 miles 
a day. He can transport, in a wheelbarrow, 150 lbs. 10 miles in a day. 

The muscles of the human jaw exert a force of 534 lbs. 

Mr. Buchanan ascertained that, in working a pump, turning a winch, in ringing 
a bell, and in rowing a boat, the effective power of a man is as the numberalOO, 
1C7, 227, and 248. 

A man drawing a boat in a Canal can transport 110000 lbs. for a distance of 7 
miles, and produce 156 times the effect of a man weighing 154 lbs. and walking 31^ 
miles in a day ; he can also produce an effect upon a tread-wheel of 30 lbs., with a 
velocity of 2.3 feet in a second, for 8 hours in a day, and can draw or push on a 
horizontal plane 30 lbs. with a velocity of 2 feet in a second, for S hours in a day. 
He can raise by a single pulley 38 lbs., with a velocity of .8 of a foot per second, for 
8 hours in a day, and he can pass over 12^ times the space horizontally that he can 
vertically. 

Ll* 



402 



ANIMAL STRENGTH. 



A foot-soldier travels in 1 minute, in common time, 90 steps = 70 yards. 

«■ " " " in quick-tima, 110 " = S6 u 

" " " " in double quick-time, 140 M = 110 " 

He occupies in the ranks a front of 20 inches, and a depth of 13, without a knap- 
sack : the interval between the ranks is 13 inches. 

Average weight of men, 150 lbs. each. 

Five men can stand in a space of 1 square yard. 

Table of* the Effective Power of Men for a Sliort IPeriod. 

Manner of Application. Force Manner of Application. Force. 



Bench-vice or chisel 

Brace-bit 

Drawing-knife or auger . 

Hand-plane 

Hand-saw 



Lbs. 
72 
16 

100 
50 
36 



Screw-driver, one-hand . 

Small screw-driver 

Thumb and fingers 

Thumb-vice 

Windlass or pincei'3 



Lbs. 
S4 
14 
14 
45 
60 



HORSES. 



A Horse can travel 400 yards, at a walk, in 434 minutes ; at a trot, in 2 
minutes ; and at a gallop, In 1 minute. He occupies in the ranks a front 
of 40 inches, and a depth of 10 feet ; in a stall, from 3% to 4)^ feet front ; 
and at a picket, 3 feet by 9 ; and his average weight = 1000 lbs. 

A Horse, carrying a soldier and his equipments (225 lbs.), can travel 25 
miles in a daj r (8 hours). 

A Draught-horse can draw 1600 lbs. 23 miles a day, weight of carriage 
included. 

The ordinary work of a horse may be stated at 22500 lbs., raised 1 foot 
in a minute, for 8 hours a day. 

In a horse-mill, a horse moves at the rate of 3 feet in a second. The diameter of 
the track should not be less than 25 feet. 

A horse-power in machinery is estimated at 33000 lbs., raised 1 foot in a minute; 
but as a horse can exert that force but 6 hoars a day, one machinery horse-power is 
equivalent to that of 4)4 horses. 

The expense of conveying goods at 3 miles per hour per horse teams being 1, the 
expense at 4% miles will be 1.33, and so on, the expense being doubled when the 
speed is b% miles per hour. 

The strength of a horse is equivalent to that of 5 men. 

The daily allowance of water for a horse should be 4 gallons. 

Ta"ble of the -A.moxi.rLt ofLabor a Horse of average Strength, 
is capable of performing, at different "Velocities, on Ca- 
nals, Hail-roads, and Turnpikes. 

Force of Traction estimated at 83.3 lbs. 







Useful Effect for One Dav, drawn 


] 




Useful Effect for One Dav, drawn 


Veloci- 


Dura- 




1 Mile. 


Veloci- 


Dura- 




1 Mile. 


ty per 
Hour. 


tion of 
Work. 






ty per 
Hour. 


tion of 
Work. 






On a Ca, 


On a Rail- 


On a Turn- 


On a Ca- 


On a Rail- 


On a Turn- 






nal. 


road. 


pike. 






nal. 


road. 


pike. 


Mi'es. 


Hours. 


Tons. 


Tons. 


Tons. 


Miles. 


Hours. 


Tons. 


Tons. 


Tons. 


2* 


11.5 


5-20 


115 


14 


6 


2. 


30 


4S 


6 


3 


8. 


243 


92 


12 


7 


1.5 


19 


41 


5.1 


4 


4.5 


102 


72 


9 


8 


1.125 


12. S 


3G 


4.5 


5 


2.9 


52 


5T 


7.2 


10 


.75 


6.6 


28. S 


3.6 



The actual labor performed by horses is greater, but they are injured by it. 

A Horse in a mill can produce an effect of 106 lbs., at a velocity of 3 feet in a sec- 
ond, for 8 houra in a day. A Mule can produce, under a like velocity and time, an 
effect of 71 lbs. ; and an Ass, 37 lbs. 

An Ox, walking at a velocity of 2 feet in a second (1.34 miles per hour), will draw 
154 lbs. for S hours in a day. 

A Horse requires a space 7 feet by 2*£ for transportation in a vessel ; and a Beeve 
requires 6}£ feet by 26 ins., without manger, and 2 feet additional length with one. 
3 Beeves or 15 Sheep require the food of 2 Horses. 



ANIMAL STRENGTH. 



403 



T'able showing the Amount of Labor produced "by Ani- 
mal IPower under different Circumstances. 



MANNER OF APPLICATION. 



10 Hours per Day. 

Man, throwing earth with a shovel, a height of 
5 feet 

Man, wheeling a loaded barrow up an inclined 
plane, height one-twelfth of length 

Man, raising and pitching earth in a shovel 
to a horizontal distance of 13 feet 

Man, pushing and drawing alternately in a 
vertical direction 

Man, transporting weight upon a barrow, and 
returning unloaded 

Man, walking upon a level 

Horse, drawing a 4-wheeled carriage at a walk 

Horse, with load upon his back, at a walk . . 

Horse, transporting a loaded wagon, and re- 
turning unloaded at a walk 

Horse, drawing a loaded wagon at a walk . . . 

8 Hours per Day. 

Man, ascending a slight elevation, unloaded . 
Man, walking, and pushing or drawing in a 

horizontal direction 

Man, turning a crank 

Man, upon a tread-mill 

Man, rowing 

Horse, upon a revolving platform, at a walk . 
Ox, upon a revolving platform, at a walk . . . 
Mule, upon a revolving platform, at a walk. . 
Ass, upon a revolving platform, at a walk . . . 

7 Hours per Day 

Man, walking with a load upon his back 

6 Hours per Pay. 

Man, transporting a weight upon his back, 

and returning unloaded 

" Man, transporting a weight upon his back up 
a slight elevation, and returning unloaded . 

Man, raising a weight by the hands 

AV 2 Hours per Day. 

Horse, upon a revolving platform at a trot. . 

Horse, drawing an unloaded 4-wheeled car- 
riage at a trot 

Horse, drawing a loaded 4-wheeled carriage 
at a trot 



Power. 



Veloci- 
ty per 
SecoDd 



Lbs. 

6 

132 

6 

13 

132 
143 
154 

264 

1540 
1540 

143 

26 

18 

140 

26 

100 

132 

66 

32 



140 

140 

44 

66 

97 

770 



% 

VA 
^A 
l 

5 
3 

3% 

2 



2 

5^ 
3 
2 
3 

2X 



1M 

.2 

H 

6% 
1% 



Weight 
raised. 

Foot per 
Minute. 

Lbs. 


Horses* 
Power 
for the 
Period 
given. 


No. 


480 


8.7 


4950 


90. 


810 


14.7 


1950 


35.5 


7920 
42900 
27720 
59400 


144 

780 

504 

1080 


184800 
346500 


3360 

6300 


4290 


62 


3120 

2790 

4200 

7800 

18000 

15840 

11880 

5280 


45.2 
39 
60.9 
113 

260.8 

229.5 

172.2 

76.5 


13200 


167.9 


14700 


160.5 


1680 
1320 


19 
14.4 


26730 


218.7 


43195 


353.5 


334950 


2741 



How many men are required upon a tread-mill, 20 feet in diameter, in 
order to raise a weight of 900 lbs., the crank being 9 inches in length? 

The weight of the wheel and its load is estimated at 5000 lbs., and the friction at 
.015= 75 lbs. The labor of a man upon such a mill is estimated at 25 lbs. Length 
of crank = .75 feet. 

750 

Then 900X.75 + 5000 X. 015 = 750 lbs., the resistance of the wheel; and . 



T5 lbs., the power required at the circumference of the wheel. 
Therefore, 75-=- 25 = 3 men. 



20-^-2 



404 



ANIMAL STIIEXGTH. 



The draught of Man and Animals by traces is as follows : 
Man 150 lbs. ; Horse GOO lbs. ; Male 500 lbs. ; Ass 360 lbs. 

A man rowing a boat 1 mile in 7 minutes performs the labor, while 
rowing, of 6 fully-worked laborers at ordinary occupations of 10 hours. 

Ta"ble showing the Effects of a Traction of lOO !LVbs. at 
different "Velocities on Canals. 



Velocity 
per Hour. 


Velocity per 
Second. 


Mass moved. 


Useful Effect. 


Miles. 


Feet. 


Lbs. 


Lbs. 


2K 


3.66 


55500 


39408 


3 


4.4 


38542 


27361 


&A 


5.13 


28316 


20100 


4 


5.86 


21680 


15390 


5 


7.33 


13875 


9850 



Velocitv 


Velocity per 


Mass 


Useful 


pr. Hour. 


Second. 


moved. 


Effect. 


Miles. 


Feet. 


Lbs. 


Lbs. 


6 


8.8 


9635 


6840 


7 


10.26 


7080 


5026 


8 


11.73 


5420 


3848 


9 


13.2 


4282 


3040 


10 


14.66 


3468 


2462 



The load carried, added to the weight of the vessel which contains it, 
forms the total mass moved, and the useful effect is the load. 

The force of traction on a railroad or turnpike is constant, but the me- 
chanical power necessary to move the carriage increases' as the velocity ; 
on a canal the force of traction varies as the square of the velocity. 

Labor upon Embankments.-[EuwooD Mobkis.] 

Single Horse and Cart. — A horse with a loaded dirt-cart, employed in 
excavation and embankment, will make 100 lineal feet of trip, or 200 feet 
in distance per minute, while moving. The time lost in loading, dump- 
ing, awaiting, etc. = 4 minutes per load. 

A medium laborer will load with a cart in 10 hours, of the following 
Earths, measured in the bank : 

Gravelly Earth, 10 ; Loam, 12 ; and Sandy Earth, 14 cubic yards. 

Earth from a natural excavation occupies J more space than when 
transported to an embankment. 

Carts are loaded as follows : Descending Hauling, J of a cubic yard in 
bank ; Level Hauling, & of a cubic yard in bank ; Ascending Hauling, \ 
of a cubic yard in bank. 

Loosening, etc. — In Loam, a three-horse plow will loosen from 250 to 800 
cubic yards per day of 10 hours. 

The cost of loosening earth to be loaded will be from 1 to 8 cents per 
cubic yard when wages are 105 cents per day. 

The cost of Trimming and Bossing is about 2 cents per cubic yard. 

Scooping.— A Scoop load will measure ^ of a cubic yard, measured In 
excavation. 

The time lost in loading, unloading, and turning, per load, is 1% minute. 

The time -lost for ever}- 70 feet of distance, from excavation to bank, 
and returning, is 1 minute. 

In Double Scooping, the time lost in loading, turning, etc., will be 1 
minute ; and in Single Scooping it will be 1^ minutes. 

Volumes of Excavation and. Embankment. 

The volume of earth in embankment is less than in excavation, as the 
compression of earth in an embankment is in excess of the expansion of its 
volume in a natural state, the proportion being as follows : 

Sand i ; Clay § • Gravel J. 

The volume of rock in bank exceeds that in excavation in the propor- 
tion of 3 to 2. r r 



ANIMAL STRENGTH. 405 

Stone. 

Hauling Stone. — A cart drawn by horses over an ordinary road will 
travel 1.1 miles per hour of trip. 

A four-horse team will haul from 25 to 36 cubic feet of limestone at each 
load. 

The time expended in loading, unloading, etc., including delays, aver- 
ages 35 minutes per trip. The cost of loading and unloading a cart, using 
a horse-crane at the quarry, and unloading by hand, when labor is $1 25 
per da} r , and a horse 75 cents, is 25 cents per perch=:24.75 cubic feet. 

The work done by an animal is greatest when the velocity with which 
he moves is % of "the greatest with which he can move when not im- 
peded, and the force then exerted .45 of the utmost force the animal can 
exert at a dead pull. 



PERFORMANCES OF MEN, HORSES, ETC. 

The following notes are designed to furnish an authentic summary of 
the fastest or most successful recorded performances in each of the feats, 
matches, or races, etc., etc., given. 

Note.— Parties desirous of maintaining such a record by the contribution of results are requested 
to address them to the Author. 



WALKING. 
Mian. 

1865, — Hill, Brooklyn, Long Island, N. Y., % mile, backward, in 7 min. 
1874, Wm. Perkins, London, Eng., % mile, in 2 min. 56 sec, % in 4 min. 40 sec.; 
and 1875, 1 mile, in 6 min. 23 sec., 2 in 13 min. 30 sec., 3 in 20 min. 47 sec, 4 in 
28 min. 59 sec, 5 in 36 min. 32 sec, 6 in 44 min. 24 sec, 7 in 51 min. 51 sec ; 1876, 
8 in 58 min. 28 sec. ; 1877, 9 in 1 hour 8 min. 7.4 sec, 10 in 1 hour 15 min. 57 sec, 
12 in 1 hour 31 min. 42.4 sec, 15 in 1 hour 56 min. 13 sec, and 20 in 2 hours 39 
min. 57 sec 

1877, D. O'Leary, London, Eng., 75 miles, in 13 hours 25 min. 44 sec, 200 in 45 
hours 21 min. 23 sec, and 519 in 140 hours 29 min. 50 sec 
1868, K Young, Mansfield, Ohio, 1% miles, in 56 min. 58 sec 
1851, J. Smith, London, Eng., 25 miles, in 3 hours 42 min. 16 sec 
1876, C. G. Ide, London, Eng., 50 miles, in S hours 19 min. 55 sec 
1801, Capt. R. Barclay, Eng., country road, 110 miles, in training, in 19 hours, ex- 
clusive of rests, and 90 in 20 hours 22 min. 4 sec, including rests; 1S03, % mile 
in 56 sec, and Charing Cross to Newmarket, 64 in 10 hours, including rests ; 1806, 
Ury to Crathynaird and back, 28 in 4 hours, and 100 in 19 hours, including 1 hour 
30 min. in rests; 1809, 1000 in 1000 consecutive hours, walking a mile only at the 
commencement of each hour. 
1830, — Newsam, Philadelphia, Penn., 1 000 miles, in 18 days. 
1818, Jos. Eaton, Stowmarket, Eng., 4 032 quarter miles, in 4 032 consecutive quar- 
ter hours; 1846 (70 years of age), Canada, 1000 miles, in 1000 consecutive hours , 
and 1847, Boston, 1 000 quarter miles, in 1 000 consecutive quarter hours. 

1876, H. Vaughan, London, Eng., 100 miles, in 18 hours 51 min. 35 sec 

1877, Wm. Gale, London, Eng., 1 500 miles, in 1 000 consecutive hours, 1)4 miles 
each hour. 

1879, E. P. Weston, of New York, London, Eng., 550 miles 110 yards, in 142 hours.* 
1879, Chas. Powell, of Eng., New York, N.Y., 124 mile3 3S3 yards, in 24 hours, 
and 518 miles 1671 yards, in 189 hours, 25 min.* t 

* Walk and run. t By corrected length of track. 



405* ANIMAL STRENGTH. 

RUNNING. 
]VtarL. 

18 — , Wm. Bingham, Toronto, Can., 75 yards, in 7 sec. 
18G9, P. Perry, Trenton, N. J., 75 yards, in 1% sec. 

1844, Geo. Seward, of U. S., Manchester, Eng., 100 yards, in 9% sec. ; 150 in 14 j^ 
sec. ; 200, running start, in 19% sec. ; and 1847, London, Eng. , 120 in 11J£ sec. 
1868, John Thomas, Philadelphia, Penn.,100 yards, in 9% sec. 
1868, J. W. Cozad, Long Island, N. Y., 125 yards, in 12% sec. 

1851, CJms. Westhall, Manchester, Eng., 200 yards, in 19 *£ sec. 

1863, Jas. Nuttall, Manchester, Eng., 300 yards, in 31% sec. ; 1S64, 600 in 1 min. 
13 sec. ; and 1872, 1 000 in 2 min. 19% sec. 

1873, R. Buttery, Newcastle, Eng., % mile, in 4S^ sec. 

1871, Frank Hewitt, Australia, % mile, in 1 min. 53% sec. 

1861, — White, Long Island, N.Y., 1 mile, drawing a sulky, in 6 min. 24% sec. 

1863, Wm. Lang, Newmarket, Eng., 1 mile, in 4 min. 2 sec, descending ground, 
Manchester, 2 in 9 min. 11% sec. ; and 1S65, 1 mile, in 4 min. 17% sec. , a dead heat with 
Richards ; 10 in 51 min. 26 sec, and 11 miles 1660 yards, in 1 hour 2 min. 2% sec. 

1S67, James Fleet, Manchester, Eng., 1% miles, in 6 min. 50 sec. 

1764, (a man), Barnet Course, Eng., 2 miles, in a sack, in 56 min. 

1710, Levi Whitehead, Branham Moor, Eng. , 4 miles, in 19 min. 

1863, J. White, Hackney Wick, Eng., 3 miles, in 14 min. 36 sec; Manchester, 5 
in 24 min. 40 sec ; and Hackney Wick, 7 in 34 min. 45 sec 

1852, William Howitt, alias Jackson, "American Deer," London, Eng., 10 miles, 
in 51 min. 34 sec, walking in last 200 yards, computed time, if run, 51 min. 20 sec. ; 
and 15 in 1 hour 22 min. 

1S52, 1. Howitt, London, Eng., 8 miles, in 40 min. 20 sec, and 9 in 45 min. 21 sec 

1863, L. Bennett, ll Deerfoot," Hackney Wick, Eng., 10 miles, in 51 min. 26 sec, 
and 11 miles 790 yards, in 59 min. 44 sec=ll% miles per hour; and Dublin, Ire- 
land, 12 in 1 hour 5 min. 6 sec. 

1851, R. Manks, London, Eng., 20 miles, in 1 hour 5S min. 18 sec, and 40 in 4 
hours 51 sec. 

1S60, H. Howard, Bridge water, Eng., 41 miles, public road, in 5 hours 36 mift., 
1 hour IS min. in rests. 

1863, G. Martin, London, Eng., country road, 50 miles, in 6 hours 17 min. 

1749, J. Manser, Peterborough to Lincoln, Eng., 50 miles, in 7 hours 30 min. 

1801, Capt. R. Barclay, Eng., Hull road, 90 miles, in 20 hours 22 min. 4 sec. 

17 — , (a courier), East Indies, 102 miles, in 24 hours. 



JUMPING, LEAPING, ETC. 
UVEan.. 

1774, A. Thorpe, Artill'y Ground, London, Eng., 1 mile, in a sack, in 11 min. 30 sec. 

1S29, Samuel Patch, over Genesee Falls, N. Y., 125 feet perpendicular (lost). 

1S48, P. M^Neely, Petersburg, Ky., 10 jumps, standing, 110 feet 4 ins. 

1854, J. Hoivard, Chester, Eng., 1 jump, board raised 4 inches in front, running 
start, with dumb-bells, 5 lbs., 29 feet 7 ins.; and 1S65, over a hurdle 3 feet 6 in. in 
height, 26 feet ; and without aid, 20 feet 6 ins. 

1854, — Kelson, New York, N.Y., 10 hops, running start, 112 feet. 

1856, TJws. King, San Francisco, Cal., spring board, running start, leaped over 9 
horses, 31 feet 7% ins. 

1865, D. Anderson, Eyemouth, Eng., hop, skip, and jump, standing, 40 feet 2 ins. 

186S, Geo. M. Kelley, Corinth, Miss., running, and from a spring board, leaped over 
17 horses standing side by side. 

1869, J. P. Naylor, Manchester, Eng., 6 jumps backward, standing, 54 feet 5 ins. 

1869, J. Parker, Leeds, Eng., hopped on one leg 100 yards in 14 sec. 

1870, R. Knox, Leith, Eng., hop, skip, and jump, running, 47 feet 7 ins. 
1872, P. P. Mulgrave, Titusville, Penn., 3 jumps, standing, 35 feet S ins. 

1877, John West, Philadelphia, Penn., running leap, 5 feet \0% ins. 

1575, J. Greaves, Ilazlehurst, Eng., dumb-bells, 23 lbs., standing jump, 13 feet 7 ins. 

1576, M. I. Brooks (amateur), London, Eng., running leap, 5 feet 11 ins. 

1878, E. W. Johnson, Baltimore, Md., standing leap, 5 feet 3 ins., and standing 
jump, 11 feet 2% ins. 

1878, F. Davis, Philadelphia, Penn., running jump, 21 feet 5 ins. 

1879, Alex. Dobson, Toronto, Can., hop, skip, and jump, running, 46 feet 8 ins, 
1879, T. Ray, Ulverstone, Eng., pole jump, 10 feet 2% ins. 



ANIMAL STKENGTH. 406 

LIFTING. 

1825, Thomas Gardner, of New Brunswick, N. S., a barrel of pork, 320 lbs., under 
each arm ; also transported across a pier an anchor, 1 200 lbs. 

1868, Wm. B. Curtis, New York, N. Y„ 3239 lbs., in harness, and 1230 lbs. by 
hands alone. 

1873, R. A. Pennell, New York, N. Y., 1 210 lbs., and 18T4 raised dumb-bell, 201^ 
lbs., by one hand. 

THROWING WEIGHTS. 

1870, D. Dinnie, New York, N. Y., light stone, 18 lbs., 43 feet; heavy stone, 24 
lbs., 34 feet 6 ins. ; heavy hammer, 24 lbs., 83 feet 8 ins. ; at Coupar Angus, Scot* 
land, light hammer, 16 lbs., 138 feet. 

1870, Jos. Steward, Virginia, Nev. Ter., a piece of lead, 3 oz., 100 yards 18 ins. 

1874, C. Wadsworth, Dublin, Ireland, 56 lb. weight, 29 feet 2 ins. 

1879, Geo. Davison, Philadelphia, Penn., light hammer, 12 lbs., 121 feet 5 ins., and 
56 1b. weight, 27 feet. 
1879, D. C. Boss, Philadelphia, Penn., heavy hammer, 16 lbs., 109 feet 9 ins. 

FLY ROD CASTING. 

1860, Seth Green, Rochester, N. Y., rod 12 feet 6 inches in length, standing 1% feet 
above the water, wind calm, 100 feet. 

SWIMMING. 

1S35, S. Bruck, 15 miles, in a rough sea, in 7 hours 30 min. 

1846, A Native, off Sandwich Islands, 7 miles at sea, with a live pig under one arm. 

1870, Chas. Whyte, London Bridge to Clock at Greenwich Hospital, Eng., favorable 
current, first 3 miles, in 35 min. 28 sec., and 5 miles, in 1 hour 4 min. 23 sec. 

1870, Pauline Rohn [young woman], Milwaukee, Wis., 650 feet, still water, in 2 
min. 43 sec. 

1874, H. Parker, London, Eng., 500 yards, in 7 min. 27.4 sec. 

1874, E. T. Jones, Leeds, Eng., 1000 yards, in 15 min. 30 sec, and 1 mile, still 
water, in 30 min. 3 sec. 

1872, J. B. Johnson, English Channel, 7 miles, in 1 hour 5 min. ;* 1874, New York 
Bay, N.Y., 3 miles (defined by estimate), smooth water, in 1 hour 10 min. 30 sec. 

1875, Agnes Beckwith [young woman], London Bridge to Greenwich Hospital, Eng., 
favorable current, 5 -f- miles, in 1 hour 9 min. 

1875, Capt. M. Webb, Dover, Eng., to Calais, France, 23 miles, crossing two full 
and two half tides = 50 miles, in 21 hours 45 mm., and Blackwall to Gravesend, 
Eng., 20 miles, first of ebb-tide, in 4 hours 53 min. 

SKATING. 

18 — , Wm. Clarke, Madison, Wis., 1 mile, in 1 min. 56 sec. 

1867, Chas. Ochford, Detroit, Mich., for 60 consecutive hours, stopping 12 minutes 
in each 12 hours. 

1867, T. Prentiss, Quincy to Lagrange, 111., 15 miles, in 50 min. 
1S6S, John Conyers, Lake Simcoe, Can., 8 miles, in 18 min. 40% sec. 

1868, E. St. Clair Milliard, Cincinnati, Ohio, 100 miles, in 11 hours 46 min., and 
for 24 hours with 20 minutes rest ; and 1876, Chicago, 111., 50 miles, in 4 hours 57 
min. 3 sec 

1S68, Annie C. Jagerisky [young woman, 17 yrs.], 30 hours, with 30 minutes rest. 

1870, — Hills, Chetney Wade, Eng., IX miles, in 3 min. 6 sec. 
1877, John Ennis, Chicago, 111., 145 miles inside of 19 hours. 

Note.— The Sporting Magazine, London, vol. ix., p. 135, reports a man in 1767 to have skated a 
mile upon the Serpentine, Hyde Park, London, in 57 seconds. 

SNOW SHOES. 

1869, J. James, Montreal, Can., % mile, 2d heat, in 1 min. 15 sec. 

1871, J. F. Scholes, Montreal, Can., % mile, in 2 min. 39% sec, and 1 in 5 min. 
393^ sec 

1871, J. D. Armstrong, Montreal, Can., % mile, in 1 min. 5 sec. 
1871, Kerar-onwe (Indian), Montreal, Can., 2 miles in 11 min. 30 sec, 3 in 17 min. 
52 sec, and 4 in 24 min. 4 sec 

* In an attempt to cross the British Channel, in which he failed. 



406* ANIMAL STRENGTH. 

RUNNING. 

Horse. 

ONE MILE. 

1850, " Black Doctor," Doncaster, Eng., 2 years, 87 lbs., in 1 min. 40 sec. 

1854, " Lecomte," New Orleans, La., 4 years, 103 lbs., 3d mile of a 2d 4-mile heat, 
in 1 min. 46 sec. 

1855, " Henry Perritt," New Orleans, La., 4 years^ 83 lbs., 1st mile of 2d heat of 

2 miles, in 1 min. 42>£ sec. 

1862, " Buccaneer," Salisbury, Eng., 5 years, 131 lbs., in 1 min. 38 sec* 
186S, " Climax," Jerome Park, N.Y., 9 years, 14S% lbs., in 1 min. 48% sec. 
, 1869, " Lobelia," Fashion Course, L. I.", 6 years, 143 lbs., 4 hurdles, in 1 min, 

51% sec. 
1873, " Thad. Stevens," Sacramento, Cal., 7 years, 115 lbs., 3d heat, in 1 min. 

43% sec. 

1874, " Springbok," Utica, N. Y., 4 years, 108 lbs., 2d heat, in 1 min. 42% sec. 

1875, M Searcher," now " Leander," Lexington, Ky., 3 years, 90 lbs., in 1 min. 41% 
sec. 

1875, u Kadi," Hartford, Conn. , 6 years, (catch) 82 lbs., 1st heat, in 1 min. 42J£ 
Sec., and 2d heat, in 1 Tain. 41% sec. 
1877, "Ten Broeck," Louisville, Ky., 5 years, 110 lbs., in 1 min. 39% sec. 

TWO MILES. 

1S_, " Child of the Islands" (Arabian), India, in 3 min. 48 sec. 
1834, "Inheritor," Liverpool, Eng., 3 years, 87 lbs., in 3 min. 25 sec* 
1847, ""Inheritress," Liverpool, Eng., aged, 115 lbs., in 3 min. 27 sec* 
1867, u Blackbird," Saratoga, N. Y., aged, 161 lbs., 8 hurdles, in 3 min. 57% sec. 
186S, " Jonesboro," New Orleans, La., 4 years, 132 lbs., 8 hurdles, in 3 min. 51% 
sec. 

1872, "Mickey Free,'* Long Branch, N. J., 154 lbs., 8 hurdles, in 3 min. 52% sec. 
1875, " Tom Leathers," New Orleans, La., 117 lbs., 8 hurdles, in 3 min. 41% sec. 

1875, " Arizona," Louisville, Ky., aged, 111 lbs., 2d heat in 3 min. 35% sec. 
1877, u Ten Broeck," Louisville, Ky., 5 years, 110 lbs., in 3 min, 21% sec. 

THREE MILES. 

1854* " Virago," Warwick, Eng., 3 years, 101 lbs., in 5 min, 29 sec. 
1855, "Brown Dick," New Orleans, La., 3 years, 86 lbs., in 5 min. 30% sec., and 
2d heat, in 5 min. 28 sec. 
1855, " Rataplan," Warwick, Eng., 5 years, 127 lbs., in 5 min, 27 sec. 
1861, " Mollie Jackson," Louisville, Ky., 4 yrs., 101 lbs., 3d heat, in 5 win. 28% sec 
1865, " Fleetwing," Saratoga, N. Y., 5 years, 114 lbs., in 5 min. 31% sec. 

1876, u Ten Broeck," Louisville, Ky., 4 years, 104 lbs., in 5 min. 26% sec 

FOUR MILES. 

1710, u Bay Bolton," York, Eng., 5 years, 168 lbs., in 7 min. 43 sec 
1752, " Skewball," Kildare, Ireland, 11 years, 119 lbs., in 7 min. 51 sec. 
1760, " Bay Malton," York, Eug., 6 years, 119 lbs., in 7 min. 43% sec. 
1767, " Selim," Philadelphia, Penn., 8 years, 140 lbs., in 8 min. 2 sec. 
1769, "Eclipse," Winchester, Eng., 5 years, 168 lbs., in 8 min. 
1823, " Sir Henry," Long Island, N. Y., 4 yeai'3, 108 lbs., in 7 min. 37% sec. 
1828, " Ariel," Newmarket, Va., 6 years, 118 lbs., 4th heat, in S min. 4 sec. 

1832, " Black Maria," Long Island, N.Y., 6 years, 118 lbs., 5th heat, in 8 min. 47 
sec. , and the 5 heats, in 41 min. 40 sec. 

1833, " Lady Elizabeth," Doncaster, Eng., 5 years, 135 lbs., in 7 min. 46 sec. 
1842, " Fashion," Long Island, N. Y., 5 years, 111 lbs., in 7 min. 32% sec. 

1855, " Lexington," New Orleans, La., 5 years, 103% lbs., in 7 min. 23% sec. ; and 
with a running start, in 7 min. 19% sec. 
1S63, " Idlewild," Long Island, N. Y., 6 years, 117 lbs., in 7 min. 26% sec, and last 

3 miles in 5 min. 27% sec. ; track heavy. 

1S71, "Abd-el-Koree," Jerome Park, N.Y.,3 years, 95 lbs., in 7 min. 33 sec 
1S73, "Thad. Stevens," San Francisco, Cal., 7 years, 115 lbs., 2d heat, in 7 min. 
30 sec. 
1876, u Ten Broeck," Louisville, Ky., 4 years, 104 lbs., in 7 min. 15% sec 

* When it is considered that in England no official record of time is taken, and that usually the start 
p.nd finish are at different points, the accuracy of the times here given is much doubted. 



ANIMAL STRENGTH. 401 

"Various Distances and Performances. 
English. 
1701, Mr. Sinclair, on the Swift at Carlisle, a gelding, 1000 miles in 1000 consecu- 
tive hours. 

1721,"Childers" (Flying), Newmarket, R. C, 6 years, 128 lbs., 3 miles, 6* fur- 
longs, and 93 yards (3.8029 miles) in 6 min. 43* sec.=3 miles in 5 min. 18 sec. 

Note.— It is related he ran 4 miles, 1 furlong, 138 yards (4.204 miles) in 7 min. 30 aec.=4 miles in 7 
min. 8.2 sec. There is a well-founded doubt about the accuracy of this performance, arising from ita 
unequaled time, the relation being more of a traditionary character than a record, and the circum- 
stance that at that date timing watches were not in use ; added to which, Bay Malton's time of 7 min. 
43>£ sec, in 1766, is recorded as 7% sec. less than was ever before accomplished. 

1745, Cooper Thornhill, between Stilton and London 3 times, 213 miles, by 14 horses, 
in 12 hours 7 min., including 33 min. S sec. in rests. 

1751, Samuel Bendell (aged 76), 1 horse, 1000 miles in 1000 consecutive hours. 

1752, Spedding's Mare, 100 miles, in 12 hours 30 min., for two consecutive days. 
1754, A Galloway mare of Daniel Corker's, Newmarket Heath, 300 miles, by one 

rider, 67 lbs., in 64 hours 20 min. 

1759, J. Shafto, Newmarket Heath, 50 miles, by 10 horses, in 1 hour 49 min. 17 
sec. 

1761, John Woodcock, Newmarket Heath, 100 miles per day, by one horse each 
day, for 29 consecutive days, 14 horses ; one day 160 miles, a horse breaking down 
at the 60th mile. 

1786, " Quibbler," 6 years, Newmarket, R. C, 77 lbs., 23 miles in 57 min. 10 sec. 

1791, Mr. Wilde, Curragh, Ireland, 127 miles, by 10 horses, in 6 hours 21 min. 

1793, — Delme, Jr., Colnbrook to London, 17 miles, in less than 44 min. 

1801, Capt. Newland, Longdown Hill, 100 miles, by hack horses, in 5 hours 5 mzw., 
and 140 miles, in 7 hours 34 min. 

1814. An Officer of 14th Dragoons, Blackwater,12 miles, 1 horse, in 25 min. 11 sec. 

1823, " Hampden," Newmarket, R. C, 4 years, 144 lbs., 3 miles, 4 furlongs, 187 
yards (3.606 miles), in 7 min. 4 sec. =4 miles in 7 min. 50 sec. 

1831, Geo. Osbaldeslon, Newmarket, 156 lbs., 60 miles, by 11 horses in 2 hours 33 
min., 100 by 16 horses in 4 hours 19 min. 40 sec., aud 200 by 28 horses in 8 hours 
39 min., including 1 hour 2 min. 56 sec. in rests; 1 horse, u Tranby," 16 miles (4 
times 4 miles) in 33 min. 15 sec. 

1846, " Sir Tatton Sykes," Doncaster, " St. Leger," 3 years, 122 lbs., 1 mile, 6 fur- 
longs, 132 yards (1.775 miles), in 3 min. 16 sec.=.l% miles in 3 min. 13 sec. 

1847, "The Widow," Newmarket, "Cambridgeshire," 8 years, 98 lbs., 1 mile, 
240 yards (1.136 miles), in 1 min. 58 sec.=l mile in 1 min. 43.8-J- sec. 

1S54, *« Stock well," Newmarket, B. C, 5 y^ars, 140 lbs., 4 miles, 1 furlong, 138 
yards (4.204 miles) in 7 min. 52 sec. =4 miles in 7 min. 29 sec. 

1S54, " West Australian," Ascot Heath, 4 years, 119 lbs., and "Kingston," 5 years, 
126 lbs., 2)4 miles in 4 min. 27 sec. =2 miles in 3 min. 33.6 sec. 

1855, "Mr. Sykes," Newmarket, "Cesare witch," 5 years, 92 lb3„ 2 miles, 2 fur- 
longs, and 28 yards (2.266 miles), in 3 min. 55 sec.=& miles in 3 min. 27+ sec. 

1857, " Saunterer," Newmarket, 3 years, 119 lbs., 1 mile, 2 furlongs, and 73 yards 
(1.292 miles) in 2 min. 10 sec.=l mile in 1 min. 40.6 sec. 

1861, " Diophantus," Newmarket, R. M., 3 years, 119 lbs., 1 mile, 17 yards (1.0097 
miles) in 1 min. 43 sec.=l mile in 1 min. 42-f- sec. 

1864, "Blair Athol," Epsom, " Derby," 3 years, 122 lbs., and 1861, " Kettledrum," 
3 years, 119 lbs., 1% miles, in 2 min. 43 sec.=l mile in 1 min. 4S.6-J- sec. 

1866, " Sultan," Goodwood, 4 years, 115 lbs., % mile in 1 min. 15 sec. 

American. 
1874, " Olitipa," Saratoga, N. Y., 2 years, 97 lbs., % mile, in 47% sec. 
1870, " Enchantress," Reading, Penn., aged, 100 lbs., % mile, 3d heat, in 51 sec. 
1878, "Bonnie Wood," Saratoga, N. Y., 3 years, 102 lbs., % mile, in 1 min. 2% 
sec. 

1876, "First Chance," Philadelphia, Penn., 5 years, 110 lbs., % mile, in 1 min. 
15 sec. 

1875, " Bob Wooley," Lexington, Ky., 3 years, 90 lbs., 1% miles, in 1 min. 54 sec. 

1877, "Charlie Gorham," Lexington, Ky., 3 years, 87 lbs., and 1879, "Mollie 
McCarthy," San Francisco, Cal., 6 years, 100 lbs., 1% miles, in 2 min. %% sec. 

* Whyte gives the distance in furlongs as 4, the Sporting Magazine in yarda as 103 ; Johnson the 
time as 48 seconds, and the Sporting Magazine as 40. 

Mm 



407* ANIMAL STKENGTH. 

1874, "Tom Bowling," Lexington, Ky., 4 years, 104 lbs., 1)4 miles, in 2 rain. 
34M s^. 

1S74, "Springook," Jerome Park, N. Y., 4 years, 114 lbs., 1% miles, in 2 ram. 
53 sec, and 1875, u Ten Broeck," Lexington, Ky., 3 years, 90 lbs., in 2 min. 49^ sec. 

1S77, "Courier," Louisville, Ky., 4 years, 101 lbs., 1879, "One Dime," Lexing- 
ton, Ky., 3 years, 100 lbs., and "Irish King," Louisville, Ky., 3 years, 100 lbs., 1% 
miles, in 3 ram. 5% sec 

1876, " Aristides," Lexington, Ky., 4 years, 104 lbs., 2% miles, in 3 ram. 45>£ sec. ; 
and 2% miles, in 4 ram. 21% sec. 

1875, "Preakness," aged, and "Springbok," 5 years, Saratoga, N.Y., 114 lbs., 2\ 
miles, in 3 mill. 56% sec. 

1876, " Aristides," Lexington, Ky., 4 years, 104 lbs., 2)4 miles, in 4 min. 21)4 sec. 
1876, " Ten Broeck," Lexington, Ky., 4 years, 108 lbs., 2% miles, in 4 ram. 5S}£ 

1S73, " Hubbard," Saratoga, N.Y., 4 year?, 108 lbs., 2% miles, in 4 ram. 5S% sec. 

1875, u Red Lad," Houston, Texas, 48 miles, without rests, in 5 hours 30 min. 
1S70, John Faylor, Carson City, Nevada, 50 miles, 18 horses, in 1 hour 58 ram. 33 

sec. ; and Omaha, Neb., 56 miles, in 2 hours 26 rain., including rests. 

1876, Jose Perez-, Los Angelos, Cal., 50 miles, 10 horses, changing eveiy mile, in 2 
hours 1 min. 30 sec. 

1869, Nell Coher, San Pedro, Texas, 61 miles, in 2 hours 55 min. 15 sec, including 
rests. 

1846, J. F. Tyler, Alabama, 14 years, 70 lbs., 188 miles (2 miles in a row-boat), by 
13 horses, country road, in 12 hours 30 min. 

1858, J. Powers, San Francisco, Cal., 150 miles, 25 horses, in 6 hours 43 min. 34 
• sec, including rests. 

1868, N. H. Mowry, San Francisco, Cal., race track, 160 lbs., 300 miles, by 30 horses 
(Mexican), in 14 hours 9 min. , including 40 minutes for rests; the first 200 in 8 
hours 2 min. 48 sec, and the fastest mile in 2 min. S sec 

1876, John Murphy , New York, N. Y., 155 miles, 20 horses, in 6 hours 45 min, 
7 sec 

A ustralian. 
187S, "First King," Melbourne, 3 year.*, 101 lbs., in 5 min. 20 sec. 

1877, " Great Eastern," Morrisania, saddle, 3d heat, in 2 min. 15% sec. 

French. 
1865, u Gladiateur," Doncaster, "St. Leger," Eng., 3 years, 122 lbs., 1 mile, 6 
furlongs, and 132 yards (1.775 miles), in 3 min. 20 sec.=zl% miles in 3 min. 17.2 sec, 

Arabian. 

1828, " Chapeau de Paille," India, 1% miles, 115 lbs., in 2 ram. 53 sec=l mile in 
1 min. 55.33 sec; "Patrician," 2 miles, 6 furlongs, and 160 yards (2.841 miles), 
126 lbs., in 5 min. 34 sec=3 miles in 5 min. 52.7 sec 

183-, Capt. Home, Madras to Bungalore, India, 200 miles, in less than 10 hours. 

1S69, A barb, New South Wales, 4 years, 2 miles, 14S lbs., in 3 min. 40>£ sec, and 
3 miles, 139 lbs., in 5 min. 53 sec 



TROTTING. 
One UMile. 
1796, A gelding of Mr. Jex, Denham and Norwich road, Eng., in 2 ram. 49 sec* 
1818, "Boston Blue," Boston, Mass., sulky, in less than 3 min. — the exact time is 

not now attainable. 

. 1824, " Albany Pony," Long Island, N. Y., harness, turnpike, in 2 ram. 40 sec 
1849, "Lady Suffolk," Cambridge, Mass., saddle, 7 heats, in 17 min. 43 sec 
1860, "Cora" (3 years), Louisville, Ky., harness, 2d heat, in 2 min. 37% sec 
1867, "Ethan Allen" and running mate, Long Island, N.Y., wagon, in 2 min, 15 

sec, 2 min. 16 sec, and 2 min. 19 sec ; 3d quarter of 3d heat, in 31 sec. 
1869, " Dexter,"t Prospect Park, L. I., road-wagon, driver and wagon 319 lbs., in 

2 min. 2\% sec. 

* Sporting Magazine, London, vol. ix., p. 46. 
t Public performance, but not recorded. 



ANIMAL STRENGTH. 408 

1S72, " Lady Stout" (1 year), Lexington, Ky., harness, in 3 min. 4% sec. 
1S72, " Doble" (2 years), Lexington, Ky., harness, in 2 min. 40% sec. 

1576, " Goldsmith Maid," Hartford, (Jonn., harness, 5th and 6th heats in 2 min. 
18 sec, and 2 min. 19% sec. 

1876, "Smuggler," stallion, Hartford, Conn., harness, in 2 min. 15% sec, 2 min. 
17 sec, and 2 min. 16% sec. ; and Philadelphia, Penn., 4 heats, in 2 min. 17% sec, 
2 min. IS sec, 2 min. 17 sec, and 2 mm. 20 sec.:=9 mm. 12% sec. 

1879, u St. Julien," Oakland, Cal., harness, 1st and 2d heats, in 2 min. 12% sec, 
and 2 mm. 15% sec. 

1878, "Hopeful," Chicago, 111., wagon, in 2 mm. 16% sec, 2 mm. 17 sec, and 2 
win. 17 sec 

1577, u Great Eastern," Fleetwood, N. Y., saddle, 3d heat, in 2 mm. 15% sec. 

1878, "Rarus," Buffalo, N.Y., harness, 2d heat, in 2 mm. 13% sec; Hartford, 3 
heats, in 2 min. 15 sec, 2 min. 13% sec, and 2 mm. 13% sec=6 min. 42% sec, 
and 1879, Rochester, N. Y., 3d heat, in 2 min. 13% sec 

1879, u Steinway" (3 years), Lexington, Ky., harness, 150 lbs., in 2 min. 25% sec 
1879, "Trinket" (4 years), Louisville, Ky., harness, 150 lbs., 4th heat, in 2 min. 

19% sec 

Two IMiles. 
1852, "Tacony," Long Island, N. Y., saddle, in 5 min. 2 sec. 
1859, "Flora Temple," Long Island, N. Y., harness, in 4 min. 50% sec 
1863, " General Butler," Long Island, N. Y., wagon, in 4 min. 56% sec. 
1865, " Dexter," Long Island, N. Y., wagon, 2d heat, in 4 min. 56% sec 

Three UVXiles. 
1S39, "Dutchman," Beacon Course, N. J., saddle, in 7 min. 32% sec 
1861, " Flora Temple," Long Island, N. Y., wagon, in 7 min. 47 sec 
1861, " Ethan Allen" and running mate, Long Island, N. Y., wagoo, in 7 min. 3% 
sec, and 1 mile in 2 min. 19% .sec. 

1868, "Longfellow," Sacramento, Cal., wagon, in 7 min. 53 sec 
1872, "Huntress," Long Island, N.Y., harness, in 7 min. 21% sec 

Four HVEiles. 
1828, "Top Gallant," Philadelphia, Penn., saddle, 4th heat, in 12 min. 15 sec 

1836, "Dutchman," Long Island, N.Y., saddle, in 10 vtin. 51 s c. 
1S49, "Trustee," Long Island, N.Y., harness, in 11 min. 6 sec 

1869, "Longfellow," San Francisco, Cal., wagon, 2d heat, in 10 min. 34% sec. 

Five IVIiles. 

1837, "Dolly," Long Island, N.Y., wagon; driver and man 310 lbs., in 16 min. 
45 sec 

1S63, "Little Mac," Long Island, N. Y., wagon, in 13 min. 43% sec 
1874, " Lady Mac," San Francisco, Cal., harness, in 13 min. 

Various Distances and. Performances. 
1S65, "Young Pocahontas," Long Island, N. Y., % mile, harness, in 33 sec 



1853, " Kemble Jackson," Long Island, N. Y., 1 mile, wagon, 250 lbs., in 8 min. 
3 sec 

1865, " Mountain Maid," Long Island, N. Y., 1 mile, wagon, 205S lbs., in 3 min. 
24% sec 

1860, " Lady Palmer," Long Island, N.Y., 2 miles, wagon and driver 335 lbs., 2d 
heat in 5 min. 7 sec 

1850, " Sally Green," Long Island, N. Y., 4 miles, wagon, 255 lbs., in 13 min. 56 sec. 

1830, "Whalebone," Long Island, N.Y., 6 miles, saddle, in 18 van. 52 sac. 

1878, "Controller," San Francisco, Cal., 10 miles, harness, in 27 min. 27% sec, 
and 20 miles, wagon, in 58 min. 57 sec 

1875, " Steel Grey," Yorkshire, Eng., 10 miles, saddle, in 27 min. 56% sec. 

1867, "John Stewart," Boston, Mass., half-mile track, 20 miles, harness, in 58 
min. 5% sec, and 20% miles in 59 min. 31 sec ; 1S08, half-mile track, 10 miles, 
wagon, in 28 min. 2% sec. 

* Public performances, but not recorded. 



408* ANIMAL STRENGTH. 

1S30, M Top Gallant," Philadelphia, Penn., 12 miles, harness, in 38 min. 

1S29, "Tom Thumb," Sunbury Common, Kng., 16>£ miles, harness, 248 lbs., in 56 
min. 45 sec, and 100 miles in 10 hours 7 min., including 37 min. in rests. 

1800, " Phenomenon," Cambridge and Huntingdon road, Eng., 12 years, saddle, 
feather (70 lbs.), 17 miles, in less than 53 min. 

1823, " Boston Blue," , Eng, 18 miles, harness, in 1 hour. 

1869, "Morning Star," Doncaster, Eng., 18 miles, harness (sulky 100 lbs.), in 57 
min. 27 sec. 

1833, " Paul Pry," Long Island, N.Y., IS miles 36 yards, saddle, 13S lbs., in 5S min. 
52 sec. 

1S4S, "Trustee," Long Island, N.Y., 20 miles, harness; sulky and driver 295 lbs., 
in 59 min. 35% sec. 

1831, "Chancellor," Philadelphia, Penn., 32 miles, saddle, 90 lbs., in 1 hour 5S 
min. 31 sec. 

1831, " Whalebone," Philadelphia, Penn., 32.3 miles, harness, in 1 hour 5S min. 
5 sec. 

1S32, "Rattler," Newmarket road, Eng., 34 miles, saddle, 154 lbs., in 2 hours 18 
min. 56 sec. 

1846, "Ariel," Albany, N. Y., 50 miles, harness, driver 60 lbs., in 3 hours 55 min. 
40% sec. 

1835, " Black Joke," Providence, R. I., 50 miles, saddle, 175 lbs., in 3 hours 57 min. 

1855, "Spangle," Long Island, X.Y., 50 miles, wagon and driver 400 lbs., in 3 
hours 59 min. 4 sec. 

1S37, "Mischief," Jersey City, N. J., to Philadelphia, Penn., 84^ miles, harness, 
very hot day and sandy road, in S hours 30 min. 

1853, " Conqueror," Long Island, N. Y., 100 miles, harness, in S hours 55 min. 53 
sec. , including 15 short rests. 

1845, "Fanny Jenks," Albany, N.Y., 101 miles, harness, in 9 hours 42 min. 57 
sec, including 18 min. 27 sec in rests. 101st mile in 4 min. 23 sec 

17S3, 5. Halliday, Leeds to York, Eng., and return, 110 miles, saddle, 196 -j- lbs., in 
less than 18 hours. 

1S73, M. Delaney^s mare, St. Paul's, Minn., 200 miles, race track, harness, in 44 
hours 20 min., including 15 hours 49 min. in rests. 

DOUBLE TEAMS. 

1867, "Bruno" and "Brunette," Long Island, N. Y., % mile, road-wagon, in 1 
min. 10% sec, and 1 mile, in 2 min. 2b% sec* 

1854, "Cinderella" and "Tom Wonder," Long Island, N. Y., 1 mile, wagon, 2d 
heat in 2inm. 32 sec. 

1862, "Lady Palmer" and "Flatbush Maid," Long Island, N. Y., 1 mile, road 
wagon, in 2 min. 26 sec, 2 miles in 5 min. 1% sec ; 2d quarter of 2d mile, in 33 sec* 

1870, "Idol" and " Kirk wood," Prospect Park, L. I.,l mile, wagon, in 2 min. 29 sec. 
1870, "Kirkwood" and "License," Prospect Park, L. I., 1 mile, wagon, 4th and 

5th heats, in 2 min. 28% sec each. 

1870, "Jesse Wales" and "Darkness," Narragansett Park, R. L, 1 mile, wagon, 
3d heat, in 2 min. 21% sec. 

1834, "Master Burke" and "Robin," Long Island, N. Y.,100 miles, wagon, in 10 
hours 17 min. 22 sec, including 2S min. 34 tec. in rests. 



PACING. 

One IVIile. 

1855, " Pocahontas," Long Island, N. Y., wagon and driver 265 lbs., in 2 min. 11% 
sec 

1S6S, "Billy Boyce," Buffalo, N. Y., saddle, in 2 min. 14% sec, 3d heat, in 2 min. 
14% sec. 

1879, "Sleepy Tom," Chicago, 111., harness, 5th heat. in 2 min. 12% sec. 

1ST9, " Rowdy Boy," Rochester, N. Y., harness, 1st and 6th heats, in 2 min. \Z% 
sec, and 2 min. 14 sec. 

Two IVIiles. 

1850, "James K. Polk," Philadelphia, Penn., saddle, 2d heat, in 4 min. 57% sec 

1853, " Hero," Long Island, N. Y., harness, in 4 min. 56% sec 

1859, " Young America," San Francisco, Cal., wagon, 2d heat, in 4 min. 58% sec 

* Public performances, but not recorded. 



ANIMAL STRENGTH. 409 



Three IMiles. 

1843, " Oneida Chief," Hobokeu, N. J., saddle, in 7 min. 44 sec. 

1847, "James K. Polk," Long Island, N. Y., harness, in 7 min. 44 sec. 

1852, " Pet," Long Island, N. Y., wagon, 2d heat, in 7 min. 59% sec. 

Ten. IMiles. 

1860, " Mary Miller," Maysville, Cal., in less than 30 min. 

TANDEM. 

1567. " Kingston" and mate, Providence, R. I., 1 mile, in 2 min. 37 sec. 
1S39, Burke's team, Bromwich Road, Eng., 45 miles, in 2 hours 55^ sec. 

STAGE-COACHING, ETC. 

1750, Four horses, by the Duke of Queensberry, Newmarket, Eng., 19 miles, in 53 
min. 24 sec. 

1789, Messrs. Bush and Matthews, of London, Eng., a post-chaise and pair, from 
London to Bath, 108 miles, in 8 hours 40 min. 

, London to Cambridge, Eng., 52 miles, in 3 hours, including rests. 

, Leeds to London, Eng., 201 miles, in 13 hours 34 min. 

, Dover to London, Eng., Express, 72 miles, in 5 hours 15 sec. 

1830, London to Birmingham, E7ig., " Tally-ho," 109 miles, in 7 hours 50 min., 
including stop for breakfast of passengers. 

SLEIGHING. 

1568, u Black Maria," Providence, R. L, to Boston, Mass., 42 miles, in 3 hours 25 
min. 

LEAPING.* 

Horse. 

1752, Sir C. Turner, Fell near Richmond, Eng., 10 miles, in 36 min., making 40 
leaps of 4 feet 4 ins. in height. 

1S21, A horse of Mr. Mane, at Loughborough, Leicestershire, Eng., 173 lbs., over a 
hedge 6 feet in height, 35feet.i 

1821, A horse of Lient. Green, Third Dragoon Guards, at Inchinnan, Eng., ridden 
by a heavy dragoon, over a wall 6 feet in height and lfoot in width at top. 

1S39, "Lottery," Liverpool, Eng., over a wall, 33 feet. 

1847, " Chandler," Warwick, Eng., over water, 37 feet. 

18 — , " Emblem," Birmingham, Eng., 36 feet 3 ins. 

, ''King of the Valley," Leicestershire, Eng., over Wissendine brook, 35 feet. 

NOTE. — The maximum stride of a horse is estimated to be 28 feet 9 ins. ; "Eclipse" has covered 
25 feet. The maximum stride of an elk is 3-1 feet, and of an elephant 14 feet. 

FLYING. 

VELOCITY OF THE FLTGIIT OF KIEDS PER IIOUE. 

Vulture. . 150 miles ; Wild Goose and Swallow. . 90 miles ; Crow 25 mile3. 

PIGEON FLYING. 

18 — , Carrier Pigeon, 950 miles, in 14 hours— 68 miles per hour. 

1S6S, Carrier Pigeons (6), Eng., ISO miles, in 3 hours 32 min. 

1870, Carrier Pigeons, Pesth to Cologne, Germany, 600 miles, in 3 hours. 

1S75, Carrier Pigeon, Dundee Lake to Paterson, N. J., 3 miles, in 3 min. 24 sec. 

COUKSING AND CHASING. 
A Greyhound and Hare have ran 12 niilos in 30 min. 
1794, A Eox, at Brende, Eng., ran 50 miles in 6^ hours. 
A Greyhound, at Busby Park, Eng., leaped over a brook 30 feet 6 ins. 

* A Salmon can leap a dam 14 feet in height.— Sporting Magazine, London, vol. xii., p. 79. 
t Sporting Magazine, London, vol. ix., p. 143. 

M Ttf* 



410 CENTRAL FORCES^ 






CENTRAL FORCES. 

All bodies moving around a centre or fixed point have a tendency to 
fiy off in a straight line : this is termed Centrifugal Force ; it is opposed 
to a Centripetal Force, or that power which maintains a body in its curvi- 
lineal path. 

The Centrifugal Force of a body, moving with different velocities in the 
same circle, is proportional to the square of the velocity. Thus the cen- 
trifugal force of a body making 10 revolutions in a minute is 4 times as 
great as the centrifugal force of the same body making 5 revolutions in a 
minute. Hence, in equal circles, the' forces are inverseh' as the squares 
of the times of revolution. 

If the times are equal, the velocities and the forces are as the radii of 
the circle of revolution. 

The squares of the times are as the cubes of the distances of the centrif- 
ugal force from the axis of revolution. 

The centrifugal forces of two unequal bodies, having the same velocity, aud at the 
same distance from the central body, are to one another as the respective quantities 
of matter in the two bodies. 

The centrifugal forces of two bodies, which perform their revolutions in the same 
time, the quantities of matter of which are inversely as their distances from the cen- 
tre, are equal to one another. 

The centrifugal forces of two equal bodies, moving with equal velocities at differ- 
ent distances from the centre, are inversely as their distances from the centre. 

The centrifugal forces of two unequal bodies, moving with equal velocities at dif- 
ferent distances from the centre, are to one another as their quantities of matter, 
multiplied by their respective distances from the centre. 

The centrifugal forces of two unequal bodies, having unequal velocities, and at 
different distances from their axes, are. in the compound ratio of their quantities of 
matter, the squares of their velocities, and their distances from the centre. 

To Compute tlie Centrifrigal Force of* any- Body. 

Rule. — Divide its velocity in feet per second by 4.01, also the square 
of the quotient by the diameter of the circle ; this quotient is the centrifu- 
gal force, assuming the weight of the body as 1. Then this quotient, mul- 
tiplied by the weight of the body, will give the centrifugal force required. 

Example — What is the centrifugal force of the rim of a fly-wheel having a diam- 
eter of 10 feet, and running with a velocity of 30 feet per second ? 

30 -^ 4.01 == 7.43, and 7.4S2H- 10 = 5.53, or the times the weight of the rim. 

Note — The diameter of a fly-wheel should be measured from the centres of grav- 
ity of the rim. 

When great accuracy is required, ascertain the centre of gyration of the 
body, and take twice the distance of it from the axis for the'diameter. 

Rule 2. — Multiply the square of the number of revolutions in a minute 
by the diameter of the circle of the centre of gyration in feet, and divide 
the product by the constant number 5217 ;. the quotient is the centrifugal 
force when the weight of the body is 1. Then, as in the previous Rule, 
this quotient, multiplied by the weight of the bod}-, is the centrifugal force 
required. 

Example— What is the centrifugal ftrce of a grindstone weighing 1200 lbs., 42 
inches in diameter, and turning with a velocity of 400 revolutionsin a- minute? 

Centre of gyration = rad. (42 -f- 2) X.7071 = 14. S5 ins. , which -4-.12, and X 2 = 2.475 

40Q2yo 475 

feet = the diameter of the circle of gyration. Then — -^ X 1200 = 91030 lbs. 

5217 

Or, let v represent velocity of body in feet "per second, w weight of body, r radius 
of circle of revolution in feet, and c centrifugal force. 

mi v^Xw v^Xiv cX 32. 166 Xr a //rX?>2.1G6xc 

ThCn 5S53K= e; 5^105 = ''' v =W > ^ ( w = U 



CENTRE OF GYRATION". 411 

Example. — If the diameter of a grindstone is 45.254 ins., its weight 1200 lbs., and 
it revolves 1146 times in a minute, what is its centrifugal force? 

Centre of gyration --=22. G27X.T071, which -^12. and x2 =2.666 feet. 

2.666X3.1416X1146 .... . ' 160 2 xl200 TmiQ7Y0 ; ; 
— = 160//. velocity per second. Then —— — — 7011S7.72Z&S. 

Ex. 2. — If a fly-wheel, 12 feet in diameter and 3 tons in weight, revolves in 8 sec- 
onds, and another of like weight revolves in 6 seconds, what should be the diameter 
of the second when their centrifugal forces are equal ? and what would be the ratio 
of the weights of these wheels, their forces being equal? 

Note. — The centrifugal forces of two bodies are as the radii of the circles of revo- 
lution directly, and as the squares of the times inversely. 

12 x 12 X6 2 12x36 

Then 3:3:: — : — ; or x — — r- — .== — — — == 6.75 feet, x representing the un- 
b 2 o J o J o4 

known element. 

Note. — The centrifugal forces of two bodies, when the weights are unequal, are 
directly as the squares of the times. 

Then3: x : : 6 2 : 8 2 , 6ri? = ?^=^^ = 5.333 tons. 
b 2 36 

Fly-wheels For Rules for weights of, and Examples, see Steam-engine, page 416. 



CENTRE OF GYRATION 



The Centre of Gyration is that point in any revolving body or system 
of bodies in which, if the whole quantit}' of matter were collected, the 
angular velocity would be the same ; that is, the momentum of the body or 
sj'stem of bodies is centred at this point, and the position of it is a mean 
proportional between the centres of oscillation and gravity. 

If a straight bar of uniform dimensions was struck at this point, the 
stroke would communicate the same angular velocity to the bar as if the 
whole bar was collected at that point. 

The Angular Velocity of a body or system of bodies is the motion of a 
line connecting any point with the axis of motion, and is the same in all 
parts of the same revolving system. 

When a body revolves on an axis, and a force is impressed upon it suf- 
ficient to cause it to revolve on another, it will revolve on neither, but on 
a line in the plane of the axes, dividing the angle which they contain ; so 
that the sine of each part will be in the inverse ratio of the angular ve- 
locities with which the bodies would have revolved about these axes sep- 
arately. 

The weight of the revolving body, multiplied into the height due to the 
velocuy with which the centre of gyration moves in its circle, is the ener- 
gy of the body, or the mechanical power which must be communicated to 
it to give it that motion. 

To Compute tlie Elements of GJ-yration. 

GxWXv _ PXrX*X32.166 "> GxWxd 

- = P. == = ix. 



rXJX32.166 WXv PX^X 32.166 

PXrX*X32.166 - GxWXd PXrX*X32.1G6 

zn XV m t := V ' 

GXv PXrx32.166 ' GxW ' 

G representing the distance of the centre of gyration from the axis of rotation, W 
the weight of the body, P the power acting upon the body, t the time the power acts 
in seconds, v the velocity in feet per second acquired by the revolving body in that 
time, and r the distance of the point of application of the power from the axis of the 
body, as the length of the crank, etc. 



412 CENTRE OF GYRATION. 

Illustration.— What is the distance of the centre of gyration in a fly-wheel, the 
power being 2-4 lbs., the length of the crank 7 feet, the time of rotation 10 seconds, 
the weight of the wheel 56l>0 lbs., and the velocity of it S feet per second? 
224x1x10x32.166 504373 „ 

5o00^8 = SW!=T tLT8 ^' 

2.— "What should he the weight of a fly-wheel making 12 revolutions per minute, 
its diameter S feet, the power applied at 2 feet from its axis S4 lbs., the time of rota- 
tion 6 seconds, and the distance of the centre of gyration of the wheel 3.5 feet? 
SX3.1416X12 S4x?x6v3'> 166 

■ g^ = 5.0265 feel = velocity. Then g^^g = 1S29.9 lbs. 

To Compute tlte Centre of GJ-j-ratioii. 
When the Body is a Compound one. 
Rule.— Multiply the weight of the several particles or bodies by the 
squares of their distances in feet from the centre of motion or rotation, 
and divide the sum of their products by the weight of the entire mass ; the 
square root of the quotient will give the distance of the centre of gyration 
from the centre of motion or rotation. 

Example. — If two weights, of 3 and 4 lbs. respectively, be laid upon a lever (which 
is here assumed to be without weight) at the respective distances of 1 and 2 feet, 
what is the distance of the centre of gyration from the centre of motion (the fulcrum) ? 

3X I 2 = 3 ; 4X22 = 16 ; |±-^ = ~ = 2.71, and -^2.71 = \Mfeet. 

That is, a single weight of 7 lbs., placed at 1.64 feet from the centre cf motion, and 
revolving in the same time, would have the same momentum as the two weights in 
their respective places. 

When the Centre of Gravity is given. 

Rule.— Multiply the distance of the centre of oscillation, from the cen- 
tre or point of suspension, by the distance of the centre of gravity from 
the same point, and the square root of the product will give the distance 
of the centre of gyration. 

Example. — The centre of oscillation of a body is 9 feet, and that of its gravity 4 
feet from the centre of rotation or point of suspension ; at what distance from this 
point is the centre of gyration? 

9x4 = 36, and V36 = 6/e^. 

To Compute the Centre of Grx-ration of a "Water- wheel. 

Rule. — Multiply severally twice the weight of the rim, as composed of 
buckets, shrouding, etc., and twice that of the arms and that of the water 
in the buckets (when the wheel is in operation) by the square of the radius 
of the wheel in feet ; divide the sum by twice the sum of these several 
weights, and the square root of the product will give the distance in feet. 

Example.— In a wheel 20 feet in diameter, the weight of the rim is 3 tons, the 
weight of the arms 2 tons, and the weight of the water in the buckets 1 ton ; what is 
the distance of the centre of gyration from the centre of the wheel? 

Kim =3tonsxl02x2z=600 3 + 2 + 1x2 = 12. 

Buckets = '2 tons x 102x2 = 400 

Water ±= 1 ton X102 . . = mo TT /1100 ^ 

— Hence / — = VM.Gi = 9 51 feet. 

The following are the distances of the centres of gyration from the cen- 
tre of motion in various revolving bodies, as given by Mr. Farey : 

In a straight, uniform Rod or Cylinder, revolving about one end ; length 
of rod x.5773, and revolving about its centre ; length x.2886. 

In a Circular Plane, revolving on its centre ; radius of the circle X .7071 ; 
revolving about one of its diameters as an axis ; radius x.o. 



CENTRES OF OSCILLATION AND PERCUSSION. 413 

In a Wheel of uniform thickness, or in a Cylinder revolving about its 
axis ; radius x .7071. 

In a Solid Sphere, revolving about one of its diameters as an axis ; ra- 
dius x. 6325. 

In a thin, holloiv Sphere, revolving about one of its diameters as an axis ; 
radius x .8164. 

In a Sphere, at a distance from the axis of revolution = V l 2 -\- % r 2 , I 
representing the length of the connection to the centre of the sphere ; and in a 
Cylinder = V> + | ~r*. 

In a Cone, revolving about its axis; radius of base x.5447; revolving 
about its vertex= v / (12/i 2 + 3r 2 -^-20), h representing the height, and r ra- 
dius of the base ; revolving about its base= V (2 A 2 + 3 r 2 H-20). 

In a Circular Ring, as the Rim of a Fly-wheel, revolving about its diam- 
eter = -/R 2 -[- r 2 -r- 2, R representing radius of periphery of ring. 

* ™ , , /6W(R 2 + r 2 ) + ^(4r- 2 + ^ 2 ) ™ 7 

In a Fly-wheel = w 19 TW 4- — ^ ' a w re P resentin 9 

the weights of the rim and of the arms and hub, and I the length of the arms 
from the axis of the wheel. 



m 3 _j_ r 3 
In a Straight Lever = w ^— :. 



Illustration. — In a solid sphere revolving about its diameter, the diameter being 
2 feet, the distance of the centre of gyration is 12x.6325:= 7.59 inches. 

General Formulae. — Let P represent power, H horses' power, F the force applied 
to rotate the body in lbs , M mass of the revolving body in lbs., r radius upon which 
F acts in feet, d distance from axis of motion to centre of gyration in feet, t time the 
force is applied in seconds, n number of revolutions in time t. v angular velocity, or 

32.166 Fr2 
number of revolutions per minute at the end of time t, and G == - 



■J- 



Md2 
4prn_ . 2pmx_ ^ Mad* _ 



G * COG ? 153.5 tr 

Mnd2 2.56 PYr 153.5 tFr 



2.56«2F"~ ? Mrf2 — > Md2 ~~ ' 

244£P_ x*Md2 __ x* M rf2 _ 

a?2 d2 — • 244 1 ~~ ; 1341007 ~~ " * 

Illustration. — The rim of a fly-wheel weighing 7000 lbs. has radii of 6.5 and 
$.75 feet ; what is its centre of gyration, and what force must be applied to it 2 feet 
from the axis of motion to give it an angular velocity of 130 revolutions per minute 
in 40 seconds ? and how many revolutions will it make in 40 seconds ? 

„ 1 , ,. 76.52 + 5.752 130X7000X6.142 

Centreof gyration^ = 6.14 /erf. Then F= ^^^ =■ 

34406636 nort „ Tt ,2.56x402x2802x2 M Aw 

-T2280- = 28 ° 2 "»" aM -^0 00X6.142 = 86 ' 97 ^voluUons. 



CENTRES OF OSCILLATION AND PERCUSSION. 

The Centric of Oscillation of a body, or a system of bodies, is that 
point in the axis of vibration, of a vibrating body, in which, if the whole 
matter of the body were collected, and it was acted upon b}^ a like force 
to that acting upon the body, it would, if suspended or supported from the 
same axis of motion, perform its oscillations or vibrations in the same 
time and with the same angular velocity. 



414 CENTRES OF OSCILLATION AND PERCUSSION". 

It is in a right line passing through the centre of gravity of the body, 
and perpendicular to the axis of motion. 

The angular velocity of a body or system of bodies is the motion of a 
line connecting any point and the centre or axis of motion : it is the same 
in all parts of the same revolving body. 

In different unconnected bodies, each oscillating about a common cen- 
tre, their angular velocity is as the velocity directly, and as the distance 
from the centre inversely. Hence, if their Velocities are as their radii, or 
distances from the axis of motion, their angular velocities will be equal. 

The Centre of Percussion of a body, or a system of bodies, revolv- 
ing about a point or axis, is that point at which, if resisted by an immov- 
able obstacle, all the motion of the body, or system of bodies, would be 
destro} T ed, and without impulse on the point of suspension. 

The Centres of Oscillation and Percussion are in the same point. 

As in bodies at rest, the whole weight may be considered as collected in 
the centre of gravity; so in bodies in vibration, the whole force may be 
considered as concentrated in the centre of oscillation ; and in bodies in 
motion, the whole force may be considered as concentrated in the centre 
of percussion. 

If the centre of oscillation is made the point cf suspension, the point of 
suspension will become the centre of oscillation. 

The Angle of Oscillation or Percussion is determined by the angle delin- 
eated by the vertical plane of the body in vibration, in "the plane of mo- 
tion of the body. 

The Velocity of a Body in Oscillation or Percussion through its vertical 
plane is equal to that acquired 03- a bod}- freely falling through a vertical 
line equal in height to the versed sine of the arc. 

The centre of percussion is also that point of a revolving body which 
would strike any obstacle with the greatest effect, and from this property 
it has received the name of percussion. 

To Compute the Centre of Oscillation or 3?ercu.ssicn of a 
Body of Uniform Density and. Figure. 

Rule. — Multiply the weight of the bod}- b} T the distance of its centre 
of gravity from the point of suspension; multiply also the weight of the 
body by "the square of its length, and divide the product by 3. 

Divide this last quotient by the product of the weight of the body and 

the distance of its centre of gravhy, and the quotient is the distance of 

the centres from the point of suspension. 

WxZ 2 
Or, — rWxg'^ distance from axis. 

Example. — Where is the centre of oscillation in a rod 9 feet in length from its 
point of suspension, and weighing 9 lbs. ? 

9 9x9 2 

9x,: = 40.5 =: product of the weight and its centre of gravity ; — : — =: 243 = quo- 
& '6 

tient of product of weight of body and the square of its length -s- 3; -— — =z6feet, 

Ir*oint of Centres of Oscillation and Percussion in T3odies 
of Various Figures. 

When the Axis of Motion is in the Vertex of the Figure. 

When the Oscillation or Motion is Facewise. 

1. In a Right Line, on any figure of uniform shape and density = .60 1. 

2. In an Isosceles Triangle = .75 h. 3. In a Circle = 1.25r. 
4. In a Parabola = .714 h. 



CENTRES OF OSCILLATION AND PERCUSSION - . 415 

When the Oscillation or Motion is Sideuise. 

1. In a Right Line, or any figure of uniform shape and density = .66 I. 

2. In a Circle = .75 d. 

3. In a Rectangle, suspended at one angle = .66 of diagonal. 

4. In a Parabola, if suspended by its vertex = .714 of axis -(-.33 parameter; if 
suspended by the middle of its base — .57 of axis -f- .5 parameter. 

3 X arc X r 

5. In a Sector of a Circle = — — — , c representing chord of the arc, and r 

4 X C 

4 r 2 

the radius of the base. 6. In a Cone — — axis -f- - . . 

J 5 ' 5 X axis 

2Xr 2 
7. In a Sphere = ^ 4" r ~l~ c > c representing the length of the cord by which 

o (c + r) 

it is suspended. 

To .A-scertain tlie Centre of Oscillation, and. Percussion ex- 
perimentally. 

Suspend the body very freely by a fixed point, and make it vibrate in small arcs, 
counting the number of vibrations it makes in a minute, and let the number of vi- 
brations made in a minute be called n; then will the distance of the centre of cscil- 

, . . ^ 140S50 . ; 
lation from the point of suspension be = — - inches. 

For the length of a pendulum vibrating seconds, or 60 times in a minute, being 
39% inches, and the lengths of the pendulums being reciprocally as the squares of 

the number of vibrations made in the same time, therefore n 2 : 60 2 : : 39% : — 

n 2 

= — , being the length of the pendulum which vibrates n times in a minute, or 

n 2 
the distance of the centre of oscillation below the axis of motion. 
Illustration. — Where is the centre of percussion of a rod 23 inches in length? 

.66 of 23 = 15.18 inches. 
2. — Tn a sphere 10 inches in diameter, suspended by a cord 20 inches in length, 
where is the centre of percussion or oscillation ? 
•2 v 52 50 

5(W+5, + 5 + 20 = aT5 + £5 = 25 - 4 ^- 

To Compute the Centres of Oscillation or Percussion of* a 
System ofiParticles or Bodies. 

Rule. — Multiply the weight of each particle or bod}- by the square of 
its distance from the point of suspension, and divide the sum of their 
products by the sum of the weights, multiplied by the distance of the 
centre of gravity from the point of suspension, and the quotient will give 
the centre required measured from the point of suspension. 

Or, — , , lr , == distance of centre. 

Example. —The length of a suspended rod being 20 feet, and the weight of a foot 
in length of it equal 100 oz., has a ball attached at the under end weighing 100 oz., 
at what point of the rod from the point of suspension is the centre of percussion? 

20 
100x 20 == 2C00 — weight of rod ; 2000X -^ = 20000 — momentum of rod, or product 

2000 x20 2 
of its weight, and distance of its centre of gravity; — — = 26Q6QG.GQ = 

force of rod ; 1000x202 == 40COOO = force of ball. 
2G6666.66 + 400000 *„'-* 
Th6D 80000 + 20000 = 16 -<W 
l-lbXl 2 -\-ct 2 

Or, ~ = centre of percussion, etc. ; I representing length of rod, b weight 

J lbXl-\-lc 

of a foot in length of rod, and c weight suspended from end. 



416 FLY-WHEELS. 

Ex. 2. — Assume a rod 12 feet In length, and weighing 2 lbs. for each foot of its 
length, with 2 balls of 3 lbs. each — one fixed 6 feet from the point of suspension, and 
the other at the end of the rod; what is the distance between the points of suspen- 
sion and percussion ? 

24X12 2 3456 
12x?xhf = 144 = momentum of rod. — - — = — r— = 1152 = force of rod, 

^1 = H = I ffll^n' 3X 62=3x 36 = ™s= « oflstball. 
3X12 = J6 = of U ball. 3xl2 2 = 3 X 144=432= " ofUball. 

19S sum of moments. j^ sum offorces . 

1602 
Then — =S.545/e<tf. 






FLY-WHEELS. 

A Fly-wheel should always have high velocity. 

The diameter should be from 3 to 4 times that of the stroke of the driv* 
ing engine. 

The weight of the rim should be about 85 to 95 lbs. per actual horse- 
power, the momentum of the wheel being 4% times that of the piston. 

When the Engine to which a Fly-wheel is to be attached is single-acting. 
it is customary to make the weight of the wheel 5 times greater than 
when it is to be attached to a double-acting engine. 

The weight of a fly- wheel in engines that are subjected to irregular mo- 
tion, as in a cotton-press, rolling-mill, etc., must be greater than in others 
where so sudden a check is not experienced. 

To Compute tlie "Weight of tlie Him of* a Fly- wheel. 

Rule. — Multiply the mean effective pressure upon the piston in lbs. by 
its stroke in feet, and divide the product by the product of the square of 
the number of revolutions, the diameter of the wheel and .00023. 

Note If a light wheel is required, multiply by .0003 ; and if a heavy one, by 

.00016. 

To Compute the Dimensions of the Rim. 

Rulk. — Multiply the weight of the wheel in lbs. bj* .1, and divide the 
product by the mean diameter of the rim in feet ; the quotient will give 
the sectional area of the rim in square inches of cast iron. 

Example.— A non-condensing engine, having a diameter of cylinder of 14 inches, 
and a stroke of piston of 4 feet, working full stroke, at a pressure of 65 lbs. per mer- 
curial gauge, and making 40 revolutions per minute, develops about 65 horses' power : 
what should be the dimensions of its fly-wheel, adapted to ordinary work ? 

Area of cylinder, 154 ins. stroke, 4X3X = 14 feet = diam. of wheel ; mean press- 
ure =50 lbs. ; 50 X 154x4 = 30S00 = product of pressure upon piston in lbs. ; and 
the stroke of the piston, which 4- 40 2 X 14 X. 00023 = 597S lbs., weight of the wheel. 

Assume the mean diameter of the wheel 13^ feet. Then 597S X .1 4- 13.25 = 
45.12 square ins. in the rim. 

Ex. 2. — If a fly-wheel, 16 feet in diameter and 4 tons in weight, is sufficient to 
regulate an engine when it revolves in 4 seconds, what should be the weight of a 
second fly-wheel, 12 feet in diameter, revolving in 2 seconds, so that it may have 
like centrifugal force ? 

Note. — The centrifugal forces of two bodies are as the radii of the circles of revo- 
lution directly, and as the squares of the times inversely. 

_, 4X16 ;rXl2 ^ 4x16x22 4x16x4 ^ ooo , 



IMPACT OR COLLISION. 417 

IMPACT OR COLLISION. 

Impact is Direct or Oblique. Bodies are Elastic or Inelastic. The 
division of them into hard and elastic is wholly at variance with these 
properties ; as, for instance, glass, which is among the hardest of bodies, 
is most elastic of all. 

Product of mass and velocity of a body is the momentum of the body. 

Principle upon which motions of bodies from percussion or collision are 
determined belongs both to elastic and inelastic bodies ; thus, there exists 
in bodies the same momentum, or quantity of motion, estimated in any 
one and same direction, both before collision and after it. 

Action and reaction are alwaj T s equal and contrary. If a body impinge 
obliquely upon a plane it rebounds at an angle equal to that at which it 
impinged, that is, angle of reflection is equal to that of incidence, and 
force of blow is as sine of angle of incidence. 

Effect of a blow of an elastic body upon a plane is double that of an in- 
elastic one, velocity and mass being equal in each ; for force of blow from 
inelastic body is as its mass and velocitj-, which latter is only destroyed 
by resistance of the plane ; but in an elastic bod} T that force is not only 
destroyed, being sustained by the plane, but another, also equal to it, is 
sustained by plane, in consequence of the restoring force, and by which 
the body is repelled with an equal velocity ; hence intensity of the blow 
is doubled. 

If two perfectly elastic bodies impinge on one another, their relative 
velocities will be the same, both before and after impact ; that is, they will 
recede from each other with same velocity with which they approached 
and met. 

Effect of collision of two bodies, as B and 6, velocities of which are dif- 
ferent, as v and v', is given in following formulas, in which B is assumed 
to have greatest momentum before impact. 

If bodies move in same direction before and after impact, sum of their 
moments before will be equal to their sum after impact. 

If bodies move in same direction before and in opposite direction after 
impact, sum of their moments before will be equal to difference of their 
sums after impact. 

If bodies move in opposite directions before and in same direction after 
impact, difference of their moments before will be equal to their sum 
after impact. 

If bodies move in opposite directions before and in opposite directions 
after impact, difference of their moments before will be equal to their 
difference after impact. 

When bodies are inelastic, their velocities after impact will be alike. 

If two bodies are imperfectly elastic, sum of their moments will be the 
same, both before and after collision ; but velocities after will be less than 
in case of perfect elasticity, in ratio of imperfection. 

To Compute "Velocities of* Inelastic Bodies after Impact. 

When Impelled in Same Direction, — — - = r. B and b representing weights of 

B -f~ b 
the two bodies, V and v their velocities before impact, and R and r velocities of bodies 

after impact, all in feet per second. Consequently, ~ v Xb = velocity lost by 

V — v B-f-6 

B, and --— - X B = velocity gained by b t 

U -J- o 

Note — In these formulas it is assumed that V > v. If V < % results will be 
negative, but may be read as positive if lout and gained are reversed in places. 

Nn 



418 IMPACT OR COLLISION. 

Illustration. — An inelastic body, 6, weighing 30 lbs., having a velocity of 3 feet, 
is struck by another body, B, of 50 lbs., having a velocity of 7 feet; velocity of b 
after impact will be 50 x 7 . 30 x 3 440 

! == =5.5 feci. 

50 + 30 80 J 

BV b v 

When Impelled in Opposite Directions, j •— = r. 

B -f- p 

Illusteation.— Assume elements of preceding case: 

50X7-30X3 ^260 

50 + 30 80 J 

B V 

When one Body is at Rest, == r. 

B + 

Illusteation. — Assume elements as preceding: 

50 X 7 350 •._! ; • 

50T30 = 80 =4 ' 75/ ^ 

To Compute "Velocities of Elastic Bodies after Impact. 

When Impelled in One Direction, B ~ bV + 2&y = R, and 2 BV— B— ftp = r. 

B+& B + & 

Illustration. — Assume elements as preceding: 
50-30 X 7 + 2 X30X_3 = 320 

50 + 30 80 J ' 

2 X 50 X 7 — 50 — 30 X 3 _ 640 _ , 
50+^ SO" - /ee# 



2& s — r_i' .i^L.^i. ::;: . 2 b 



Or, V V — i; = velocity of R ; and v + — — - V — v = velocity of r. 

B + b B+ b 

When Impelled in Opposite Directions. 

B — bVcoZbv _ , 2 B V — B — bv 

———- = R; and „ , ,_ = r. 

B + b B + & 

Illusteation. — Assume elements as preceding : 

50 — 30 X T co 2 X 30 X 3 _ 140 co 180 _ _ & - ^ . &nd 

50 + 30 2_ 80 ' ' 

2 X 50 X 7 - 50 - 30 X 3 _ 700 - 60 _ g 

50 + 30 _80 

2 & (V + ,) = _ 2X30X7 + 3 ^600 = 

1 B + 6 '50 + 30 80 

-« » ' j \f ' il f » •„« ' V B — 6 „ . 2 B V 

F7i£>i one Body is at Rest, n , , = K, and — — — - = r. 
B + B + 

Illusteation. — Assume elements as preceding: 

LX^EM = !i? = 1.75/^, and 2 X 5 ° X 7 = ™ = 8.T5/e* 

50 + 30 80 ' 50 + 30 80 

To Compute "Velocities of Imperfect Elastic Bodies after 

Impact. 

Effect of collision is increased over that of perfectly elastic, but not 

doubled, as in that case it must be multiplied by — — — , when — repre- 

' mm 

sents degree of elasticit}- relative to both perfect elasticity and inelasticity : 

Moving in Same Direction^ — m + n X — ^— (V — v) = R ; and v + m + n X 
* m B + m 

(V — v) = r, m and n representing ratio 0/ perfect to imperfect elasticity. 



PILE-DRIVING. 419 

Illustbation.— Assume elements as preceding, m and n = 2 and 1 : 
7 - 1£* X g^^ X 7~^~3 = 7 - 1.5 X |? X 4 = 7 - 2.25 = 4.75 feet ; and 

JF/iew. Moving in Opposite Directions, — * — ~^ ' °° - = R, and 

2BV — v(b — nB) __ r 



B+& 



v(B-^b) Bv(l+-^ 



When one Body is at Rest, lt , . — = K, and 

U -j- b B -\- b 

Illusteation. — Assume elements of p receding case: 

7 X (50 -5X30) = TX 50-15 62 

50 + 30 80 J ' 

«>XTxq + .5) = 350 X 1.5 6>5625/ ^ 

50 + 30 80 ' 



PILE-DRIVING. 

The effect of the blow of a ram, or monkey, of a pile-driver, is as 
the square of its velocity ; but the impact is not to be estimated di- 
rectly by this rule, as the degree and extent of the yielding of the pile 
materially affects it. The rule, therefore, is of value in application 
only as a means of comparison. 

By my experiments in 1852, to determine the dynamical effect of a 
falling body, it appeared that while the effect was directly as the ve- 
locity, it was far greater than that estimated by the usual formula 

V s2g, which, for a weight of 1 lb. falling 2 feet, would be 11.34, 
giving a momentum of 11.34 ft. lbs. ; whereas, by the effect shown by 
the record of actual observations, it would be vW 4.426 == 50 /6s. 

Piles are distinguished according to their position and purpose: thus, 
Gauge Piles are driven to define the limit of the ground to be inclosed, or 
as guides to the permanent piling. 

Sheet or Close Piles are driven between the gauge piles to form a con- 
tinuous inclosure of the work. 

The weight which is required of each pile to sustain should be computed 
as if it stood unsupported by an} T surrounding earth. 

When the length of an oak pile does not exceed 16 times its diameter, it 
may be loaded permanently with a weight of 450 lbs. per square inch of 
its sectional area. 

A heavy ram and a low fall is the most effective condition of operation 
of a pile-driver, provided the height is such that the force of the blow will 
not be expended in merely overcoming the inertia of the pile, and at the 
same time not from such a height as to generate a velocity which will be 
expended in crushing the fibres of the head of the pile. 

The refusal of a pile intended to support a weight of 133^ tons can be 
safely taken at 10 blows of a ram of 1350 lbs., falling 12 feet, and depress- 
ing the pile .8 of an inch at each stroke. 

Pneumatic Piles. — A hollow pile of cast iron, 2)^ feet in diameter, was 
depressed into the Goodwin Sands 33 feet 7 inches in 5)^ hours. 



420 PILE-DRIVING. 

Nasmyth's Steam Pile-hammer has driven a pile 14 inches square, and 
18 feet in length, 15 feet into a coarse ground, imbedded in a strong clay, 
in 17 seconds, with 20 blows of the hammer, or monkey, making 70 strokes 
per minute. 

To ComTaute "Weight tliat a 3?ile -will sustain, -with. 
Safety. 

(Maj. John Sanders, U. S. E.) 

Rule. — Divide height of fall by distance the pile has sunk by last 
blow, both in inches ; multiply result by weight of ram in lbs., and di- 
vide product by 8. 

Or, - = W. 

R Representing weight of ram in lbs., h height of fall, and d distance 
pile is depressed by the blow, both in inches. 

Illustratio n.— A ra m weighing 3500 lbs., falling 3.5 feet, depressed a pile 

_. 42-^4.2 X 3500 35000 AonK „ 
4.2 ins. Then = — - — = 4375 I bs. 

o o 

By the ordinary formula, -\/s2gW = 15 X 3500 = 52750 lbs., the 
computed force ; hence, assuming rule of Major Sanders as a guide, 

„ = .0814, which may be taken as the coefficient whereby to re- 

52750 ' J J 

duce the momentum of a ram, to the weight a pile can bear with safety. 

When. File is Driven to " Refusal." 

Rule. — Multiply weight of ram in lbs. by cube root of fall in feet and 
by a range of factors from 15 to 25, according to the character of the 
ground and the stress to be borne. 

Illtjstkation.— -Assume preceding case with a factor of 20. 
3500 X -y 7 3^5 X 20 = 106400 lbs. 

Dr. Whewell deduced : 

1. A slight increase in the hardness of a pile or in the weight of a ram 
will considerably increase the distance a pile may be driven. 

2. The resistance being great, the lighter a pile the faster it may be 
driven. 

3. The distance driven varies as the cube of the weight of the ram. 

To Compute tlie Space through ^which a 3?ile is driven. 
RA -4- C = s, C representing resistance of earth. Hence, b}^ inversion, 

To Compute Coefficient of Resistance of Earth.. 

Weisbach gives following formula : The resistance of the earth be- 
ing constant, the mechanical effect expended in the penetration of a 

R 2 h 
pile will be „ p = W. Taking elements of preceding case, with ad- 

r -f- Ks 
dition of weight of the pile at 1500 lbs., the result would be 
35 00* x 42 _ 513500000 _ 
1500 + 3500 x 4.2 ~~ 21000 ~~ 



PENDULUMS. 421 



PENDULUMS. 

Pendulums are Simple or Compound, the former being a material point, 
or single weight suspended from a fixed point, about which it oscillates, 
or vibrates, by a connection void of weight ; and the latter, a like body or 
number of bodies suspended by a rod or connection. Any such bod}^ will 
have as many centres of oscillation as there are given points of suspen- 
sion to it, ancl when any one of these centres are determined the others 
are readity ascertained. Thus, soxsg = a constant product, and sr = 
Vsoxsg, sg o and r representing the points of suspension, gravity, oscilla- 
tion, and gyration. 

Or, any body, as a cone, a cylinder, or of any form, regular or irregu- 
lar, so suspended as to be capable of vibrating, is a compound pendulum, 
and the distance of its centre of oscillation from any assumed point" of 
suspension is considered as the length of an equivalent simple pendulum. 

All vibrations of the same pendulum, whether great or small, are per- 
formed very nearly in the same time. 

The Number of Oscillations of two different pendulums in the same time 
and at the same place are in the inverse ratio of the square roots of the 
lengths of these pendulums. The Length of a Pendulum vibrating sec- 
onds is in a constant ratio to the force of gravity. 

The Times of the Vibration of pendulums are proportional to the square 
roots of their lengths. Consequently, the lengths of pendulums for differ- 
ent vibrations are as follows : 

Latitude of Washington. 

39.0958 ins. for one second. I 4.344 for third of a second. 

9.774 ins. for half a second. 2.4435 for quarter of a second. 

Lengths of JPendulums vibrating Seconds at the Level 
of the Sea in several Places. 

Ins Ins I Ins. 

Equator 39 .0152 New York .... 39. 1017 Paris 39. 1284 

Washington . . . 39.0958 | London 39.1393 J Lat. 45° 39.127 

vVl-t- g = t, I representing the length of a pendulum vibrating seconds in 

inches, g the measure of the force of gravity (32.155 at Washington, and 

32.191 at London), and t the time of one oscillation. 

Illustration. — The length of a simple pendulum vibrating seconds, and the 
measure of the force of gravity at Washington, are 3^.0C5S ins., and 32.155/<ee£. 

3.1416 / ° ' '°° _ 3.1416xVl-013z=3.1416x.31S3=:lseccmd. 
V 32.155X12 

To Compute the Length of a Simple Pendulum for a 
given Latitude. 

39.12T — .09932 cos. 2 L = I, L representing the latitude. 

Illustration. — Required the length of a simple pendulum vibrating seconds in 

the latitude of 50° 31'. 

L = 50° 31' cos. 2 L— .2x50° 31'=: cos. ISO — 50° 3l / x2 = cos. 78° 5S'=r.l913S — 

39.127 -f-7l913SX-09LS2 (the two — or negative == an affirmative or -f) = 39.1461 

ins. 

To Compute the Length of a Simple Pendulum for a 

given Number of Vibrations. 

L^2 

— - = I, L' representing the length for the latitude, t the time in seconds, and n the 

number of vibrations. 

Nn* 



422 PENDULUMS. 

To Compute the Length of a Simple 3Pend.nru.rn, trie Al- 
terations of* >vhich Avill "be trie same in ZNTumloer as tlie 
Inches in its Length. 

V (6<VL') 2 = I in inches. 

Example. — What will be the length of a pendulum in New York, the vibrations 
of which will be the same number as the inches in its length ? 

^/(v/3y.l013x6J)2 — T.2112 = 52 inches. 

To Compute the Number of ^iterations of a Simple IPen- 
dulum in a given Time. 

/L. t 

I ~t = n, - representing time of one vibration in seconds. 



Vi 



To Compute the Time of Vibration of a Simple IPenclu- 
lum, the Length "being given. 

y/l -r- L' = t in seconds. 

Example.— The length of a pendulum is 156.S inches ; what is the time of its vi- 
bration in New York ? 

156 - 8 

= 2 seconds. 



/- 



&U017 

To Compute the Measure of G-ravity, the Length of the 
Pendulum and. the Number of its Vibrations "being given. 

.82246 In* ,. _. , , ■ * * 

= g, g representing the measure of gravity in feet. 

t 2 

To Compute the Centre of GJ-ravitv of a Compound. Pen- 
dulum of Two "Weights connected in a Right Line. 

When the Weights are both on one Side of the Point of Suspension, — = 

W -\-w 

o = distance of centre of gravity from the point of suspension. When the Weights 

I W — Vw 
are on opposite Sides of the Point of Suspension, = o = distance of cen- 

tre of gravity of the greater weight from the point of suspension. 

Illustration. — A compound pendulum, composed of two weights alike to two 
cannon-balls, on the opposite sides of the point of suspension, and connected by a rod 
in a right line between them, has the following elements : 

Lengths of pendulum from centre of points of suspension 25 and 2S inches. Weights 
of balls and connections from points of suspension, for 25 inches 38 lbs., and for IS 
inches 12 lbs. 
The length of it as a simple pendulum in latitude of New York, and that number 

25x38 18x12 

of vibrations in one second, would be - = 14. 6S inches = distance of 

3S + 12 J 

. ^ M Z2W+r2u> 252X3S4-1S2X12 

centre of gravity from point of suspension ; and — — — — - = — /o _i_ -..^ — 

( \Y -j- W) 14. bo (ob -j— 1-) 

= 47.06 inches, the length of it as a simple pendulum; that is, from the point of 
suspension to a point extending below the greater weight. 



,„.>.0958X1 6.252T 
Hence . / — . .. .- = n . . = .911 vibrations. 
;.Uo o.8b 



/ 39. 095$ 

2 y v 4T - ( 



2. — If the two weights were both on the same side of the point of suspension. 
Lengths of pendulum being 15 -j- IS = 43 and IS ins. 

43 X 38 -4- IS X 12 

\,^T 19 = 46.25 ins. z= distance of centre of gravity from point of suspen* 

432 v 38 -4- 182 v"i 
sion ; and — — — — ^— — — — =40. OS ins., the lengtk of it as a simple pendulum. 
46.25X(3S-|- 12) 



GOVERNORS. — PNEUMATICS. — AERODYNAMICS. 423 



GOVERNORS. 

The operation of the Governor or Conical Pendulum depends upon the 
principles of Central Forces. 

When in a Ball Governor the Balls diverge, the ring on the vertical 
shaft raises, and in proportion to the increase of the velocity of the balls 
squared, or the square roots of the distances of the ring from the fixed 
point of the arms, corresponding to two velocities, will be as these veloci- 
ties. 

Thus, if a governor makes 6 revolutions in a second when the ring is 16 
inches from the fixed point or top, the distance of the ring will be 5.76 
inches when the speed is increased to 10 revolutions in the same time. 

For 10 : 6 : : -/16 : 2.4, which, squared = 5.76 ins., the distance of the 
ring from the top. Or, 6 2 : 10 2 : 5.76 : 16 ins. 

A governor performs in one minute half as many revolutions as a pen- 
dulum vibrates, the length of which is the perpendicular distance between 
the plane in which the balls move and the fixed point or centre of sus- 
pension. 

To Compute tlie Number of Revolutions of a Ball Gov- 
ernor per [Minute to maintain tlie Balls at any given 
Height. 

188 

-— - — revolutions, H representing the vertical height between the plane of the Balls 

and the points of their suspension in inches. 

To Compute the Vertical Height "bet-ween the Plane of 

the Balls and. their Points of Suspension. 

/188\2 

( — ) = vertical height in ins., r representing the number of revolutions in a minute. 



PNEUMATICS.— AERODYNAMICS. 

The motion of gases by the operation of gravity is the same as that for 
liquids. The force or effect of wind increases as the square of its velocity. 

If a volume of air, and of the temperature of 32°, is heated t degrees 
without assuming a different tension, the volume becomes (1+. 002088 £) 
= V; and if it acquires the temperature t', it will then assume the volume 
(1 -f.002088 1 — 32°). When air passes into a medium of less density, its 
velocity is determined by the difference of the densities. Under like con- 
ditions, a conduit will discharge 30.55 times more air than water. 

The force of wind upon a surface perpendicular to its direction has been 
observed as high as 67% lbs. per sq. foot ; velocity = 159 feet per second. 

To Compute the "Volume of .A.ir discharged, through an 
Opening into a Vacuum per Second.. 

tt C"/2 g h=iV in cubic feet, a representing area of opening in square feet, and C 
coefficient of efflux. 

To Compute the Velocity of Air in its Passage from a 
Greater to a Lesser Density. 

1347,4 a c v'M(d+M)TrVffl cubic feet, d representing density of the atmosphere, 

M the pressure in the reservoir from which it flows in inches oj mercury, and T 
= 1 + 0020SS(* — 32°). 



424 WIND-MILLS. 

Illtjstraton. — What is the volume of air at a barometric pressure of 29.T ins. 
which a reservoir will furnish, upon which a manometer indicates 1.2 ins. through 
a cylindrical pipe 3 ins. in diameter, temperature of the air 56° ? 

C = .93,T=1-|-.0020S8 (56 — 32) = 1.05, a = .05. 
Then ^jyX.05x.93xV / l-2X2y.T-|- 1.2x1.05= 2.1096X6.24 = 13.16 cubicfeet. 

To Compute tlie Resistance of a Plane Surface to the Air, 

.0022 av 2 = P, a representing area of plane in square feet, v velocity of it in the di- 
rection of the wind in feet per second, -f- when it moves opposite, and — when it 
moves with the wind. 

Dr. Hutton deduced that the resistance of air varied as the square of 
the velocity nearly, and to an inclined surface as the 1.84 power of the 
sine X cosine. 

The figure of a plane makes no appreciable difference in the resistance, 
but the convex surface of a hemisphere, with a surface double the base, 
has only half the resistance. 

At high velocities, experiments upon railwaj-s show that the resistance 
becomes nearly a constant quantity. 

The resistance of the air to a train of cars in a dead calm was found to 
be ^g of their weight. 

The velocity of a train of cars which give a resistance of the ^ of the 
load with a fair wind was 34^ miles per hour, and only 27% with an ad- 
verse wind. 

To Compute the Resistance of* a Plane Surface when 
moving at an .A-ngle to the -A_ir. 

v 2 a sin. 2 x _ . , ./ , : l •':' '-. , 
7j-r = P in pounds, x representing the angle of incidence. 

For other elements, etc., see Treatise on Aerometry of D'Aubuisson de Voissin, 
pages 124, 1S6, 234, and .313, vol. xxxix. Journal of Franklin Institute. 

WIND-MILLS. 

The driving shaft of a wind-mill should be set at an elevating angle 
with the horizon when set upon low ground, and at a depressing angle 
when set upon elevated ground. The range of these angles is from 3° to 
35°. A velocity of wind of 10 feet per second is not generally sufficient 
to drive a loaded wind-mill, and if the velocity exceeds 35 feet per sec- 
ond the force is generally too great for the ordinary structure. 

The angle of the sails should be from 18° to 30° at their least radius, 
and from 7° to 17° at their greatest radius, the mean angle being from 
15° to 17° to the plane of motion of the sails. The length of an arm (whip) 
is divided into 7 parts, the sails extending over 6 parts. 

Deductions from. "Velocities varying from 4 to 9 Feet per 
Second..— [Mr. Smeaton.] 

1. The velocity of wind-mill sails, so as to produce a maximum effect, is 
nearly as the velocity of the wind, their shape and position being the same. 

2. The load at the maximum is nearly, but somewhat less than, as the 
square of the velocity of the wind, the shape and position of the sails be- 
ing the same. 

3. The effects of the same sails, at a maximum, are nearty', but some- 
what less than, as the cubes of the velocity of the wind. 

4. The load of the same sails, at the maximum, is nearly as the squares, 
and their effect as the cubes of their number of turns in a given time. 



WIND-MILLS. 425 

5. When sails are loaded so as to produce a maximum effect at a given 
velocity, and the velocity of the wind increases, the load continuing the 
same— 1st, the increase of effect, when the increase of the velocity of the 
wind is small, will be nearly as the squares of those velocities ; 2dly, when 
the velocity of the wind is double, the effects will be nearly as 10 to 27>£ ; 
but, Sdlj-, when the velocities compared are more than double of that when 
the given load produces a maximum, the effects increase nearly in the 
simple ratio of the velocity of the wind. 

6. In sails where the figure and position are similar, and the velocity 
of the wind the same, the number of revolutions in a given time will be 
reciprocally as the radius or length of the sail. 

7. The load, at a maximum, which sails of a similar figure and position 
will overcome at a given distance from the centre of motion, will be as 
the cube of the radius. 

8. The effects of sails of similar figure and position are as the square of 
the radius. 

9. The velocity of the extremities of Dutch sails, as well as of the en- 
larged sails, in all their usual positions when unloaded, or even loaded to 
a maximum, is considerably greater than that of the wind. 

Results of Experiments on tlie Effect of "Wind-mill Sails. 

When a vertical wind-mill is employed to grind corn, the mill-stone 
usually makes 5 revolutions to 1 of the sail. 

1. When the velocity of the wind is 19 feet per second, the sails make 
from 11 to 12 revolutions in a minute, and a mill will grind from 880 to 
990 lbs. in an hour, or about 22000 in 24 hours. 

2. When the velocity of the wind is 30 feet per second, a mill will carry 
all sail, and make 22 revolutions in a minute, grinding 1984 lbs. of flour 
in an hour, or 47609 lbs. in 24 hours. 

The velocit}- of the wind in a brisk gale is from 15 to 20 miles per hour, 
exerting a pressure of 1 to 2 lbs. per square foot. A high wind moves at 
the rate of 30 to 40 miles per hour, with a pressure of 4V£ to 8 lbs. per 
square foot; while in a storm it ma}^ vary from 50 to 60 miles per hour, 
and exert a pressure of 12^ to 18 lbs. per square foot. 

To Compute the IMechanical Effect and. Elements of 
"Wind.-m.ills. 

.00048 n v 3 a u = P, n representing number of arms, v velocity of wind per second, a 
area of sails in square feet, and u number of revolution of arms per minute. 



1n ., . , . 1144000 „ . 7 

.1047 u = angular velocity ; : *= area of sails. 



V 



K 2 + r 2 

— r'=z radius of centre of percussion of arms in feet. 



2 

A wind-mill with four arms 70 feet in extreme diameter, and 6 feet wide, 
will raise 1000 lbs. 218 feet in 1 minute, and if working on an average of 
8 hours per day, it is equal to 34 men. It is estimated that 25 square feet 
of canvas will perform the work of a man. 

From 10 to 11 yards of sail will grind and dress 11 bushels of wheat. 

3.16v 
— = proper number of revolutions, x representing the mean angle of the sails 

11 5 v 
to the plane of motion, and — '— = number of revolutions when x = 16°. 

Illustration. — The number of arms of a wind-mill is 4, the velocity of the wind 
16 feet per second, the area of the sails 250 square feet, and the revolutions of the 
arms 6 per minute. 

Then .0004S X4X 250 Xl6 3 X 6= 11796.43 ft. lbs. 



42 6 HYDRODYNAMICS. 

2. Taking tbe preceding elements, the inner radius being 4 feet, the 
length of the arms 28, and the mean angle = 16° ; then 



■j 



.1047X6 == .6282, angular velocity; — -|^? = 279.3, area of sails. 

282 + 42 ™^ _,• 3.16v 3.16x16 50.58 ;,- „ 

— T) — = 20/eetf radius ; = —-r^T = 10 - 66 revolutions. 

2 r' sm. a? 17.2x.*75b4 4.74 

^ 11.5x16 rtrt 

Or, — — = 9.2 revolutions. 



HYDRODYNAMICS. 

Hydrodynamics treats of the force of the action of Liquids or Inelastic 
Fluids, and it embraces Hydrostatics and Hydraulics : the former of which 
treats of the pressure, weight, and equilibrium of liquids in a state of rest, 
and the latter of liquids in motion, as the flow of water in pipes, the rais- 
ing of liquids by pumps, etc. 

Fluids are of two kinds, aeriform and liquid, or elastic and inelastic; 
Fluids press equally in all directions, and any pressure communicated to 
a fluid at rest is equally transmitted throughout the whole fluid. 

The Pressure of a fluid at any depth is as the depth or vertical height, 
and the pressure upon the bottom of a containing vessel is as the base 
and perpendicular height, whatever may be the figure of the vessel. The 
pressure, therefore, of a fluid upon any surface, whether Vertical, Oblique, 
or Horizontal, is equal to the weight of a column of the fluid, the base of 
which is equal to the surface pressed, and the height equal to the distance 
of the centre of gravity of the surface pressed, below the surface of the 
fluid. 

The pressure upon a number of surfaces is ascertained by multiplying 
the sum of the surfaces into the depth of their common centre of gravity, 
below the surface of the fluid. 

The side of an} T vessel sustains a pressure equal to its area, multiplied 
by half the depth of the fluid, and the whole pressure upon the bottom and 
against the sides of a vessel is equal to three times the weight of the fluid. 

When a body is partly or wholly immersed in a fluid, the vertical press- 
ure of the fluid tends to raise the bod} r with a force equal to the weight of 
the fluid displaced ; hence the weight of any quantity of a fluid displaced 
by a buoyant bod}' equals the weight of that body. 

The bottom of a Conical, Pyramidal, or Cylindrical vessel, or of onejthe 
section of which is that of an inverted frustrum of a Cone or Pyramid, 
sustains a pressure equal to the area of the bottom and the depth of the 
fluid. 

The Centre of Pressure is that point of a surface against which an}' fluid 
presses, to which, if a force equal to the whole pressure were applied, it 
would keep the surface at rest. Hence the distance of the centre of press- 
ure of any given surface from the surface of the fluid is the same as that 
of the Centre of Percussion. 

Centres of Pressure. 

Of a Parallelogram, When the Side, Base, Tangent, or Vertex of the Figure is at 
the Surface of the Fluid, is at % of the line (measuring downward) that joins the 
centres of the two horizontal sides. 

Of a Triangular Plane, When the Base is uppermost, is at the centre of a line, 
raised vertically from the vertex, and joining it with the centre of the base ; and 
When the Vertex is uppermost, it is at % of a line let fall perpendicularly from the 
vertex, and joining- it with the centre of the base. 



HYDRODYNAMICS. 427 

Of a Right-angled Triangle, When the Base is uppermost, is at the intersection of 
a line extended irora the centre of the base to the extremity of the triangle by a line 
running horizontally from the centre of the side of the triangle. When the Vertex 
or Extremity is uppermost, it is at the intersection of a line extended from the cen- 
tre of the base to the vertex, by a line running horizontally from % of the side of the 
triangle, measured from the base. 

Of a Trapezoid, When either of parallel Sides are in Surface, , x n=d, 

L 0-j- 4o 

b and b' representing breadth of figure, and n length of line joining opposite sides. 
Of a Circle, is at % of its radius, measured from the upper edge. 

3 v r 
Of a Semicircle, When the Diameter is in the Surface of the Fluid, -~r- = d,d rep- 
resenting distance from surface of the fluid, and r radius of circle. 

15 v r 32 r 

When the Diameter is downward, -— — - = d. 

12 p— lb 

When the Side, Base, or Tangent oft he Figure is below the Surface of the Fluid, 

2 h' 3 h 3 

Of a Rectangle or Parallelogram, -zXttz —■=d,handh > representing the depths 

6 a 2 — h z 
of the upper and under surfaces of the figure from the surface of the fluid. 

3 o 2 -f m 2 
Or, r= m, m representing half the depth of the figure, and o the depth 

of the centre of gravity of the figure from the surface of the fluid. 

_^ 3 m o -f- m 2 ' .' ■ m 2 . ' 

Or," — —^ ;== distance from upper side of figure. Or, — =z distance from cen- 
tre of gravity. 

Of a Trapezoid, When either of the parallel Sides are Horizontal, 

(0 / 2 + 4&?>' + Z, 2 )Xn 2 + 18(&' + o) 2 o 2 ' - . 

— : — ' '-— l ! —d, n representing height of figure. - 

18(o -J-o) 2 o 

7i 2 -f-18o 2 
Of a Triangular Plane, When the Vertex is uppermost, — = d, distance let 

fall perpendicularly from the surface of the fluid upon a line joining the vertex and 
centre of gravity of the figure. 

Or, — — distance from centre of gravity of the figure ; andh-\- — = distance 
18 o 6 

of centre of gravity of the figure below surface of fluid, h representing the depth of 
the vertex below the surface. 

n2 _|_is 2 
When the Base is uppermost, — -^ — = d. 

4 z _|_ 7-2 r 2 

Of a Circle, — — — d. Or, —- — distance below centre of the circle. 

J 4:0 4o 

Of a Semicircle, When the Diameter is Horizontal, and upward or downward, 

!^ n -4-0— d: — = distance of centre of gravity from the diameter in the 

4o 9po 3p 

first case, and pn ~~ — distance from the centre of the circumference in the sec- 
op 

md case. And — = distance of centre of pressure below centre of gravity. 

4o 9po 

PRESSURE. 

To Compute tlie Pressure of* a Fluid, upon, the Bottom of 
its Containing "Vessel. 

Rule.— Multiply the area of the base by the height of the fluid in feet, 
and the product by the weight of a cubic foot of the fluid. 



428 HYDRODYNAMICS. 

To Compute tlie Pressure of a Fluid upon a Vertical, In« 
clined., Curved, or any Surface. 

Rule. — Multiply the area of the surface by the height of the centre of 
gravity of the fluid in feet, and the product by the weight of a cubic foot 
of the "fluid. 

Example. — What is the pressure upon a sloping side of a pond of fresh water 10 
feet square, the depth of the pond being S feet ? 

Centre of gravity, S-^-2z=4feetfrom the surface. 

Then 102X4 = 400, which X 62. 5 =25000 lbs. 

Ex. 2. — What is the pressure upon the staves of a cylindrical reservoir when filled 
with fresh water, the depth being 6 feet, and the diameter of the base 5 feet? 

5x3.1416 = 15.10S feet curved surface of reservoir, which must be considered as a 
plane. 15.70Sx6x6T~2= 282.744, which X62.5 = 17671.5 lbs. 

Ex. 3. — What is the pressure of fresh water upon a gate or embankment in the 
form of a trapezoid, its breadths at top and bottom being 11 and 9 feet, and its 
depth 10 feet ? 

.333X 11 — 9 + 9 = 9.666, which -=- 2 = 4.S33 = the centre of gravity of the fluid. 

iii-?Xl0x4.S33 = 4S3.33, which X62.5 = 3020S.125 lbs. 

Sluice-gates. 

The stress upon a Sluice-gate is determined b}' its area, and the distance 
of its centre of gravity from the surface of the fluid. 

Example. — What is the pressure on a sluice-gate 3 feet square, its centre of gravi- 
ty being 30 feet below the surface of a pond of fresh water ? 

3x3x30 = 270, which X62.5 = 16S75 lbs. 

Flood. -gates. 

The stress upon a Flood-gate is identical with that upon a Sluice-gate 
or any vertical surface. 

Example. — A rectangular flood-gate in fresh water is 25 feet in length by 12 feet 
deep ; what is the pressure upon it ? 

25X12X12^2 == 1S00, which X62.5 = 112500 lbs. 
When water presses against both sides of a plane surface, there arises 
from the resultant forces, corresponding to the two sides, a new resultant, 
which is obtained by the subtraction of the former, as they are opposed to 
each other. 

Illustration. — The depth of water in a canal is 7 feet ; in its adjoining lock it is 
4 feet, aud the breadth of the gates is 15 feet ; what mean pressure have they to sus- 
tain, and what is the depth of the point of its application below the surface? 

7x15 = 105, and 4x15 = 60 sq.feet. 

Hence (105Xr — 00x2) X 62.5 — 1546.875 lbs., the mean pressure. 

Then 1546.875 -f- 62.5 = 247.5 = number of cubic feet pressing upon gates upon the 
high side, and247.5-f- 15x7 = 2.35 feet = depth of centre of gravity of mean pressure. 

To Compute the Pressure of a Column of* a Fluid, per 
Square Inch.. 

Rule. — Multiply the height of the column in feet by the weight of a 
cubic foot of the fluid, and divide the product by 144;" the quotient will 
give the weight or pressure per square inch in pounds. 

Note. — When the height is given in inches, omit the division by 144. 

Example. — The height of a column of fresh water is 23 feet; what is ita-pressure 
per .square inch ? 

23X62.5 — i48fr.fi, which -hl44= 9.9S3 lbs. 



HYDRODYNAMICS. 429 

PIPES. 
To Compute tlie required. Tliiclsness of a iPipe. 

Rule. — Multiply the pressure in pounds per square inch by the diame- 
ter of the pipe in inches, and divide the product by twice the tensile re- 
sistance of a square inch of the material of which the pipe is constructed. 

By experiment, it has been found that a cast-iron pipe 15 inches in diameter, and 
% of an inch thick, will support a head of water of 600 feet ; and that one of oak, of 
the same diameter, and 2 inches thick, will support a head of ISO feet. 

Example. — The pressure upon a cast-iron pipe 15 inches in diameter is 300 lbs. 
per square inch ; what is the required thickness of the metal ? 

300x15 = 4500, which -f- 3000x2 = .75 inch. 

Note. — Here 3000 is taken as the value of the tensile strength of cast iron in or- 
dinary .small water-pipes. This is in consequence of the liability of such castings to 
be imperfect from honey-combs, springing of the core, etc. 

Ex. 2.— The pressure upon a lead pipe 1 inch in diameter is 150 lbs. per square 
inch ; what is the required thickness of the metal ? 

150X 1 = 150, which -4-500x2 = .15 inch. 

HYDROSTATIC PRESS. 
To Compute tlie Elements of a Hydrostatic Press. 

VIA _ WZ'a : W/'a _ PA/ 

— — =z W ; = A ; — - — = P ; = a, P representing the power or pressure 

L CL x L A. VV L 

applied, W the weight or resistance in pounds, I and I' the lengths of the lever and 
fulcrum in inches or feet, and A and a the areas of the ram and piston in square 

inches. 

Illustration The areas of a ram and piston are 86.6 and 1 square inches, the 

lengths of tlie lever and fulcrum 4 feet and 9 inches, and the power applied 20 lbs. ; 
what is the weight that may be borne ? 

20X4XS6.6 692S OMB _ 

— — - — =— — =9237 lbs. 

.75X1 .75 

To Compute tlie Thickness of tlie !3VEetal to resist a given 

Pressure. 

Rulk. — Multiply the pressure per square inch in pounds by the diam- 
eter of the cylinder in inches, and divide the product by twice the esti- 
mated practical tensile resistance oy value of the metal in pounds per square 
inch, and the quotient will give the thickness of the metal required. 

Example. — The pressure required is 9000 lbs. per square inch, and the diameter 
of the cylinder is 5.3 inches ; what is the required thickness of the metal or cast iron? 

~^t^ — t— -o^/>^ — 3.$15 ins. The value of the metal is here taken at 6000. 
0000x2 12000 

HYDRAULIC RAM. 

The useful effect of a Hydraulic Ram, as determined by Eytelwein, va- 
ried from .9 to .18 of the power expended. When the height to which the 
water is raised compared to the fall is low, the effect is greater than with 
any other machine ; but it diminishes as the height increases. 

To Compute tlie Efficiency of a Hydraulic Ram. 

gX y-r- V and v representing the volumes of water expended and raised in feet, 

and h and h' the heights of the fall and of the water raised. 
Illustration. — The heights of a fall and of the elevation are 10 and 26.3 feet, 
and the volumes expended and raised are 1.71 and .543 cubic feet per minute. 
5 543X26.3 _ 71.40 _ 

6 A 1.71X10 102.6 

Oo 



430 



HYDRODYNAMICS. 



To Compute tlie Vohime to "be expended, "Wlieri tlie 
Heiglit and Volume required, are given. 



;A'-r(A-2VAA') „ 13.SG5-7- (10 — 2^10x26.3) 

•=V= — — — =1.71 feet. 



1.2 



1.2 



To Compute tlie Heiglit of Fall required, When tlie "Vol- 
umes and lieiglit of Elevation are given. 

Table of Results of Operations of Hydraulic Rams. 



Number of 
Strokes. 


Height of Fall. 


Height of Ele- 
vation. 


Water Expended. 


Water Raised. 


Useful Effect. 


Min. 


Feet. 


Feet. 


Cubic Feet. 


Cubic Feet. 




66 


10.06 


26.3 


1.71 


.543 


.9 


50 


9.93 


38.6 


1.93 


.421 


.85 


36 


6.05 


38.6 


1.43 


.169 


.75 


31 


5.06 


38.6 


1.29 


.113 


.67 


15 


3.22 


38.6 


1 98 


.058 


.35 


10 


1.97 


38.6 


1.58 


.014 


.18 


— 


22.8 


196.8 


.38 


.029 


.67 



WATER rOWEE. 

Water acts as a moving power, either by its weight or by its vis viva, 
and in the latter case it acts either by pressure or by impact. 

The Natural Effect or Power of a fall of water is equal to the weight of 
its volume and the vertical height of its fall. 

If water is made to impinge upon a machine, the velocity with which it 
impinges may be estimated in tlie effect of the machine. The result or 
effect, however, is in nowise altered ; for in the first case P = \wh } and in 

the latter : 



\ 2 o 



Yw, V representing the volume in cubic feet, ic the weight in 



lbs., and v the velocity ofthejloic infect per second. 

To Compute tlie Power of a Fall of Water. 

Rule. — Multiply the volume of the flowing water in cubic feet per min- 
ute by 62.5, and this product by the vertical height of the fall in feet. 

Note. — When the Flow is over a Weir or Notch, the height is measured from the 
surface of the tail-race to a point 4-9 of the height of the weir, or to the centre of ve- 
locity or pressure of the opening of the flow. 

When the Flow is through- a Sluice or Horizontal Slit, tlie height is measured 
from the surface of the tail-race to the centre of pressure of the opening. 

Example.— What is the power of a stream of water when flowing over a weir 1 
foot in depth by 5 feet in width, and having a fall of L0 feet from the centre of 
pressure of the flow? 

2 

By Rule, page 379, - 5xW*g 1X-G23 = 16.C5 cubic feet per second, 
a 

1G.G5xG0x62.5x20 == 1243750 lbs., which -r- 33000 = 37.S4 horses' power. 
Water sometimes acts by its weight and vis viva simultaneously, bv com- 
bining the effect of an acquired velocity with the fall through which it 
flows UDon the wheel or instrument. 

In this case the mechanical effect = ( k + — ) Vx62.5. 



HYDRODYNAMICS. 431 

Slnices. 

The methods of admitting water to an Overshot or Breast Wheel are various, 
consisting of the Over/ally the Guide-bucket, and the Penstock. 

An Overfall Sluice is a saddle-beam with a curved surface, so as to direct the cur- 
rent of water tangentially to the buckets ; a Guide-bucket is an apron by which the 
water is guided in a course tangential to the buckets ; and a Penstock is the sluice- 
board or gate, placed as close to the wheel as practicable, and of such thickness at 
its lower edge as to avoid a contraction of the current. The bottom surface of the 
penstock is formed with a parabolic lip. 

WATEK-WHE3BXS. 

"Water-wiiekls are divided into two classes, Vertical and Horizontal. 
The vertical consist of the Overshot, Breast, and Undershot ; and the Hori- 
zontal of the Turbine or Reaction wheels. 

Vertical wheels are limited by construction to falls of less than 60 feet. 
Turbines are applicable to falls" of any height from 1 foot upward. 

Vertical wheels applied to a fall of from 20 to 40 feet give a greater ef- 
fect than a Turbine, and for very low falls Turbines give a greater effect. 

The order of effect of these wheels is as follows : 

Ratio of Effect to Power. 



Undershot, Poncelet's, from .0 to .4 to 1 
Undershot from .27 to .45 to 1 

Im t ? ^..^f..^ e . a . C ;} from .3 to .5 tol 



Overshot and high) ,. ~ . * -> 

breast *V j from .6 to. 8 tol 

Turbine from .6 to .8 tol 

Breast from .45 to .65 to 1 

The efficiency of Turbines for very high falls is less than for lower falls, on ac- 
count of the hydraulic resistance involved, and which increases as the square of the 
veloci;y. 

Turbines, being operated at a higher number of revolutions than Vertical Wheel?, 
are more generally applicable to mechanic.il purposes ; but in operations requiring 
but low velocities the Vertical Wheel is preferred. For variable resistances, as roll- 
ing-mills, etc., the Vertical Wheel is far preferable, as its mass serves to regulate the 
motion better than a small wheel. 

In economy of construction there is no essential difference between a Vertical 
Wheel and a Turbine. When, however, the fall of water and the quantity of it are 
great, the Turbine is the least expensive. Variations in the supply of water affect 
vertical wheels less than Turbines. 

The durability of a Turbine is less than that of a Vertical Wheel; and it is indis- 
pensable to its operation that the water should be free from sand, branches, leaves, etc. 

With Overshot and Breast Wheels, Avhen only a small quantity of water is availa- 
ble, or when it is required or becomes necessary to produce only a portion of the 
power of the fall, their efficiency is relatively increased, from the buckets being but 
proportionately filled ; but with Turbines the effect is contrary, as when the sluice is 
lowered or the supply decreased the water enters the wheel under circumstance? in- 
volving greater loss of effect. To produce the maximum effectof a stream of water 
upon a wheel, it must flow without impact upon it, and leave it without velocity ; 
and the distance between the point at which the water flows upon a wheel and the 
level of the water in the reservoir should be as small as practicable. 

Small wheels give less effect than large, in consequence of their making a greater 
number of revolutions and having a smaller water arc. 

Sliron cling. 

The Shrouding of a wheel consists of the plates at its periphery, which 
form the sides of the bucket. , 

The height of they«^ of a water-wheel is measured between the surfaces 
of the water in the penstock and in the tail-race, and ordinarily two thirds 
of the height between the level of the reservoir and the point at which the 
water strikes a wheel is lost for all effective operation. 

The velocity of a wheel at the centre of percussion cf the fluid should 
be from .5 to .G that of the flow of the water. 



432 HYDRODYNAMICS. 

By the deductions of Weisbach it appears that the effect of impact is 
only half the available effect under the most favorable circumstances ; 
hence the least practicable part of a fall should be used to produce impact. 

Under the circumstances of a variable supply of water, the Breast-wheel 
is better calculated for effective duty than the Overshot, as it can be made 
of a greater diameter ; whereby it affords an increased facility for the re- 
ception of the water into its buckets, also for its discharge at the bottom ; 
and further, its buckets more easily overcome the retardation of back-wa- 
ter, enabling it to be worked for sl longer period in the back-water conse- 
quent upon a flood. 

Friction, of* tlie Grudgeons. 

A very considerable portion of the mechanical effect of a wheel is lost in the effect 
absorbed by the fiiction of the gudgeons. 

To Compute tlie Friction of tlie Griidgeoiis of a Water- 
wheel. 

Wrn C .00S6=/, W representing the weight of the wheel, r the radius of the gud- 
geon in inches, and n the number of revolutions of the wheel. 
For well-turned surfaces and good bearings, C = .075 with oil or tallow ; when 

the best of oil is well supplied :=. 054; and, as in ordinary circumstances, when a 

black-lead unguent is alone applied = .11. 
Illustration. — A wheel weighing 25000 lbs. has gudgeons 6 inches in diameter, 

and makes 6 revolutions per minute ; what is the loss of effect ? 

Assume C^.OS. Then 25000x^X6x.OSx.COS6zr: 309.6 lbs. 

OVERSHOT-WHEELS. 

In an Over shot-wheel the flow of water acts in some degree by impact, 
but chiefly by its weight. 

The lower the speed of the wheel at its circumference, the greater will 
be the mechanical effect of the water. A proper velocity is about 5 feet 
per second. 

The number of buckets should be as great as practicable, and they 
should retain water so long as practicable. The maximum effect is at- 
tained when the buckets are so numerous and close that the water surface 
in the bucket commencing to empty itself should conle in contact with the 
under side of the bucket next above it. 

Curved buckets give the greatest effect, and Radial give but .78 of the 
effect of Elbow-buckets. A wheel 40 feet in diameter should have 152 
buckets. 

Small wheels give a less effect than large, in consequence of their great- 
er centrifugal action, and discharging the water from the buckets at an 
earlier period than with larger wheels, or when their velocity is lower. 

"When the head of water bears to the fall or the^height of the wheel a 
proportion as great as 1 to 4 or 5, the ratio of effect to power is reduced to 
!8 and even .75. The general law, therefore, is, that the ratio of effect to 
power decreases as the proportion of head to the total head and fall in- 
creases. 

A wheel with shallow Shrouding acts more efficiently than one where it 
is deep, and the depth is usually made 10 or 12 inches* but in some cases 
it has been increased to 15 inches. 

The breadth of a wheel depends upon the capacity necessary to give the 
buckets to receive the required volume of the water. 

Form of Buckets. — Radial buckets— that is, when the bottom is a right line— in- 
volve so great a loss of mechanical effect as to render their use incompatible with 



HYDRODYNAMICS. 433 

economy ; and when a bucket is formed of two pieces, the lower or inner piece is 
tunned the bottom or floor, and the outer piece the arm or wrist. The former is 
usually placed in a line with the radius of the wheel. 

The line of a circle passing through the elbow, made by the junction of the floor 
and arm, is termed the division circle, or bucket pitch, and it is usual to put this at 
one half the depth of the shrouding. 

When the arm of a bucket is included in the division angle of the buckets, that is, 

> — , 11 representing the number of buckets, the cells are not sufficiently covered, ex- 
cept for very shallow shrouding ; hence it is best to extend the arm of a bucket over 
five fourths of the division angle, so as to cover or overlap the elbjw of the bucket 
next in advance of it. 

The least section of a cell should be somewhat greater than the section of the wa- 
ter flowing on to the wheel, and the cells should be in the plane of the flow of the 
water. 

Fairbairn gives the area of the opening of a bucket in a wheel of great diameter, 
compared to the volume of it, a* 5 to 24. 

Buckets having a bottom of two planes, that is, with two bottoms, and two divi- 
sion circles or bucket pitches and an arm, give a greater effect than with one bottom. 

When an opening is made in the base of the buckets, so as to afford an escape of 
the air contained within it without a loss of the water admitted, the buckets are 
termed ventilated, and the effective power of the wheel is much greater than with 
the close buckets. 

To Compute tlie Ttadiuis of a "Wheel, the ISJ"iim"ber ofRev- 
olvvtions, and. tlie Height of tlie Fall of Water upon an 
Oversliot-Avlieel, 

When the whole Fall and the Velocity of the Flow, etc., are given. 

— = r; „„,,.,. ■ = ii, h representing the height of the whole /aZ7, h' the 

1 -j- cos. a 3. 1410 r J 

height between the centre of gravity of the discharge and the half depth of the 
bucket upon which the water flow?, a the angle which the point of entrance of the 
water into a bucket makes with the summit of the wheel, n the number of revo- 
lutions, c the velocity of the wheel at its circumference, and r its radius. 
Note.— As a proportion of the velocity of the flow is lost, it is proper to assume 

v 2 
the height h' as but = — 1.1. 

Illustration A fall of water is 30 feet, the velocity of its flow is 16, the angle 

of its impact upon the buckets is 12°, and the required velocity of the wheel is S feet 
per second ; what is the required radium, number of revolutions, and the height of 
the fall upon the wheel ? 

ft'i'^Xtl = 488/«*; cos. 12° = .97S15. Then ^ ~ \ ; 5 Q 8 ^ ^ = 12.05 feet 

%g r ■' l-f-.978 1.9 <8 

30x8 240 . 

radius. ; ^ = == 5.9, number of revolutions. 

3.1410X12.95 40.684 - 

When the Number of Revolutions and the Ratio between the Velocities of the 
Flow and at the Circumference of the Wheel are given. 

V Mmvi(xn) 2 h4- (14- cos. fl)2-(14-eos. a) v . 

I — J v "* : — ! —r, x representing -, and c — 

.000380 (x n) * fun H, c 1 

3. 1416 ?7. r 



'60 
Illustration. — If the number of revolutions are 5, x — 2, and the fall, etc., as in 
the previous case ; what is the radius of the wheel and the velocity of the flow ? 

V.(MM)i72 (2X5)^X30 4- (1.978)2 _ l.rTS _ V 2 3l0-f 3 .9 A25 - 1.078 _ .5177 _ 
.000886 (2X5)2 ~~ ^)3SJ A 380 ~~ 

13.41 feet; — *■ '- 1 — = 7.03/ee* velocity at circumference of wheel, and as 

x = 2. Hence 7.03X2 = 14.03 velocity of flow. 

Oo* 



434 HYDRODYNAMICS. 

Further, if the velocity of the flow is 14.06 feet, what is the height of it ? 
14.062 
5^X11 = 8.8T/«*. 

To Compute tlie "Width, of an Overshot-Vv^heel. 

oV 

— =ip, o representing a coefficient = 3, when the buckets are filled to an excess, and 

s c' 
5 when they are deficiently filled ; s depth of shrouding in feet ; c' the velocity of 
the wheel at the centre of the shrouding ; and w the width of the buckets in feet. 
Illustration. — A wheel is to be 30 feet in diameter, with a depth of shrouding 

of 1 foot, and is required to make 5 revolutions per minute under a discharge of 10 

cubic feet per second ; what should be the width of the buckets ? 

Assume o = 4, and c' = — = 7.854. Then — — — - = 6.00 feet. 

60 1XT.854 

To Compute the 3N"ixixi'ber of Btxckets. 

7 ( 1 -J — —J = distance between the buckets, s representing depth of shrouding in 

inches ; and - — = number of buckets. 
v 

Illustration. — Take the elements of the preceding case. 

Then 7 (l -f- i?) = 7x2.2 == 15.4 ins., and 30 X 3 ^ C >< 12 = T3<4j say T2 ouc kets; 

hence — r- = 5°, angle of subdivision of buckets. 

To Compute tlie Effect of an Overshot Water-wheel. 

™w-(fYw+f) 

-^ = P, V representing the volume of the water flowing per sec- 

V hw 
ond, w the xoeight of the water, and v' the velocity of the water discharged at the 
tail of the wheel. 

Illustration. — A volume of 12 cubic feet per second has a fall of 10 feet, the 
wheel using but 8.5 feet of it, and the velocity of the water discharged is 9 feet per 
second ; what is the effect of the fall ? 

The friction of the wheel is assumed to be 750 lbs. 

12X8.5X62.5- (-^ X 12x62.5 + 75o) _ em _ (i ,c 6x750 + 750 , 46s0 _ 

12X10X62.5 7500 7500 ' _ 

= ratio of effect tojiower; and 46S0X60 seconds -±- 33000 = 8.51 horses' power. 

To Compute tlie Power of an Oversh.ot-wh.eel. 

Rule. — Multiply the weight of water in lbs. discharged upon the wheel 
in one minute by the height or distance in feet from the centre of the open- 
ing in the gate to the surface of the tail-race ; divide the product by 33000, . 
and multiply the quotient by the assumed or determined ratio of effect to 
power. Or, for general purposes, divide the product by 50000, and the 
quotient is the horses' power. 

The Mechanical Effect of water is the product of its weight into the 
height from which it falls. 

Example. — The volume of water discharged upon an overshot-wheel is 640 cubic 
feet per minute, and the effective height of the fall is 22 feet ; what are the horses' 
power ? 

640x62 5x22 8S0000 

' — ——— = 26.67, which, X. 75= the assumed ratio of effect to i»ower 

in such a case = 20 horses. 



HYDRODYNAMICS. 435 

"Useful Effect of an Oversliot-wheel. 

With a large wheel running in the most advantageous manner, .84 of 
the power may be taken for the effect. 

The velocity of a wheel bears a constant ratio, for maximum effects, to 
that of the flowing water, and this ratio is at a mean .55. 

The ratio of effect to power with radial-buckets is .78 that of elbow-buck- 
ets. The ratio of effect decreases as the proportion of head to the total 
head and fall increases. Thus a wheel of 10 feet in diameter gave with 
heads of water above the gate, ranging from .25 to 3.75 feet, a ratio of ef- 
fect decreasing from .82 to .67 of the power. 

Note.— For the Theoretical Effect of a wheel, whether Overshot, Undershot, or 
Breast, see Formulae of Weisbach and Morin. (Weisbach, vol. ii., pages lS-i-215.) 

BREAST-WHEELS. 

Breast-ivheels are designed for falls of water varying from 5 to 15 feet, 
and for flows of from 5 to 80 cubic feet per second. They are constructed 
with either ordinary buckets or with blades confined by a Curb, 

When buckets are inclosed in a curb, they are not required to hold water ; 
hence they may be set radial. The buckets should be numerous, as the 
loss of water escaping between the wheel and the curb is less the greater 
their number ; and that they maj T not lift or cany up water with them 
from the tail-race, it is proper to give the bucket such a plane that it may 
leave the water as nearly vertical as may be practicable. 

The distance between two buckets should be equal to the depth of the 
shrouding, or at from 10 to 15 ins. 

It is essential that there should be air-holes in the floor of the buckets, 
to prevent the air from impeding the flow of water into them, as the water 
admitted is nearly as deep as the interval between them ; and the veloci^ 
ty of the wheel should be such that the buckets should be filled to^or j}£ 
of their volume. 

The inclosure within which the water flows to a breast-wheel as it leaves 
the sluice is termed a Curb or Mantle. 

When wheels are constructed of iron, and are accurately set in mason- 
ry, a clearance of .5 of an inch is sufficient. 

High Breast-wheels are used when the level of the water in the tail-race 
and penstock or forebay are subject to variation of heights, as the wheel 
revolves in the direction in which the water flows from the buckets, and 
back-water is therefore less disadvantageous, added to which, penstocks 
•can be so constructed as to admit of an adjustable point of opening for 
the water to flow upon the wheel. 

The effect of this wheel is equal to that of the overshot, and in some in- 
stances, from the advantageous manner in which the water is admitted to 
it, it is greater when both wheels have the same general proportions. 

With a wheel 30 feet in diameter, having 96 buckets, the water admitted at a 
point 50° from the summit of it, with a velocity of 8 feet per second, the wheel hav- 
ing a velocity of 5 feet, the ratio of effect was .69. When the water flows at from 
10° to 12° above the horizontal centre of the wheel, Fairbairn gives the area of the 
opening of the bucket, compared with the volume of it, as 8 to 24. 

To Compute tlie Proportion and. Effect of a IBreast- 
■wlieel. 

Illustration. — The flow of water is 15 cubic feet per second ; the height of the 
fall, measured from the centre of pressure of the opening to the tail-race, is 8.5 feet; 
velocity of revolution 5 feet per second ; and depth of buckets 1 foot, filled to .5 of 
their volume. 



436 HYDRODYNAMICS. 

V 15 

Width of wheel = — , s representing depth, and v velocity of buckets ; = 3, 

si; 1X& 

and as the buckets are but .5 filled, 3-f- .5 = 6 feet. 

Assume the water is to flow with double the velocity of the rotation of the wheel; 
hence v = 5x2 = 10 feet; and the fall required to generate this velocity = — X 1 1 
100 ^ 

Deducting this height from the total fall, there remains for the height of the curb, 
or for the fall during which the weight of the water alone acts, h — h' = 8.5 — 1 71 = 
6.19 feet. 

Making the radius of the wheel 12 feet, and the radius of the bucket circle 11.5 
feet, the whole of the mechanical effect of the flow of the water = 15x62.5xS.5-= 
7363.75 lbs. 

The theoretical effect, as determined by Weisbach, vol. ii., p. 210 = 
7273 lbs., from which are to be deducted the losses, which he computes 
as follows : 

Loss b} T escape of water between the wheel and the curb = 916 
Loss by escape at sides of wheel and the curb = 180 

Friction and resistance of the water = 160 

1256 /fo. 

* 

Assuming the weight of the wheel 16500 lbs., the radius of the gudgeons 

5 x 60 

to be 2.5 inches, C, as before, == .08, and n =-^z — a — n ., . i . . = 4. 

12x2x3.1416 

Then 16500 x 2.5 X 4x .08 X .0086 t£ 113.5 lbs., and 1256 + 113.5 = 

5903.4 
1369.5 : henc3 7273 - 1369.5 =* 5903.5 ; % hu b J* = -74, and 5903.5 X 60 -~ 
1 ' /9b8./5 

33000 = 10-73 horses 1 power. 

To Compute tlie 3?o-wer of a Breast-wheel. 

Rule. — Proceed as per rule for an overshot-wheel, using 55000 and .6 
with a high breast, and 62500 and .6 for a low breast. 

The Committee of the Franklin Institute ascertained that, with a high breast- 
wheel 20 feet in diameter, the water admitted under a head of 9 inches, and at 17 
feet above the bottom of the wheel the elbow-buckets gave a ratio of effect to power 
of .731 at a maximum, and radial-buckets .653. AVith the water admitted at a 
height of 33 feet 8 ins., the elbow-buckets gave .C5S, and the radial .628. 

At 10.96 feet above the bottom of the wheel, with a head of 4.29 feet, the elbow- 
buckets gave .544, and the radial .329. 

At 7 feet above the bottom of the wheel, and a head of 2 feet, this low breast gave . 
.62 for elbow-buckets, and .531 for radial. 

At 3 feet S inches above the bottom of the wheel, and a head cf 1 foot, the elbow- 
buckets gave .555, and the radial .533. 

UNDERSHOT-WHEELS. 

Undershot-wheels are usually set in a curb, with as little clearance for 
the escape of water as practicable ; hence a curb concentric to this wheel 
is more effective than one set straight or tangential to it. 

The computations for an undershot-wheel and the rules for construc- 
tion are nearly identical with those for a breast-wheel. 

The buckets are usually set radially, but they maj^ be inclined upward, 
so as to be more effectively relieved of water upon their return side, and 
they are usually filled from .5 to .6 of their volume. The depth of the 
shrouding should be from 15 to 18 inches, in order to prevent the overflow 
of water within the wheel, which would retard it. 



HYDRODYNAMICS* 437 

The si nice-gate should be set at an inclination to the plane of the curb, 
or tangential to the wheel, in order that its aperture may be as close to 
the wheel as practicable ; and in order to prevent the partial contraction 
of the flow of water, the lower edge of the sluice should be rounded. 

The effect of undershot-wheels is less than that of breast-wheels, as the 
full available as weight is less than with the latter. 

To Compute tlie Power of* an. TTnd.ersliot-'wlieel. 

Proceed as per rule for an overshot-wheel, using 93750 for 50000, and 
A for .75. 

poncelet's wheel. 

In a Poncelet Wheel the buckets are curved, so that the flow of water is 
along their concave side, pressing upon them without impact ; and the 
effect is greater than when the water impinges at nearly right angles to 
plane-surfaced buckets. 

This wheel is advantageous for application to falls under 6 feet, as their 
effect is greater than that of other undershot-wheels with a curb, and for 
falls from 3 to 6 feet their effect is equal to that of a Turbine. 

In their arrangement, the aperture of the sluice should be brought close 
to the face of the wheel. The first part of the course should be inclined 
from 4° to 6° ; the remainder of the course, which should cover or em- 
brace at least 3 buckets, is carried concentric to the wheel, and at the end 
of it a quick fall of 6 ins. is made, to guard against the effect of back- 
water. The sluice should not be opened over 1 foot in any case, and 6 ins. 
is a suitable height for falls of 5 and 6 feet. 

The distance between two buckets should not exceed 8 or 10 ins., and 
the radius of the wheel should not be less than 40 ins., or more than 8 feet. 

The plane of the stream or head of water should meet the periphery of 
the wheel at an angle of from 24° to 30°. The space between the wheel 
and its curb should not exceed .4 of an inch. 

The depth of the shrouding should be such as to prevent the water from 
flowing through it and over the buckets, and the width of the wheel should 
be equal to that of the stream of flowing water. 

The effect of this wheel increases with the depth of the water flow, and, 
therefore, other elements being equal, as the filling of the buckets, to ob- 
tain the maximum effect, the water should flow to the buckets without 
impact, and the velocity of rotation of wheel should be onl} T a little less 
than half the velocity of the water flowing upon the wheel. The effect is 

a maximum when c = ,5v cos. y, and then ' • Vw — L. 

Zg 

To Compute tlie Proportions of a Poncelet 'Wheel. 

Note. — As it is impracticable to arrive at the results by a direct formula, they must 
be obtained by gradual approximation.. 

Example — The height of the fall is 4.5 feet ; the volume of the flowing water 40 
cubic feet per second : the radius of the wheel = 2 h, or 9 feet ; the depth of the 

stream = - h = .75 feet ; and C assumed to be .9. 

1 v* 
h representing- height of fall in feet, d depth of shrouding = t . <- \- d', 

V volume of flowing water in cuoicfee:. rf , ^ ^ . %g of ^.^ ' 

n number of revolutions s= , e width of sluice, 

c velocity of circumference If wheel, r radius ^ curvature of buckets s^g-j 

a radius of wheel, v velocity of flowing water, 

C coefficient of resistance of flow of water, 



438 HYDRODYNAMICS. 

x angle between plane of flowing water and that of the circumference of the wheel 

at the point of contact, sin. of -==z 'v/cos. z. 

z angle made by circumference of wheel with end of buckets = 2 tang, y, 

y angle of direction of flowing water from circumference of wheel =.'-— / -• 

- a V >4--- 

Then v=.d< s /2glh — — \ =.9XlG.29 = 14C6 feet — velocity of flowing watel 

.-. the velocity of the wheel, being less than half the velocity of the flowing water; 

14.G6 — .66 m ^ J ' ' t „ t r _,.. 1 14662 214.92 

c = = 7 feet ; and the depth of the shrouding = - X -jj J- . 1 5 = tt^z-tt-; 

^ 4 & g LDi do 

+.75 =1.5S feet; y = ^^- T X. / L5b =1.22x\/^2 = .25, the angle 

JXJ V 32.106 + ^- 

roxT 

corresponding to which = 14° 30' ; n = — — — = 7.43 revolutions ; z = 2 tang, y 

40 1 *>9 

= 2x.£5SG2 = .51724 .-. z=-27° 20': e= ^ — 3.63 /g^; r = 



75X14.^6 ' r cos. 27° 20' 

- QO ,L = l>7S/eef; a? = sin. f = V cos. z = V cos. 27° '20' = .C43 = sin. of 70° 34' 
:. j? = 141°S'. 

IMPACT AND REACTION WHEELS. 

If the buckets are given increased length, and formed to such a hollow 
curve that the water leaves the wheel in nearly a horizontal direction, 
the water then both impinges on the buckets and exerts a pressure upoxi 
them ; therefore the effect is greater than with an impact wheel alone. 

Impact Wheels. — Impact Turbines are the most simple but least efficient 
form of impact wheel. They consist of a series of rectangular buckets or 
blades, set upon a wheel at an angle of 50° to 70° to the horizon ; the wa- 
ter flows to the blades through a pyramidal trough set at an angle of 20° 
to 40°, so that the water impinges nearly at right angles to the blades. 
The effect is estimated at about .5 the entire mechanical effect, which is 
increased by inclosing the blades in a border or frame. 

Reaction Wheels. — The reaction of water issuing from an orifice of less 
capacity than the section of the vessel of supply is equal to the weight of a 
column of water, the basis of which is the area of the orifice or of the stream, 
and the height of which is twice the height due to the velocity of the water 
discharged. 

v 2 

Hence the expression is 2 . ~— a w = L, to representing the weight of a 

cubic foot of water, and a the area of the opening. 

If the water flows out at the side of the vessel, the direction of the re- 
action is horizontal ; and if the water vein is contracted, C, a coefficient 

of contraction, must be introduced, as 2 . — C w a = L. 

' 2g 

In the discharge through a thin plate, Mr. Ewart, of Manchester, de- 
termined C = .96 ; and when the orifice was provided with a mouth-piece 
in the form of the vena contracta, it was .94. 

To Compute tlie Effect of a Reaction-wheel. 

v 2 
h 4- - — = li == height determining the pressure of water upon the orifice, h 

representing the height of the centre of the discharge from the centre of 
pressure of the opening of the svpply-vcssel. 



HYDRODYNAMICS. 439 



Hence C V % g h-\- v 2 — v' = s — absolute velocity of the water, v' repre 
senting velocity of rotation of the icheel in the opposite direction to the efflux 

of the water , and — i - Y w = L. 

This efflux increases as v, the maximum effect depending upon the 
wheel acquiring an infinite velocity* , and as the velocity increases the re 
sistances increase. 

When this wheel is loaded, so that the height due to the velocity, corre- 

v 2 " 
sponding to the velocity of rotation v, is equal to the fall, or ^— = h, or 

^9 

v=Y2gh, there is a loss of 17 per cent, of the available effect; and 

v 2 v 2 

when x — = 2h, there is a loss of but 10 per cent. : and when -r— — 4^, 
2 9 2# ' 

there is a loss of but 6 per cent. Consequent!} 7 , for moderate falls, and 
when a velocity of rotation exceeding the velocity due to the height of the 
fall may be adopted, this wheel works very effectively. 
The efficiency of the wheel is but one half that of an undershot-wheel. 
When the sluice is lowered, so that only a portion of the wheel is open- 
ed, the efficiency of a Reaction-wheel is less than that of a Pressure Tur- 
bine. 

TURBINES. 

In High-pressure Turbines the reservoir (of the wheel) is inclosed at top, 
and the water is admitted through a pipe at its side. In Low-pressure, 
the water flows into the reservoir, which is open. 

In Turbines working under water, the height Qi) is measured from the 
surface of the water in the supply to the surface of the discharged water 
or race ; and when they work in air, the height is measured from the sur- 
face in the supply to the centre of the wheel. 

In order to obtain the maximum effect from the water, the velocity of 
it, when leaving a Turbine, should be the least practicable. 

The efficiency is greater when the sluice or supply is wide open, and it 
is less affected by head than by variations in the supply of water. It va- 
ries but little with the velocity', as it was ascertained by experiment that 
when 35 revolutions gave an effect of .64, 55 gave but .66. 

When Turbines operate under water, the flow is always full through 
them ; hence they become reaction-wheels, which are the most efficient. 

The experiments of Morin gave results of the efficiency of Turbines as 
high as .75 of the power expended. 

The angle of the plane of the water entering a Turbine with the inner 
periphery of it should be greater than 90°, and the angle which the plane 
of the water leaving the reservoir makes with the inner circumference of 
the Turbine should be less than 90°. 

When Turbines are constructed without a guide curve* the angle of 
plane of flowing water and inner circumference of wheel = 90°. 

Great curvature involves greater resistance to the efflux of the water; 
and hence it is advisable to make the angle of the plane of the entering 
water rather obtuse than acute, say 100° ; the angle of the plane of the 
water leaving, then, should be 50°, if the internal pressure is to balance 
the external ; and if the wheel operates free of water, it may be reduced 
to 25° and 30°. 

* Guide curves are plates upon the centre body of a Turbine, which give direction to the flowing 
water, or to the blades of the wheel which surround them. 



±40 HYDRODYNAMICS. 

The angle made by the plane of the discharged water with the water pe* 
riphery should never exceed 20°. 

Fourneyron 's work either in or out of water, are applicable to high and 
low falls, and are either high or low pressure Turbines. They are best 
adapted for very low falls, and those of moderate height, say up to 30 feet, 
with large supplies of water. The pressure upon their step is confined to 
the weight of the wheel alone. 

Fourneyron makes the angle of the plane of the water entering a Tur- 
bine == 90°, and the angle of the plane of the water leaving = 30°. 

JonvaVs.— This wheel is essential^ alike in its principal proportions to 
Fontaine's, and in the principle of operation it is the same. The water in 
the race must be at a certain depth below the wheel. 

The efficienc3 T of this wheel decreases as the volume of water is dimin- 
ished, or as the sluice is contracted. 

Fontaine's. — In the operation of this wheel the water in the race is in 
immediate contact with the wheel, and its efficiency is greatest when the 
sluice is fully opened. Its efficiency, also, is less affected by variations 
of the head of the flow than in the volume of the water supplied ; hence 
they are adapted for Tide-mills. 

The pressure upon the step, in addition to the weight of the wheel, in- 
cludes that of the contained water. 

Whitelaiv's. — This wheel is best adapted for high falls and small vol- 
umes of water. 

Poncelefs. — This wheel is alike to one of his undershot-wheels set hori- 
zontally, and it is the most simple of all the horizontal wheels. 

The Ratio of Effect to Power of the several Turbines is as follows : 



Jonval 6 to .7 to 1 

Fontaine 6 to .7 to 1 



Poncelet 65 to .75 to 1 

Fourneyron 6 to .75 to 1 

Whitelaw 6 to .75 to 1 

A Tremont Turbine, as observed by Mr. Francis, in his experiments at 
Lowell, Mass., gave a ratio of effect to power as .79375 to 1. 

To Compute the Horses' Power of a Turbine. 

.0425 D 2 h Vh = P, D representing the diameter of the wheel at the outer extremities 
of the buckets. Or, .0S5 V h = P, V representing the volume of water expended in 
cubic feet per second. 

Example.— The diameter of a Turbine is S.3 feet, and the fall of the flowing wa- 
ter is 16 feet ; what is the poAver of it ? 

.0425x8. 32xl6Xv /16 = - 0425 X6S.S9xl6x4 = 187.38 horses. 

DIMENSIONS OF WATER-WHEELS AND TURBINES. 

To Compute the Diameter of* a Shaft, When the Stress is 
considered as heing laid ixnifbrmly along its Length. 

When of Cast Iron. 
?C— — = d, W representing the weight or load in lbs. , I the length of the shaft be- 
tween the journals in feet, and d the diameter of the shaft in its body in inches. 
When the Shaft has to resist both Lateral and Torsional Stress, Ascertain 
the diameter for each stress, and the cube root of the sum of their cubes 
will give the diameter required. 

To Compute tlie Diameter of the G-udgeons, 

The Length being considered 1.25 times its Diameter. 

.04S V W =zd, W representing the weight upon each gudgeon. 
For ?ther Rules, see Strength of Materials, page 447. 



HYDRODYNAMICS. 44 1 

To Compute tlie Dimensions of tlie ^rms. 

When of Cast Iron, 

- — =zw, d representing the diameter of the shaft in inches, n the number of arms, 
y n w 

and w the width of the arm in inches ; - = t, t representing thickness of the arm. 

When the Arm is of Oak, w should be 1.4 times that of iron, and the thickness .7 
ohat of the width. 

For the Elements of a Turbine, see Experiments of J. B. Francis, pages 44, 54, 
and yystrom's Mechanics, p. 226, 6th Edition, omitting No. 25. 

Barker's Mill. — The effect of this mill is considerably greater than that 
which the same quantity of water would produce if applied to an under- 
shot-wheel, but less than that which it would produce if properly applied 
to an overshot-wheel. 

For a description of it, see Grier's Mechanics' Calculator, page 234 ; and for its 
formulae, see London Artisan, 1845, page 229. 

The higher an overshot-wheel is, in proportion to the whole descent of 
the water, the greater will be its effect. The effect is as the product of 
the volume of water and its perpendicular height. 

The weight of the arch of loaded buckets in an overshot-wheel in lbs. is 
ascertained by multiplying | of their number by the number of cubic feet 
in each, and that product by 40. 

MEMORANDA. 

A volume of water of 17.5 cubic feet per second, with a fall of 25 feet, 
applied to an undershot-wheel, will drive a hammer of 1500 lbs. in weight 
from 100 to 120 blows per minute, with a lift of from 1 to 1.5 feet.* 

A volume of water of 21.5 cubic feet per second, with a fall of 12.5 feet, 
applied to a wheel having a great height of water above its summit, being 
7.75 feet in diameter, will drive a hammer of 500 lbs. in weight 100 blows 
per minute with a lift of 2 feet 10 inches. Estimate of power 31.5 horses. 

A Stream and Overshot Wheel at Fishkill Creek, N. Y., of the following 
dimensions — viz., Height of head to centre of opening, 24% ins. ; open- 
ing* 1% D 7 80 ins. ; wheel, 22 feet diameter by 8 feet face ; 52 buckets, 
each 1 foot in depth, making 3)^ revolutions per minute — drives 3 run of 
i.% feet stones 130 revolutions in a minute, with all the attendant machin- 
ery, and grinds and dresses 25 bushels of wheat per hour. 

A Breast-wheel and Stream of the following dimensions — viz., Head, 20 
feet ; height of water upon wheel, 16 feet ; opening, 18 feet by 2 ins. ; diam- 
eter of wheel, 26 feet 4 inches ; face of wheel, 20 feet 9 inches ; depth of 
buckets, 15% ins. ; number of buckets, 70 ; revolutions, 4^ per minute- 
drives at the Rocky Glen Factory, Fishkill, N. Y., 6144 self-acting mule 
spindles ; 160 looms, weaving printing-cloths 27 ins. wide of No. 33 yarn 
(33 hanks to a lb.), and producing 24000 hanks in a day of 11 hours. 

4% bushels Southern and 5 bushels Northern wheat are required to make 1 barrel 
of flour. 

IMPULSE AND RESISTANCE OF FLUIDS. 

Impulse and Resistance of Water. — Water or any other fluid, when flow- 
ing against a body, imparts a force to it by which its condition of motion 
is altered. The resistance which a fluid opposes to the motion of a body 
does not essential^ differ from Impulse. 

t The Impulse of one and the same mass of fluid under otherwise similar 
circumstances is proportional to the relative velocities o^fv of the fluid. 

* The volume of water required for a hammer increases in a much greater ratio than the velocity 
to be given to it, it being nearly as the cube of the velocity. 

Pp 



442 HYDRODYNAMICS. 

For an equal transverse section of a stream, the impulse against a sur- 
face at rest increases as the square of the velocity of the water. 

Impulse against Plane Surfaces. — The impulse of a stream of water de- 
pends principally upon the angle under which, after the impulse, it leaves 
the water ; it is nothing if the angle is 0, and a maximum if it is deflected 

back in a line parallel to that of its flow, or 180°, P = 2°-^-Yw. 

When the Surface of Resistance is a Plane, and —90°, then P = Vw. 

If the surface is at rest, P = 2 A h iv, c and v' representing the velocities 
of the water and of the surface upon which it impinges, w the weight of the 
'fluid per cubic foot, A the transverse section of the stream in sq. ins., and 
c if v' the relative motions of the water and surface. 

The normal impulse of water against a plane surface is equivalent to 

the weight of a column which has for its base the transverse section A of 

v 2 . 
the stream, and for altitude twice the height due to its velocity, 2h = 2 — 

The resistance of a fluid to a body in motion is the same as the impulse 
of a fluid moving with the same velocity against a body at rest. 

Maximum Effect of Impulse. — The effect of impulse depends principally 
on the velocity v of the impinged surface. It is, for example, 0, both when 
v — o and v = ; hence there is a velocity for which the effect of impulse 

is a maximum = (o — v)v ; that is, v — «, and the maximum effect of the 

impulse of water is obtained when the surface impinged moves from h 
with half the velocity of the water. 

Illustration A stream of water having a transverse section of 40 square inches 

discharges 5 cubic feet per second against a plane surface, and flows off with a ve- 

5^144 
locity of 12 feet per second ; the effect of its impulse, then, is, o = = 18 ; 

-JQ _ -lO 

g = 32.166, w = 62.5 ; \j 2 1C6 X5X62.5 = .1865x5x62.5 = 5S.28 lbs. 

Hence the mechanical effect upon the surface = Pu = 5S.2Sxl2 = 690.36 lbs. 

The maximum effect would tev = -— -X . n =z9feet, and -x— X 5x62.5 = 
L 2> 40 & l g 

1 7S6.92 

-X5.0363x312.5 = 7S6.92 lbs. ; and the hydraulic pressure = — — = 87.44 lbs. 
i " 

When the Surface is a Plane and at an Angle, then P = (1 — cos. a) - V w. 

Illustration A stream of water having a transverse section of 64 square inch- 
es discharges 17.778 cubic feet per second against a fixed cone, having an angle of 
convergence from the flow of the stream of 50°, the hydraulic pressure in the direc- 
tion of the stream ; then o = 1 AA = 40 ; cos. 50° = .64279. 
1 64-^144 

4.0 
(1 — .64279) — - — x 17. 77SX62.5 = . 35721X1375. 431 = 491.318 lbs. 
32.166 

When the Surface of Resistance is a Plane at 90°, and has Borders added 
to its Perimeter, the effect will be greater, depending upon the height of 
the border and the ratio of the transverse section between the stream and 
the part confined. 

Oblique Impulse. — In oblique impulse against a plane, the stream may 
flow in one, two, or in all directions over the plane. 

o — v 

When the Stream is confined at three Sides, P = (1 cos. a) Vw. 

o — v 
When the Stream is confined at two Sides, P = sin. a 2 V w. 



HYDRODYNAMICS. 443 

The normal impulse of a stream increases as the sine of the angle of in- 
cidence ; the parallel impulse as the square of the sine of the angle, and 
the lateral impulse as double the angle. 

When an Inclined Surface is not Bordered, then the stream can spread 
over it in all directions, and the impulse is greater, because of all the an- 
gles by which the water is deflected ; a is the least ; hence each particle 
that does not move in the normal plane exerts a greater pressure than the 

2 sin. a 2 o — v 

particle in that plane, and P == ^—. — : r, X V w. 

r * ' 1 -|- sin. cr g 

Impulse and. Resistance against Sxirfaces. 

The coefficient, of resistance, C, or the number with which the height 
due to the velocity is to be multiplied, to obtain the height of a column 
of water measuring this hydraulic pressure, varies for bodies of different 
figures, and only for surfaces which are at right angles to the direction of 
motion is it nearly a definite quantit}^. 

Illustration. — If a wind impinges with a velocity of 20 feet per second against 
the four arms of a wind-mill, each having an area of '^00 square feet, and an angle 
of inclination of 75° to the direction of the wind, then is the impinging force of the 

2 (sin 75°) ^ 20^ 

wind in the direction of the axis of the wheel === 1.85x^— i .* „ rn n Xx — X 4 X 200 

l + (sm. <5°)2 2g- 

X-0S1; .0SI — the assumed density of the wind, as given by Weisbach, § £01. 

According to the experiments of Du Buat and Thibault, C = 1.85 for 
the impulse of air or water against a plane surface at rest, and for the re- 
sistance of air or water against a surface in motion, C == 1.4. In each 
case about .66 of the effect is expended upon the front surface, and .34 
upon the rear. 

The resistance of air to a surface revolving in a circle has been found 

by Button and others to vary, but it may be expressed by C = 1.5. If 

the surface is not at right angles to the direction of the motion, but makes 

. ' . t , •'' i"" « 2C sin. a 2 

with it an acute angle a, then for C put r— : — : ,. 

r 1 -f sin. a 2 

PERCUSSION OF FLUIDS. 

When a stream strikes a plane perpendicular to its action, the force 
with 'which it strikes is estimated by the product of the area of the plane, 
the density of the fluid, and. the square of its velocity. 

Or, A d V 2 = P, A representing the area in square feet, d the weight of the 
fluid in lbs., and V the velocity in feet per second. 

If the plane is itself in motion, then the force becomes Ad (V — v) 2 = T, 
v representing the velocity of the plane. 

If C represent a coefficient to be determined by experiment, and h the 
height due to the velocitj- V, then V 2 — 2gh, and the expression for the 
force becomes A C 2 g h = P. 



444 



MOTION OF BODIES IN FLUIDS. 



MOTION OF BODIES IN FLUIDS. 



m. 



From the following Table several practical inferences may be drawn, 

1. That the resistance is nearly as the surface, the resistance increasing 
but a very little above that proportion in the greater surfaces. 

2. The resistance to the same surface is nearly as the square of the ve- 
locity, but gradually increasing more and more* above that proportion as 
the velocity increases. 

3. When the after parts of the bodies are of different forms, the resist- 
ances are different, though the fore parts be alike. 

4. The resistance on the base of a hemisphere is to that on the convex 
side nearly as 2.4 to 1, instead of 2 to 1, as theory assigns the proportion. 

5. The resistance on the base* of a cone is to that on the vertex nearly 
as 2.3 to 1. And in the same ratio is radius to the sine of the angle of the 
inclination of the side of the cone to its path or axis. So that, in this in- 
stance, the resistance is directly as the sine of the angle of incidence, the 
transverse section being the same, instead of the square of the sine. 

6. Hence we can ascertain the altitude of a column of air, the pressure 
of which is equal to the resistance of a body moving through it with any 
velocity. 

Thus, let a = area of the section of the body, similar to any of those in the 
Table, perpendicular to the direction of motion ; R = the resistance to the ve- 
locity in the Tab\e; and x — the altitude sought, of a column of air the base 
of which = a and its pressure R. Then ax~ the volume of the column 
in feet, and 1.2 ax its weight in ounces. 

5 R 

Therefore, 1.2aa; = R, and x = -X — is the altitude sought in feet, viz.. 

5 ' 6 a & ' 

- of the quotient of the resistance of a bod} T divided by its transverse sec- 

6 

tion, which is a constant quantity for all similar bodies, however different 

in magnitude, since the resistance R is as the section a. 

Illustration. — The convex side of the large hemisphere, the resistance of which = 
.634, or at a velocity of 16 feet per second : then Rz=.634, and x— T X — 2.38 feet, 

the altitude of a column of air, the pressure of which is equal to the resistance upon 
a spherical surface, at a velocity of 16 feet. 

Resistance of different Figures at different "Velocities in 

^\.ir. 



Veloci- 
ty per 


Cone. 


Sphere. 


Cylinder. 


Hemis 


phere. 


Small 
Hemi- 










Second. 


Vertex. 


Base. 






Flat. 


Round. 


sphere. 


Feet. 


Oz. 


Oz. 


Oz. 


Oz. 


Oz. 


Oz. 


Oz. 


3 


.028 


.064 


.027 


.05 


.051 


.<)2 


.028 


4 


.048 


.109 


.047 


.09 


.096 


.039 


.048 


5 


.071 


.162 


.068 


.143 


.148 


.063 


.072 


8 


.168 


.382 


.162 


.36 


.368 


.16 


.184 


9 


.211 


.478 


.205 


.456 


.464 


.199 


.233 


10 


.26 


.587 


.255 


.565 


.573 


.242 


.287 


15 


.589 


1.316 


.581 


1.327 


1.336 


.5.V2 


.6*61 


20 


1.069 


2.54 


1.057 


2.528 


2.542 


1.033 


1.196 



The diameter of all the figures but the small hemisphere was 6% ins., and the al- 
titude of the cone 6% ins. The small hemisphere was 4% ins. 
The angle of the side of the cone and its axis is, consequently, 25° 42' nearly. 

* This is a refutation of the popular assertion that a taper spar can be towed in water easiest 
when the base is foremost. 



MOTION OF BODIES IN FLUIDS. 



445 



If a body move through a fluid at rest, or the fluid move against the 
body at rest, the resistance of the fluid against the body is as the square 
of the velocity and the density of the fluid ; that is, R = d v 2 . For the re- 
sistance is as the quantity of matter or particles struck, and the velocity 
with which they are struck. But the quantity or number of particles 
struck in any time are as the velocity and the density of the fluid ; there- 
fore the resistance of a fluid is as the density and square of the velocity. 

The resistance to a plane is as the plane is greater or less, and therefore 
the resistance to a plane is as its area, the density of the medium, and the 
square of the velocity ; that is, R = a d v 2 . 

If the motion is not perpendicular, but oblique to the plane or to the 
face of the body, then the resistance in the direction of the motion will be 
diminished in the triplicate ratio of radius to the sine of the angle of in- 
clination of the plane to the direction of the motion, or as the cube of ra- 
dius to the cube of the sine of that angle. So that R =a adv 2 s 3 , 1 == radius, 
and s — sine of the angle of inclination. 



Ta"ble of* the Resistance to an Area of one Square Foot 
moving throngh "Water, or contrari^^rLse. 



Angle of 
Surface 


Pressure per 


Square 


Foot for toe following Velocities per Foot 


with 








per Minute 








Plane of 


















Current. 


60 


120 


180 


240 


300 


480 


600 


900 


Degrees. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


6 


.022 


.09 


.202 


.359 


.561 


1.435 


2.242 


5.046 


7 


.027 


.109 


.246 


.437 


.682 


1.747 


2 .73 


6.142 


8 


.033 


.133 


.298 


.53 


.829 


2.122 


3.315 


7.459 


9 


.039 


.156 


.351 


.624 


.975 


2.496 


3.9 


8.775 


10 


.045 


.179 


.404 


.718 


1.121 


2.87 


4.485 


10.091 


15 


.089 


.355 


.798 


1.42 


2.218 


5.678 


8.872 


19.963 


20 


.152 


.608 


1.369 


2.434 


3.802 


9.734 


15.21 


34.222 


25 


.235 


.94 


2.115 


3.76 


5.874 


15.038 


23.497 


52.869 


30 


.338 


1.353 


3.045 


5.413 


8.458 


21.653 


33.832 


76.123 


35 


.449 


1.798 


4.045 


7.192 


11.237 


28.766 


44.947 


101.132 


40 


.565 


2.258 


5.081 


9.032 


14.113 


36.13 


56.452 


127.018 


45 


.665 


2.66 


5.985 


10.639 


16.624 


42 . 557 


66.495 


149.614 


50 


.749 


2.995 


6.739 


11.981 


18.72 


47.923 


74.88 


168.48 


55 


.812 


3.249 


7.31 


12.995 


20.304 


51.979 


81.217 


182.739 


60 


.864 


3.455 


7.775 


13.822 


21.596 


55.286 


86.385 


194.366 


65 


.902 


3.607 


8.117 


14.43 


22.547 


57.72 


90.187 


202.922 


70 


.932 


3.728 


8.389 


14.914 


23.302 


59.654 


93.21 


209.722 


75 


.953 


3.81 


8.573 


15:241 


23.814 


60.965 


95.257 


214.329 


80 


.966 


3.857 


8.678 


15.428 


24.107 


61.714 


96.427 


216.926 


85 


.973 


3.892 


8.757 


15.569 


24.326 


62.275 


97.305 


218.936 


90 


.975 


3.9 


8.775 


15.6 


24.375 


62.4 


97.5 


219.375 



The resistance to a plane, from a fluid acting in a direction perpendicu- 
lar to its face, is equal to the weight of a column of the fluid, the base of 
which is the plane and altitude equal to that which is due to the velocity 
of the motion, or through which a heavy body must fall to acquire that 
velocity. 

The resistance to a plane running through a fluid is the same as the 
force of the fluid in motion with the same velocity on the plane at rest. 
But the force of the fluid in motion is equal to the weight or pressure 
which generates that motion, and this is equal to the weight or pressure 
of a column of the fluid, the base of which is the area of the plane, and its 
altitude that which is due to the velocity. 

Pp* 



446 MOTION OF BODIES IN FLUIDS. 

1. If a = the area of a plane, v its velocity, n the density or specific gravity 
of the fluid, % g = 16.0833 feet, and the altitude due to the velocity v, be- 

v* , v 2 anv 2 . . _ 

ing x-, then axnx^- = -~ — = Me resistance K. 

2. If the direction of motion be not perpendicular to the face of the 

a n v s 
plane, but oblique to it, then — t — R. 

3. If W represent the weight of a body, a being resisted by the absolute 
force R, then the retarding force f or — = -^ — ■ — . 

Illustration. — If a plane 1 foot square be moved through water at the rate of 

32.166 feet per second, then * ■ z=16.0S3, the space a body would require to fall 

to acquire a velocity of 32.166 feet per second; therefore 1x62.5 (weight of a cubic 

32.166 2 
foot of water) X ^-nr^r = IMS lbs. = resistance of the plane. 

. u4. odd 

The resistance to a sphere moving through a fluid is but half the resist- 
ance to its great circle, or to the end of a cylinder of the same diameter, 

moving with an equal velocity, — j = R, being the half of that of a 

C} T linder of the same diameter, r representing radius. 

Illustration — An iron ball of 9 lbs., having a diameter of 4 ins., when projected 
at a velocity of 1600 feet per second, will meet a resistance which is equal to a weight 
of 132.68 lbs. over the pressure of the atmosphere. 

The resistance that a body sustains in moving through a fluid is in propor* 
tion to the square of the velocity ; and it is the same, whether the plane 
moves against the fluid or the fluid against the plane. 

The progression of a solid floating body, as a boat in a channel of still 
water, gives rise to a displacement of the water surface, which advances 
with an undulation in the direction of the body, and this undulation is 
termed the Wave of Displacement. 

The resistance of a fluid to the progression of a floating body increases 
as the velocity of the body attains the velocity of the wave of displace- 
ment, and it is greatest when the two velocities" are equal. 

In the motion of elastic fluids, it appears from experiments that oblique 
action produces nearly the same effect as in the motion of water, in tne 
passage of curvatures," apertures, etc. 



STRENGTH OF MATERIALS. 447 

STRENGTH OF MATERIALS. 
ELASTICITY AND STRENGTH. 

The component parts of a rigid body adhere to each other with a 
force which is termed Cohesion. 

Elasticity is the resistance which a body opposes to a change of form. 

Strength is the resistance which a body opposes to a permanent sep- 
aration of its parts. 

Elasticity and Strength, according to the manner in which a force is 
exerted upon a body, are distinguished as Tensile Strength, or Abso- 
lute Resistance ; Transverse Strength, or Resistance to Flexure ; Crush- 
ing Strength, or Resistance to Compression ; Torsional Strength, or Re- 
sistance to Torsion ; and Detrusive Strength, or Resistance to Shearing. 

The limit of Stiffness is flexure, and the limit of Strength or Resist- 
ance is fracture. 

Resilience, or toughness of bodies, is strength and flexibility com- 
bined ; hence any material or body which bears the greatest load, aud 
bends the most at the time of fracture, is the toughest. 

The Specific Gravity of Iron is ascertained to indicate very correct- 
ly the relative degree of its strength. 

The Neutral Axis, or Line of Equilibrium, is the line at which ex- 
tension terminates and compression begins. 

The resistance of cast iron to crushing and tensile strains is, as a 
mean, as 4.3 to 1.* 

English cast iron has a higher resistance to compression, and a less tensile resist- 
ance, than American. 

The mean tensile strength of American cast iron, as determined by 
Major Wade for the U. S. Ordnance Corps, is 31829 lbs. per square 
inch of section ; the mean of English, as determined by Mr. E. Hodg- 
kinson for the Railway Commission, etc., in 1849, is 19484 lbs. ; and by 
Col. Wilmot at Woolwich, in 1858, for gun-metal, is 23257 lbs. 

The ultimate extension of cast iron is the 500th part of its length. 

The mean transverse strength of American cast iron, also determ-. 
ined by Major Wade, is 681 lbs. per square inch, suspended from a bar 
fixed at one end and loaded at the other ; and the mean of English, as 
determined by Fairbairn, Barlow, and others, is 500 lbs. 

The resistance of wrought iron to crushing and tensile strains is, as 
a mean, as 1.5 to 1 for American ; and for English 1.2 to 1. 

The mean tensile strength of American wrought iron, as determined 
by Prof. Johnson, is 55900 lbs. ; and the mean of English, as determ- 
ined by Capt. Brown, Barlow, Brunei, and Fairbairn, is 53900 lbs.f 

The ultimate extension of wrought iron is the 600th part of its 
length. 

The resistance to flexure, acting evenly over the surface, is nearly 
}4 the tensile resistance. 

* The experiments of Mr. Hodgkinson on iron of low tensile strength gives a mean of 6.595 to 1. 
f The results, as given by Mr! Telford, included experiments upon Swedish iron; hence they are 
omitted in this summary. 



448 STRENGTH OF MATERIALS. 

iMod/ulns of Elasticity. 

The Modulus or Coefficient of the Elasticity of any substance is the meas- 
ure of its elastic reaction or force, and is the height of a column of the 
same substance, capable of producing a pressure on its base, which is to 
the weight causing a certain degree of compression, as the length of the 
substance is to the diminution of its length. 

It is computed by this analogy : As the extension or diminution of the 
length of any given substance is to its length in inches, so is the force that 
produced that extension or diminution to the modulus of its elasticity. 
p 

Or, x : P : : I : w — — x representing the length a substance 1 inch square 

and lfoot in length would be extended or diminished by the force P, and w 
the weight of the modulus in lbs. 

To Compute tlie "Weight of tlie Modtilus of Elasticity of 
a Sifbstance. 

Rule. — As the extension or compression of the length of any substance 
is to its length, so is the weight that produced that extension or compres- 
sion to the modulus of elasticity in pounds avoirdupois. 

Example.— If a bar of cast iron, 1 inch square and 10 feet in length, is extended 
.O0S inch, with a weight of 1000 lbs., what is the weight of its modulus of elasticity ? 
.COS : 120 (10x12) : : 1000 : 15 000 000 lbs. 

Note "When the weight of the modulus of elasticity of a substance is known, the 

height of it can bs readily computed by dividing the weight by the weight of a bar 
of the substance 1 inch square and 1 foot in length. 

Ex. 2.— If a wrought-iron chain, 60 feet in length and .2 inch in diameter, is sub- 
jected to a strain of 150 lbs., what will it be extended? 

The modulus of elasticity of iron wire is 2G S^S 000 lbs., and the area of chain .2*X 
.7S54 = .31416. 

150 

— : = 417.463 lbs. per square inch, and 60x12 = 720 ins. 

.ol41b 

To Compute tlie "Weiglit -wlieix tlie Heiglit is given. 

Rule. — Multiply the weight of 1 foot in length of the material by the 
height of its modulus in feet, and the product will give the weight. 

To Conapute tlie Heiglit of tlie nVTocUilns of Elasticity. 

Rule. — Divide the weight of the modulus of elasticity of the material 
by the weight of 1 foot of it, and the quotient will give the height in feet. 

From a series of elaborate experiments by Mr. E. Hodgkinson for the 
Railway Commission, he deduced the following formulae for the extension 
and compression of cast and wrought iron : 

Cast-iron Extension : 13 934 040 j — 2 907 432 000 ^ = W. 

Cast-iron Compression : 12 931 560 -. — 522 979 200 -^ = W, e and c rep- 
resenting the extension and compression, and I the length in inches. 

Illustration. — What weight will extend a bar of cast iron, 4 inches square and 10 
feet in length, to the extent of .2 inch? 

2 .22 

13 C34 040x^ — 2 007 432 000 — — — 23223.4 — 8076.2 = 15147.2, which X 4 ins 
120 12U 2 

= C05SS.S lbs. 



STRENGTH OF MATERIALS. 



449 



Modulus of Elasticity and Weight of various Substances. 

;- Substances. 



Ash 

Brass, yellow.. . 

" wire 

Copper, cast . . . 

Elm 

Fir, red 

Glass 

Gun-metal 

Hempen fibres . 

Ice 

Iron, cast 

44 wrought. . 

u wire 

Lead, cast 



lasticitj 


r and. "W 


Height in 


Weight in 


Feet. 


Lbs. 


4 970 000 


1 656 670 


2 4GOOO0 


8464 000 


4112 000 


14632 720 


4800000 


IS 240 000 


5680000 


1490500 


8330000 


2016 000 


4440000 


5550 000 


2790000 


8 844300 


5000000 


170000 


6 000000 


2370000 


5750 000 


17 968 500 


7 550000 


25830000 


8377000 


28230500 


146000 


720000 



Lignum- vitse. 

Limestone 

Mahogany 

Marble, white 

Oak 

Pine, pitch 

u white 

Steel, cast 

u wire 

Stone, Portland . . . 

Tin, cast 

Willow 

Yellow pine, mean . 
Zinc 



Height in 
Feet. 



1850000 
2400000 
6570 000 
2 150 000 
4750 000 
8700 000 

8 970000 
8530 000 

9 000000 
1 672 000 
1 053 000 
620000*) 

105)0000 
44S0000 



Weight in 
Lbs. 



10S0400 

3 300000 

2071000 

2508000 

1710 000 

2 430000 

1S30000 

26 650 000 

28689 000 

1 718 800 

3510 000 

1 426 000 

2 100 000 

13440 000 



The elasticity of Ivory, as compared to Glass, is as .95 to 1. 



To Compute tlie Length of* a Prism of a HVIaterial widen, 
would "be severed, "by its own "Weight when. Suspended. 

Rule. — Divide the tensile resistance of the material by the weight of a 
foot of it in length, and the quotient will giv r e the length. 

Modulus of Cohesion, or Length in Feet required to Tear asunder the following 
Substances : 

Rawhide 15375 feet ; Hemp twine 75000 feet ; Catgut. . . . 25000 feet. 

TENSILE STRENGTH. 

Tensile Strength is the resistance of the fibres or particles of a body to 
separation. It is therefore proportional to their number, or to the area of 
its transverse section. 

The fibres of wood are strongest near the centre of the trunk or limb of 
a tree. 

Cast Iron. — Experiments on Cast-iron bars give a tensile strength of 
from 4000 lbs. to 5000 lbs. per square inch of its section, as just sufficient 
to balance the elasticity of the metal ; and as a bar of it is extended the 
5500th part of its length for every ton of direct strain per square inch of 
its section, it is deduced that its elasticity is fully excited when it is ex- 
tended less than the 3000th part of its length, and the extension of it at 
its limit of elasticity is estimated at the 1200th part of its length. 

The mean tensile strength, then, of cast iron being from 1G000 to 20000 
lbs., the Value of it, when subjected to a tensile strain, ma} T be safely esti- 
mated at from % to X of this, or of its breaking strain. 

A bar of cast iron will contract or expand .000006173, or the 162000th 
of its length for each degree of heat; and assuming the extreme range of 
the temperature in this country 140° (— 20° -f 120°), it will contract or 
expand with this change .0008642, or the 1157th part of its length. It 
shrinks in cooling from .0104 to .0118th of its length. 

It follows, then, that as 2240 lbs. will extend a bar the 5500th part of 
its length, the contraction or extension for the 1157th part will be equiva- 
lent to a force of 10648 lbs. (4% tons) per square inch of section. 

Cast iron (Greenwood) at three successive meltings gave tenacities of 
21300, 30100, and 35700 lbs. 

Cast iron at 2.5 tons per square inch will extend the same as wrought 
iron at 5.6 tons. 



450 



STRENGTH OF MATERIALS. 



The mean tensile strength of four kinds of English cast iron, as determ- 
ined by the Commissioners on the application of iron to Railway Struc- 
tures, was 15711 lbs. per square inch (7.014 tons) ; and the mean ultimate 
extension was, for lengths of 10 feet, .1997 inch, being the 600th part of its 
length ; and this weight would compress a bar the 775th part of its length. 

Tensile strength of the strongest piece of cast iron ever tested — 15970 
lbs. This was a mixture of grades 1, 2, and 3 of Greenwood iron, and at 
the 3d fusion. 

Wrought Iron. — Experiments on Wrought-iron bars give a tensile 
strength of from 18000 lbs. to 22400 lbs. per square inch of its section, as 
just sufficient to balance the elasticity of the metal; and as a bar of it is 
extended the 10000th part of its leng'th for every ton of direct strain per 
square inch of its section, it is deduced that its "elasticity is fully excited 
when it is extended the 1000th part of its length, and the extension of it 
at its limit of elasticity is estimated at the 152uth part of its length. 

The mean tensile strength of wrought iron being from 55000 to 65000 
lbs., the Value of it, when subjected to a tensile strain, may be safely esti- 
mated at from % to % of this, or of its breaking strain. 

A bar of wrought iron will expand or contract .000006614, or the 
151200th part of its length for each degree of heat; and assuming, as be- 
fore stated for cast iron, that the extreme range of temperature in the air 
in this country is 140°, it will contract or expand with this change .000926, 
or the 1080th of its length, which is equivalent to a force of 20740 lbs. 
(9^ tons) per square inch of section. 

Experiments upon wrought iron, to determine the results from repeated 
heating and laminating, furnished the following : 

From 1 to 6 reheatings and rollings, the tensile stress increased from 
43904 lbs. to 61824 lbs., and from 6 to 12 it was reduced to 43904 again. 

The tensile force of metals varies with their temperature, generally de- 
creasing as the temperature is increased. In silver the tenacity decreases 
more rapidly than the temperature ; in copper, gold, and platinum it de- 
creases less rapidly than the temperature. 

In iron, the tensile strength at different temperatures is as follows : 60°, 1 ; 
114°, 1.14 ; 212°, 1.2 ; 250 b , 1.32 ; 270°, 1.35 ; 325°, 1.41 ; 435°, 1.4. 

Stirling's Mixed or Toughened Iron. — By the mixture of a portion of 
malleable iron with cast iron, carefully fused in a crucible, a tensile strain 
of 25764 lbs. has been attained. This mixture, when judiciously managed 
and duly proportioned, increases the resistance of cast iron about one 
third ; the greatest effect being obtained with a proportion of about 30 
per cent, of malleable iron. 

Bronze (gun-metal) varies in tenacity from 23000 to 54500 lbs. 

Elements connected with the Tensile Resistance ofvari- 
ous Substances. 



a S3 



.3 

.34 

.35 

.35 

.20 

.46 

.45 





■--r-~ . ■-}-.- 




C X => >> Z = - 




£■=•=•1 S5f 


Substances. 






=»=s tl^ 




!££« *~ v = c 




H - =s~~ 




Lbs. 




Beech 


3355 


.3 


Cast iron, English 


4000 


.22 


u American 


5000 


.2 


Oak 


2S5(J 


.23 


Steel plates, blue temp'd . 


51720 


.02 


u wire 


55700 
8332 


.5 
.23 


Yellow pine 





| i^s 










Substances. 


9 — = s 

~ ■ - 5 




i. u «-~ 




5 2«2H 




E- ~ 




Lbs. 


Wrought iron, ordinary. 


17600 


" " Swedish . 


24400 


l - " English . 


/1SS50 
\ 22400 


14 »« American 




*« wire, No. 9, unannealed 


47532 


" u " annealed.. 


36300 



STRENGTH OF MATERIALS. 



451 



TENSILE STRENGTH OF MATERIALS. 

"Weight or Power required to Tear asunder one 
Square Inch. 



Copper, wrought 

' 4 rolled 

" cast, American 

u wire 

" bolt 

Iron, cast, Low Moor, No. 2 

" Clyde, No. 1 

" No. 3 

" Calder, No. 1 

" Stirling, mean 

44 mean of American . . . 

" mean* of English 

" Greenwood, American 

" gun-metal, mean 

" wrought wire 

44 best Swedish bar 

" Russian bar 

" English bar 

44 rivets, American ..... 

" bolts 

" hammered 

44 mean of English ..... 

44 rivets, English 

44 crankshaft ... 

44 turnings 

44 plates, boiler, Ameri-) 
can j 



34000 
36*000 
24250 
61200 
36800 
14076 
16125 
23468 
13735 
25764 
31829 
19484 
45970 
37232 
103000 
72000 
59500 
56000 
53300 
52250 
53913 
53900 
65000 
44750 
55800 
48000 
62000 



Iron, plates, mean, English 

44 44 lengthwise 

44 44 crosswise 

44 inferior, bar 

44 wire, American 

44 " 44 16diam. 

44 scrap . 

Lead, cast 

44 milled 

44 wire 

Platinum, wire 

Silver, cast 

Steel, cast, maximum .... 

44 44 mean 

44 blistered, soft j 

44 shear 

44 chrome, mean 

44 puddled, extreme . . 
44 American Tool Co .. 
44 plates, lengthwise . . 
44 44 crosswise . . . 
44 razor 

cast, block 

Banca 



Tin. 
tt 

Zinc 



sheet . 



Lbs. 



51000 

53800 • 

48800 

30000 

73600 

80000 

53400 

1800 

3320 

2580 

53000 

40000 

142000 

88657 

133000 

104000 

124000 

170980 

173817 

179980 

96300 

93700 

150000 

5000 

2122 

3500 

16000 



Lake Superior and Iron Mountain charcoal bloom iron has resisted 90000 
lbs. per square inch. 



MISCELLANEOUS SUBSTANCES. 



Brick, well burned . 
44 fire 



44 inferior , 

Cement, blue stone 

44 hj-draulic 

44 Harwich 

44 Portland, 6 mos. . . 

44 Sheppy 

44 Portland 1, sand 3. 

Chalk... 

Glass, crown 

Gutta-percha 

Hydraulic lime 

44 44 mortar .... 

Ivory 

Leather belts 




Limestone 

Marble, Italian 

44 white 

Mortar, 12 j-ears old . 

Plaster of Paris 

Rope, Manila 

4 4 hemp, tarred . . , 

44 wire 

Sandstone, fine grain 

Slate 

Stone, Bath 

44 Craigleth 

44 Hailes 

44 Portland , 

Whalebone 



Lbs. 



670 

2800 

5200 

9000 

60 

72 

9000 

15000 

37000 

200 

12000 

352 

400 

360 

857 

1000 

7600 



By Commissioners, on application of iron to Railway Structures. 



452 



STRENGTH OF MATERIALS. 



Gold 5, Copper 1 . 

Brass 

11 j'ellow 

Bronze, least 

" greatest . 



Ash 

Beech 

Box 

Bay 

Cedar 

Chestnut, sweet 

Cypress 

Deal, Christiana . . . 

Elm 

Lance 

Lignum- vitas 

Locust 

Mahogany 

" Spanish . 



COMPOSITIONS. 
Lbs. 



50000 

42000 
18000 
17698 
56788 



Copper 10, Tin 1 

44 8, 4 ' 1, gun-met. 

" 8, " 1, sm'lbars 

Tin 10, Antimony 1 

Yellow metal 



woods. 



| Lbs. 



14000 
11500 j 
20000 ! 
14000 ! 
11400 
10500 

6000 
12400 
13400 
23000 
11800 
20500 
21000 
12000 

8000 



Maple 

Oak, American white. 

44 English , 

li seasoned , 

44 African 

Pear 

Pine, pitch , 

44 larch 

44 American white 

Poplar , 

Spruce, white , 

Sycamore , 

Teak , 

Walnut 

Willow 




10500 
11500 
10000 
13600 
14500 

9800 
12000 

9500 
11800 

7000 
10290 
13000 
14000 

7800 
13000 



Results of Experiments on tlie Tensile Strength of 
Wrouglit-irou Tie-rods. 



Common English Iron, 1%. Inches in Diameter. 

Description of Connection. 



Breaking Weight, 



Semicircular hook fitted to a circular and welded eye , 

Two semicircular hooks hooked together 

Right-angled hook or goose-neck fitted into a cylindrical eye 

Two links or welded eyes connected together 

Straight rod without any connection articulation , 



Lbs. 

14000 
16220 
29120 
4S1G0 
5G000 



Iron bars when cold rolled are materially stronger than when only hot 
rolled, the difference being in some cases as great as 3 to 2. 



WIRE ROPES. 

Result of Experiments on tlie Tensile Strength of Iron 
and. Steel "Wire Ropes. 



Charcoal Iron 

Wire Rope. 

Circum. 



Ins. 



Weight per 
Foot. 



Lbs. 

X 

IK 



Breaking 
Weight. 



Lbs. 

13440 
44800 



Steel Wire 
Rope. 
Circum. 



Ins. 



Stretch in 6 Weight per 
Feet. Foot. 



Breaking 
Weight. 



Ins. 



Lbs. 

% 



Lbe. 

23600 
i -36000 



Tensile Strength of Copper at different Temperatures. 



Temp. 


Strength in Lbs. 


Temp. 


Strength in Lbs. 


Temp. 


Strength in Lba 


122° 
212° 
302° 


33079 
32187 
30872 


482 ° 
545° 
602° 


26981 
25420 

22302 


801° 

912° 
1016° 


18854 
14789 
11054 






STRENGTH OF MATERIALS. 



453 



Extension of Cast-iron Bars,When suspended Vertically. 
1 Inch Square and 10 Feet in Length. Weight applied at one End. 



Weight ap- 
plied. 


Extension. 


Set. 


Weight ap- 
plied. 


Extension. 


Set. 


Lbs. 

529 
1058 
2117 


Ins. 

.0044 

.0092 
.0190 


Ins. 

.000015 
.000059 


Lbs. 

4234 

8468 

14820 


Ins. 

.0397 

.0871 

I .1829 


Ins. 

.00265 
.00855 
.02555 



Steel. 



The tensile strength of steel increases by reheating and rolling up to 
the second operation, but decreases after that. 

Ratio of the Dnctility- and. :Mallea"bility of* Mietals. 



In the order of 

Wire-drawing 

Ductility. 


In the order of 
Laminable 
Ductility 


In the order of 

Wire -drawing 

Ductility. 


In the order of 
Laminable 
Ductility 


In the order of 

Wire-drawing 

Ductility. 


In the order of 
Laminable 
Ductility. 


Gold. 

Silver. 

Platinum. 


Iron. 

Copper. 
Zinc. 


Tin. 
Lead. * 
Nickel. 


Gold. 

Silver. 

Copper. 


Tin. 

Platinum. 

Lead. 


Zinc. 
Iron. 
Nickel. 



The relative resistance of Wrought Iron and Copper to tension and 
compression is as 100 to 54.5. 

TRANSVERSE STRENGTH. 

The Transverse or Lateral Strength of any Bar, Beam, Rod, etc., is in 

f>roportion to the product of its breadth and the square of its depth ; in 
ike-sided beams, bars, etc., it is as the cube of the side, and in cylinders 
as the cube of the diameter of the section. 

When one End is fixed and the other projecting, the strength is inversely 
as the distance of the weight from the section acted upon ; and the strain 
uporwmy section is directly as the distance of the weight from that section. 
When both Ends are supported only, the strength is 4 times greater for 
an equal length, when the weight is applied in the middle between the 
supports, than if one end only is fixed. 

When both Ends are fixed, the strength is 6 times greater for an equal 
length, when the weight is applied in the middle, than if one end only is 
fixed. 

The strength of any beam, bar, etc., to support a weight in the centre 
of it, when the ends rest merely upon two supports, compared to one when the 
ends are fixed, is as 2 to 3. 

When the Weight or Strain is uniformly distributed, the weight or strain 
that can be supported, compared with that when the weight or strain is 
applied at one end or in the middle between the supports, is as 2 to 1. 

In Metals, the less the dimension of- the side of a beam, etc., or the di- 
ameter of a cylinder, the greater its proportionate transverse strength : 
this is in consequence of their having a greater proportion of chilled or 
hammered surface compared to their elements of strength, resulting from 
dimensions alone. 

The strength of a Cylinder, compared to a Square of like diameter or 
sides, is as 6.25 to 8. The strength of a Hollow Cylinder to that of a Solid 
Cylinder, of the same length and volume, is as the greater diameter of the 
former is to the diameter of the latter. 

The strength of an Equilateral Triangle, fixed at one End and loaded at 
the other, having an edge up, compared to a Square of the same area, is as 

Qq 



454 STRENGTH OF MATERIALS. 

22 to 27 ; and the strength of an equilateral triangle, having an edge down, 
compared to one with an edge up, is as 10 to 7. 

Note. — In these comparisons, the beam, bar, etc., is considered as one end being 
fixed, the weight suspended from the other. In Barlow and other authors the com- 
parison is made when the beam, etc., rested upon supports. Hence the stress is con- 
trariwise. 

Detrusion is the resistance that the particles or fibres of materials op- 
pose to their sliding upon each other. Punching and shearing are detru- 
sive strains; 

Deflection. — When a bar, beam, etc., is deflected by a cross-strain, the 
side of the beam, etc., which is bounded by the concave surface, is com- 
pressed, and the opposite side is extended. 

In Stones .and Cast metals, the resistance to compression is greater than 
the resistance to extension. 

In Woods, the resistance to extension is greater than the resistance to 
compression. 

. The general law regarding deflection is, that it increases, ceteris pari- 
bus, directly as the cube of the length of the beam, bar, etc., and inversely 
as the breadth and cube of the depth. 

The resistance of Flexure of a body at its cross-section is very nearly 
3^ of its tensile resistance. 

The stiffest bar or beam that can be cut out of a cylinder is that of which 
the depth is to the breadth as the square root of 3*to 1 ; the strongest, as 
the square root of 2 to 1 ; and the most resilient, that which has the breadth 
and depth equal. 

Relative Stiffness of* IMaterials to Resist a Transverse 

Strain. 

Ash 089 Elm 073 

Beech .073 Oak 095 

Cast iron 1. White pine 1 

The strength of a Rectangular Beam in an inclined position, to resist a 
vertical stress, is to its strength in a horizontal position as the square of 
radius to the square of the cosine of elevation ; that is, as the square of 
the length of the beam to the square of the distance between its points of 
support, measured upon a horizontal plane. 

Experiments upon bars of cast iron, 1, 2, and 3 inches square, give a re- 
sult of transverse strength of 447, 348, and 338 lbs. respectively; being in 
the ratio of 1, .78, and .756. 

The strongest rectangular bar or beam that can be cut out of a cylinder 
is one of which the squares of the breadth and depth of it, and the "diame- 
ter of the cylinder, are as 1, 2, and 3 respectively. 

The ratio of the crushing to the transverse strength is nearly the. same in 
glass, stone, and marble,' including the hardest and softest kinds. 

Green sand iron castings are 6 per cent, stronger than dry, and 30 per 
cent, stronger than chilled ; but when the castings are chilled and an- 
nealed, a gain of 115 per cent, is attained over those made in green sand. 

Chilling the under side of cast iron very materially increases its strength. 

Woods. — Beams of wood, when laid with their annual or annular layers 
vertical, are stronger than when thev are laid horizontal, in the proportion 
of8to7. 

Woods are denser at the roots and at the centre of their trunks. Their 
strength decreases with the decrease of their density. 

Oak loses strength in drying. 



Wrought iron ... 1.3 
Yellow pine 087 



STRENGTH OF MATERIALS. 



455 



Transverse Strength, of* Materials, deduced from tlie Ex- 
periments oF TJ. S. Ordnance Department, Barlow, Ften- 
nie, Stephenson, Hodgkinson, Faii/bairn, Pasley, Hat- 
field, and the Author. 

Reduced to the uniform Measure of One Inch Square, and one Foot in Length; 
Weight suspended from one End. 



Materials. 


Break- 
ing 
Weight. 


Value for 
general Use. 


Materials. 


Break- 
ing 

Weight. 


Value for 
general Use. 


METALS. 


Lbs. 






Lbs. 




/'means of 


50T 


125 to 160 


WEOUGHT IEON. 






Cast iron, \ four divi- 


G32 


155 " 210 




(700 






733 
772 


ISO » 240 
192 « 250 


American 


•^650 
(600 


160 to 200 


(^grades . . 






u mean by Maj. Wade 


681 


170 " 225 


English 


400 


100 4t 130 


" West Ft. Foundry, 






u 


550 


135 44 180 


980 


250 " 325 


Swedish* 


665 


165 u 210 


u English, Low Moor, 


MIXTUEE OF CAST AND 




cold blast 


472 


110 » 140 


WROUGHT IRON, etc. 






44 Ponkey, cold 


5S1 


145 " 190 


Cast iron,Blaenavon . 


— 


145 


44 hot blast, mean . . . 


500 


125 " 165 


" lOperct. of wr't 


— 


175 


" cold " " ... 


516 


130 " 170 


u 30 it u 


— 


230 


M Ystalyfera, cold bl' t 


770 


195 " 255 


u 50 « « 


, — 


1S5 


" mean of 65 kinds . . 


500 


125 " 165 


" and 2% per ct. 






14 mean of 15 kinds, 






of nickel, mean 


— 


180 


direct fr. the pig, 






44 Stirling, 2d qu. 


— 


154 


cold blast 


641 


160 " 215 


44 44 3d 44 





125 


44 planed bar 

" rough bar 


518 


130 " 170 







55 


534 


133 44 175 


Brass 





5S 


Steel, greatest 


1918 


850 " 450 


stones (American). 






Steel, puddled (per- 






Flagging, blue 


31. 


10 


manent bend) 


800 


170 « 225 


Freestone, Conn 


13. 


4 


WOODS. 






44 Dorchester 


10.8 


3X 


Ash 


16S 
130 
160 
160 


55 
32 
40 
53 


lt N. Jersey. 

" N. York . . 
Granite, blue, coarse. 


(201 

1l7.S 
24. 

IS. 


6^ 
6 




Birch 


8 


Chestnut 


6 


Deal, Christiana 


137 


45 


44 Quincy, Mass. 


26. 


8# 


Elm 


125 


30 


stones (English). 
Adelaide marble .... 






Hickory 


250 


55 


45 


IX 


Locust 


295 
202 


80 
65 


Arbroath 


17. 

90 


5% 


Maple 


Bangor slate 

Bath 


30 


Norway pine 


123 
208 


40 
50 


5.2 

68. 


IX 


Oak, African 


Caithness, paving, Sc. 


22 


44 American white 


230 


50 


Cornish granite 


22. 


7 


" " live . 


245 


55 


Craigleth sandstone . 


10.7 


3^ 


44 Canadian 


143 


36 


Darley sandst., Vict 1 a 


1.3 


4 


44 Dantzic 


122 


30 


Kentish rag 


35.8 


12 


" English 


140 
18S 


35 
45 




11 

43. 


3& 


u " superior 


Llangollen slate 


14 


Pitch pine 


136 


45 


Fark Spring sandst'e 
Portland oolite 


4.3 


1.4 


44 American. 


160 


50 


21.2 


7 


Riga fir 


94 


30 


Valentia, paving, lrel. 
Welsh, 44 


68 5 


23 


Teak 


206 
92 


60 
30 


157. 
26. 


55 


White pine 


1% 


44 American 


130 


45 


44 landing . . 


22.5 


Whitewood 


116 


38 


u paving... 


10.4 


W 



Increase in Strength of several Woods "by Seasoning. 

Ash 44.7 per cent. I Elm 12.3 per cent. I White pine . 9 per cent 

Beech ...61.9 u Oak 26.1 



: With 840 lbs. the deflection was 1 inch, and the elasticity of the metal destroyed. 



456 



STRENGTH OF MATERIALS. 



Concretes, Cements, etc. 



Materials. 



concretes (English). 
Fire-brick beam, Portl'd cement 
" sand 3 parts, lime 1 part 

CEMENTS (English). 

Blue clay and chalk 

Portland \ 

Sheppy 



Breaking 
Weight. 



3.1 
.7 

5.4 
87.5 
10.2 

5. 



Materials. 



bricks (English). 

Best stock 

lire-brick 

New brick 

Old brick 

Stock-brick, well burned 

u inferior, burned 



Breaking 
Weight. 



11.8 
14. 
10.7 
9.1 

5.S 
2.5 



Transverse Strength of Cast-iron Bars and. Oals: Beams 
of* "Varioxis ITigures. 

Reduced to the uniform Measure of One Inch Square of Sectional Area, and One 
Foot in Length. Fixed at one End; Weight suspended from the other. 



Form of Bar or Beam. 



Breaking; 
Weight. 



CAST IHON. 



Square 

Square, diagonal vertical . 



Cylinder . 



Lbs. 
673 

56S 

573 



Hollow cylinder; greater 
diameter twice that of 
lesser i 794 

Rectangular prism, 2 ins. 
deep X X in. depth 145G 

' ' 3 ins. deep X >i in. d epth 2392 

"4 " X^ " I 2652 



Form of Bar or Beam. 



j Breaking 
Weight. 






Equilateral 

edge up . 
Equilateral 

edge down 

2 ins. deep X 2 ins. wide 
X .26S in>. depth 



triangle, an 
triangle, an 



2 ins. deep X 2 ins. wide 
X .26S ins. depth..., 



OAK. 

Equilateral triangle, an 
edge up 

Equilateral triangle, an 
edge down 



Lbs. 
560 

95S 
2068 

555 

114 

130 



Transverse Strength of Solid and Hollo^w Cylinders of 
various !M!aterials. 

One Foot in Length. Fixed at one End; Weight suspended from the other. 



Materials. 


Solid 
External 
Diameter. 


Hollow 
Internal 
Diameter. 


Breaking 
Weight. 


Breaking Weight 
for 1 Inch extern- 
al Diam ,and pro- 
portionate intern- 
al Diam. 


WOODS. 

Ash 


Ins. 

2. 
2. 
2. 
1. 
2. 

3. 

2.87 


Ins. 
1. 

1.928 


Lfet. 

685 
004 
772 
75 
610 

12000 

190 


Lbs. 

86 




75 


Fir* 


97 


White pine 


75 
76 


METALS. 

Cast iron, cold blast 


444 


STONE-WARE. 

Rolled pipe of fine clay 


8 



Brick-Avovk. 

A brick arch, having a rise of 2 feet, and a span of 15 feet. 9 inches, and 
2 feet in width, with a depth at its crown of 4 inches, bore 358400 lbs. 
laid along its centre. 

* An inch-square batten, from the same plank as this specimen. broke at 139 lbs. 



STRENGTH OF MATERIALS. 457 

To Compute tlie Transverse Strength, of a Rectangular 
Beam or Bar. 

When a Beam or Bar is Fixed at one End, and Loaded at the other. 
Rule. — Multiply the Value of the material in the preceding Tables, or, 
as may be ascertained, by the breadth and square of the depth in inches, 
and divide the product by the length in feet. 

Note.— When the beam is loaded uniformly throughout its length, the result must 
be doubled. 

Example. — What are the weights each that a cast and wrought iron bar, 2 inches 
square and projecting 30 ins. in length, will bear without permanent injury ? 

The values for cast and wrought iron in this and the following calculations are 
assumed to be 225 and 180. 

Hence 225x2x22 = 1S00, which. H- 2.5 = 720 lbs.; and 180x2x22 = 1440, which, 
-f- 2.5 = 576 lbs. 

If the Dimensions of a Beam or Bar are required to Support a given 
Weight at its End. 'Rule. — Divide the product of the weight and the 
length in feet by the Value of the material, and the quotient will give the 
product of the breadth and the square of the depth. 

Example What is the depth of a wrought-iron beam, 2 inches broad, necessary 

to support 576 lbs. suspended at 30 ins. from the fixed end? 

576x2 5 

— "* = 8, which, -r- 2 ins. for the breadth == 4, and y/A. = 2 ins., the depth. 

When a Beam or Bar is Fixed at both Ends, and Loaded in the Middle. 
Rule. — Multiply the Value of the material by 6 times the breadth and 
the square of the depth in inches, and divide the product by the length in 
feet. 

Note. — When the beam is loaded uniformly throughout its length, the result must 
be doubled. 

Example.— What weight will a bar of cast iron, 2 ins. square and 5 feet in length, 
support in the middle, without permanent injury? 

225X2X6X22 = 10800, which, -h 5 = 2160 lbs. 

Or, If the Dimensions of a Beam or Bar are required to Support a given 
Weight in the Middle, between the Fixed Ends. Rule. — Divide the prod- 
uct of the weight and the length in feet by 6 times the Value of the mate- 
rial, and the quotient will give the product of the breadth and the square 
of the depth. 

Example. — What dimensions will a cast-iron square bar 5 feet in length require 
to support without permanent injury a stress of 2160 lb.s. ? 

-r— — - = ——— = 8, which, -r- 2 ins. for the assumed breadth, = 4, and \f 4 = 2 ins.. 
225x6 1350 7 v 

the depth. 

When the Breadth or Depth is required. Rule. — Divide the product ob- 
tained by the preceding rules by the square of the depth, and the quotient 
is the breadth ; or by the breadth, and the square root of the quotient is 
the depth. 

Illustration.— If 128 is the product, and the depth is 8 ; then 12S -4- 8 2 = 2, the 
breadth. Also, 128 H- 2 = 64, and y/Gi = 8, the depth. 

When the Weight is not in the Middle between the Ends. Rule. — Multi- 
ply the Value of the material by 3 times the length in feet, and the breadth 
and square of the depth in inches, and divide the product by twice the 
product of the distances of the weight, or stress from either end. 

Example. — What is the weight a cast-iron bar, fixed at both ends, 2 ins. square 
and 5 feet in length, will bear without permanent injury, 2 feet from one end? 
225X3X5X2X22 ^2_7000^ 22 
2X2X3 12 

Qq* 



458 STRENGTH OF MATERIALS. 

When a Beam or Bar is Supported at both Ends, and Loaded in the Mid' 
die. Rulk. — Multiply the Value of the material by 4 times the breadth 
and the square of the depth in inches, and divide the product by the length 
in feet. 

Note. — \Vhen the beam is loaded uniformly throughout its length, the result musk 
be doubled. 

Example. — What weight will a cast-iron bar, 5 feet between the supports, aud 2 
ins. square, bear in the middle, without permanent injury ? 

225x?xlx22 — 72000, which, -f5-. 1440 lbs. 
Or, If the Dimensions are required to Support a given Weight. Rulk. — 
Divide the product of the weight and length in feet by 4 times the Value of 
the material, and the quotient will give the product of the breadth, and 
the square of the depth. 

When the Weight is not in the Middle between the Supports. Rulk. — 
Multiply the Value of the material by the length in feet, and the breadth 
and the square of the depth in inches, and divide the product by the prod- 
uct of the distances of the weight, or stress from either support/ 

Example. — What weight will a cust-iron bar, 2 ins. square and 5 feet in length, 

support without permanent injury, at a distance of 2 feet from one end, or support? 

225x5x2x22 0000 



2X(5-2) "~ 



- = 1500 lbs. 



To Compute tlie Pressvire upon tlie Ends or npon tlie 
Supports. 

Rule. — 1. Divide the product of the weight and its distance from the 
nearest end or support by the whole length, and the quotient will give 
the pressure upon the end or support farthest from the weight. 

2. Divide the product of the weight and its distance from the farthest 
end, or support, by the whole length, and the quotient will give the press- 
ure upon the end or support nearest the weight. 

Example.— What is the pressure upon the supports in the case of the preceding 
example ? 

1500x2 1500x3 

— - — - =. GOO lbs. upon support farthest from the weight ; — - — = COO lbs. upon 
5 5 

support nearest to the weight. 

When a Beam or Bar, Fixed or Supported at both Ends, bears two 
Weights at unequal Distances from the Ends, Let m and n represent dis- 
tances of greatest and least iceights from their nearest end, W and w great- 
est and least iceights, L whole length, I distance from least weight to farthest 
end, and l distance of greatest weight from farthest end. 

_. raXW Ixw A nxw /'XW .„ 

Then — - 1 — - — = pressure at w end, and — 1 = — = pressure at W end. 

L L L L 

Illustration. — A beam 10 feet in length, having both ends fixed in a wall, bears 
two weights, viz., one of 1000 lbs. at 4 feet from one of its ends, and the other of 
2000 lbs. at 4 feet from the other end ; what is the pressure upon each end? 

4X2000.6X1000 4x1000,6x2000 .^1 
— 1 — — = 1400 lbs., pressure upon w end ; — — 1 — — = 1600 lbs.,. 

pressure at W end. 

When the Plane of the Beam or Bar projects obliquely Upward or 
Downward. When fixed at one End and Loaded at the other. Rulk. — 
Multiply the Value of the material by the breadth and square of the depth 
in inches, and divide the product by the product of the length in feet and 
the cosine of the angle of elevation "or depression. 

Note.— When the weight is laid uniformly along its length, the result must be 
doubled. 



' 



STRENGTH OF MATERIALS. 459 

Example.— What is the weight an ash beam, 5 feet in length, 3 inches square, 
and projecting upward at an augle of 7° 15', will bear without permanent injury? 

55 X 3 X 3 2 = 1485, which, -f- 5x cos. T°T5' = 1485 -=- 5X.992 = 299. 39 lbs. 

To Compute tlie Transverse Strength, of Cylinders, El- 
lipses, etc. 

When a Cylinder, Square {the diagonal being vertical), Hollow Cylin- 
der, or Beams having sections of an Ellipse, are either Fixed at one End and 
loaded at the other, or Supported at both Ends, the Load applied in the Mid- 
dle, or between the Supports. Rule. — Proceed in all cases as if for a rect- 
angular beam, taking for the breadth and depth, and Value of the material, 
as follows : 

Cylinder, diameter 3 x.6; Rectangle,* side 3 ; Hollow Cylinder (di- 
am/ 3 — diam. 3 ) x .6 ; Ellipse, transverse diam. vertical conj. X transverse 2 , 
X.6; and conj. diam. vertical transverse X conj. 2 X.6 of Value. 

When an Equilateral Triangle, or T Beam. Rllk. — Proceed in all 
cases as if for a rectangular beam, taking the following proportions of the 
Value of the material. 

r,. j . { Equilateral triangle, edge up, bxd 2 , x.2 of Value. 

Fixed atone or \ K * uilateral trian | le ed | e d * bxd *\ xM " 
bom Ends. | T beam or bai% edge dowri) bxd ^ x .42 

o * j * { Equilateral triangle, edge up, bxd 2 , X.34 " 

Supported at > F ^ uilateral triangle, ed|ed6wn, bxd 2 , X .2 
both Ends. j T beam or bar? edge up? hxd ^ xA2 

To Compute tlie Diameter of* a Solid. Cylinder to Support 
a given. "Weiglit. 

When Fixed at one End, and Loaded at the other. Rune. — Multiply the 
weight to be supported in pounds b} T the length of the cylinder in feet ; 
divide the product by .6 of the Value of the material, and the cube root 
of the quotient will give the diameter. 

Note — When the cylinder is loaded uniformly throughout its length, the cube 
root of half the quotient will give the diameter. 

Example. — What should be the diameter of a cast-iron cylindrical beam, 8 ins. in 
length, to support 15000 lbs. without permanent injury? 

8 ins. =t Mfeet; ^°^ =T4.07 ; and 3 /74.07 = 4.2. 

When Fixed at both Ends, and Loaded in the Middle. Rule. — Multiply 
the weight to be supported in pounds by the length of the cylinder be- 
tween the supports in feet; divide the product by .6 of the Value of the 
material, and the cube root of % of the quotient will give the diameter. 

Note. — When tlie cylinder is loaded uniformly along its length, the cube root of 
half the quotient will give the diameter. 

Example. — What should be the diameter of a cast-iron cylinder, 2 feet between 
the supports, that will support 19335 lbs. without permanent injury? 

19305X2 no „ ' , /2SC ' „„ . 
-——--. = 2S6, and 3 / __ — 3.61 ins. 
.6x225 ' V 6 

When Supported at both Ends, and Loaded in the Middle. Rule. — Mul- 
tiply the weight to be supported in pounds by the length of the cylinder 
between the supports in feet; divide the product by .6 of the Value of the 
material, and the cube root of X of the quotient will give the diameter. 

Note. — When the cylinder is loaded uniformly, along its length, the cube root of 
half tlie quotient will give the diameter. 

* The strength of a Square, the diagonal being vertical, compared to that of its circumscribing 
rectangle, is as 1 to 2.8. 



460 STRENGTH OF MATERIALS. 

Example. — What should be the diameter of a cast-iron cylinder, 2 feet between 
the supports, that will support 54000 lbs. without permanent injury? 

54000x2 _ V- . /S00 m ^ . 

~ — SCO, and 3 / — = 5.85 ins. 

.6X225 ' V 4 

And what its diameter if loaded uniformly along its length ? 
800 - 1 - 2 
— — — — 100, and ^100 = 4.64 ins. 

To Compute tlie Relative Value of* IVlaterials to resist a 
Transverse Strain. 

Let V represent this value in a Beam, Bar, or Cylinder, one foot in length, and one 
inch square, side, or in diameter; W the weight; I the length in feet; b the breadth, 
and d the depth in inches ; m the distance of the weight from one end ; and n the 
distance of it from the other in feet. . 

Note. — In cylinders, for b d 2 put d 3 . 

IW 

1. Fixed at one End, weight suspended from the other, =z V. 

bd 2 

/ W 

2. Fixed at both Ends, weight suspended from the middle, = V. 

6 bd 2 

/W 

3. Supported at both Ends, weight suspended from the middle, - — =-. V. , 

4. Supported at both Ends, weight suspended at any other point than the middle, 
m nW 

Ibd 2 ~ ' 

5. Fixed at both Ends, weight suspended at any other point than the middle, 
2 m n W _ 

Slbd* ~~ , 

From which formulae, the weight that ma}' be borne, or any of the di- 
mensions, may be computed by the following : 

. Ybd 2 „, vbd 2 , nv . /z\v 

1 . — - — = W ; is I ; — — —b;/ — r =± rf. In rectangular beams, etc., 

/ Z W 

fcandd = 3/— . 

2. — — & W ; - w - = I ; ^^ = 6 ; ^ — = d. In rectangular beams, 

/ Z W 
etc., b and d = 3 / -— -. 

4&rf2 V 46rf2 V ' HV , />W . _ . . 

3. .— - z= W ; —^r- =» I ; j^ = * >' V 46V w rf In rectan S ular ^^ 

etc., Z) and d= 3 / ——. 

. lbd 2 V < ranW T mnW /mnW 

4. = W ; = Z ; — — z = fc ; / -7— rr = rf. In rectangular beams, 

mn bd 2 Y /rf 2 V \ loV 

lm n W 



I mi 
etc., Z> andd=3/-7 

V * 



v 



r Slbd 2 V ' 2mnW , 2w.nW . /2mnW _ T 

5- = W ; , lo , r = Z , -^-tt7T7 = »\ / » , . „ = d. In rectangular 

2 7n ?i 3 </2 y 3 Z d* y V 3 I b V 



/2m? 
beams, etc., Z> and d = 3 / —^j 



2mnW 
V ' 



When the weight is uniformly distributed, the same formulae will ap- 
ply, W representing only half the required or given weight. 



STRENGTH OF MATERIALS. 461 

GrircLers, Beams, Hiiritels, etc. 

The Transverse or Lateral Strength of any Girder, Beam, Brest-summer, 
Lintel, etc., is In proportion to the product of its breadth and the square 
of its depth, and also to the area of its cross-section. 

The best form of section for Cast-iron girders or beams, etc., is deduced 
from the experiments of Mr. E. Hodgkinson, and such as have this form 
of section X are known as Hodgkinson's. 

The rule deduced from his experiments directs that the area of the bot- 
tom flange should be 6 times that of the top flange — flanges connected by 
a thin vertical web, sufficiently rigid, however, to give the requisite lat- 
eral stiffness, and tapering both upward and downward from the neutral 
axis ; and in order to set aside the risk of an imperfect casting, by any 
great disproportion between the web and the flanges, it should be tapered 
so as to connect with them, with a thickness corresponding to that of the 
flange. 

As both Cast and Wrought iron resist crushing or compression with a 
greater force than extension, it follows that the flange of a girder or beam 
of either of these metals, which is subjected to a crushing strain, accord- 
ing as the girder or beam is supported at both ends, or jixed at one end, 
should be of less area than the other flange, which is subjected to exten- 
sion or a tensile strain. 

When girders are subjected to impulses, and are used to sustain vibra- 
ting loads, as in bridges, etc., the best proportion between the top and 
bottom flange is as 1 to 4 : as a general rule, they should be as narrow 
and deep as practicable, and should never be deflected to more than one 
five-hundredth of their length. 

In Public Halls, Churches, and Buildings where the weight of people 
alone are to be provided for, an estimate of 175 pounds per square foot of 
floor surface is sufficient to provide for the weight of flooring and the 
load upon it. 

In Churches, Buildings, etc., the weight to be provided for should be 
estimated at that which may at any time be placed thereon, or which at 
any time may bear upon any portion of their floors ; the usual allowance, 
however, is for a weight of 280 lbs. per square foot of floor surface for 
stores and factories, and 175 lbs. per square foot when the weight of peo- 
ple alone is to be provided for. 

In all uses, such as in buildings and bridges, where the structure is ex- 
posed to sudden impulses, the load or stress to be sustained should not 
exceed from i to J- of the breaking weight of the material employed ; but 
when the load is uniform or the stress quiescent, it may be increased to 
J and ^ of the breaking weight. 

An open-web girder or beam, etc., is to be estimated in its resistance on 
the' same principle as if it had a solid web. In cast metals, allowance is 
to be made for the loss of strength due to the unequal contraction in cool- 
ing of the web and flanges. 

In cast iron, the mean resistance to Crushing or Extension is as 3.6 to 1, 
and in wrought iron as 1 to 1.3; hence the mass of metal below the 
neutral axis will be greatest in these proportions when the stress is inter- 
mediate between the ends or supports of the girders, etc. 

Wooden Girders or Beams, when sawed in two or more pieces, and have 
slips set between them, and the whole bolted together, are made stiffer by 
the operation, and are rendered less liable to decay. 

Girders cast with a face up are stronger than when cast on a side, in 
the proportion of 1 to .96, and they are strongest also when cast with the 
bottom flange up. 



462 STEEXGTH OF MATERIALS. 

The following results of the resistances of metals will show how the 
material should be distributed in order to obtain the maximum of strength 
with the minimum of material : 



To Tension. 



Cast iron 

Copper 

Wrought iron 



To Crushing. 



(21000 
(32000 


90300 


140500 


24250 


117000 


("45000 
\72000 


40000 


83000 



The best iron has the greatest tensile strength, and the least compressive or 
crushing. 

The most economical construction of a Girder or Beam, with reference to attaining 
the greatest strength with the least material, is as follows : The outline of the top, 
bottom, and sides should be a curve of various forms, according as the breadth or the 
depth throughout is equal, and as the girder or beam is loaded only at one end, or 
in the middle, or uniformly throughout. 

To Compute the Dimensions and Form o£ a GJ-irder or 

Beam. 

When a Girder or Beam is Fixed at one End, and Loaded at the other. 

1. When the Depth is uniform throughout the entire Length, The section at every 
point must be in proportion to the product of the length, breadth, and square of the 
depth, and as the square of the depth is in every point the same, the breadth must 
vary directly as the length ; .consequently, each side of the beam must be a vertical 
plane, tapering gradually to the end. 

2. When the Breadth is uniform throughout the entire Length, The depth must 
vary as the square root of the length ; hence the upper or lower sides, or both, must 
be determined by a parabolic curve. 

3. When the Section at every point is similar — that is, a Circle, an Ellipse, a 
Square, or a Rectangle, the sides of which bear a fixed proportion to each other, 
The section at every point being a regular figure, for a circle, the diameter at ev- 
ery point must be as the cube root of the length; and for an ellipse, or a rectang.e, 
the breadth and depth must vary as the cube root of the length. 

When a Girder or Beam is Fixed at one End, and Loaded uniformly through- 
out its Length. 

1. When the Depth is uniform throughout its entire Length, The breadth must in- 
crease as the square of the length. 

2. When the Breadth is uniform throughout its entire Length, The depth will vary 
directly as the length. 

3. When the Section at every point is similar, as a Circle, Ellipse, Square, and 
Rectangle, The section at every point being a regular figure, the cube of the depth 
must be in the ratio of the square of the length. 

When a Girder or Beam is supported at both Ends. 

1. When Loaded in the Middle, The constant of the beam, or the product of the 
breadth and the square of the depth, must be in proportion to the distance from the 
nearest support; consequently, whether the lines forming the beam are straight or 
curved, they meet in the centre, and of course the two halves are alike : the beam, 
therefore, may be considered as one of half the length, the supported end correspond-' 
ing with the free end in the case of beams, one end being fixed, and the middle of 
the beams similarly corresponding with the fixed end. 

1. When the Depth is uniform throughout, The breadth must be in the ratio of the 
length. 

2. When the Breadth is uniform throughout, The depth will vaiy as the square root 
of the length. 

3. When the Section at every point is similar, as a Circle, Ellipse, Square, and 
Rectangle, The section at every point being a regular figure, the cube of the depth 
will be as the square of the distance from the supported end. 






STRENGTH OF MATERIALS. 463 

When a Girder or Beam is Supported at both Ends, and Loaded uniformly 
throughout its Length. 

1. When the Depth is uniform, The breadth will be as the product of the length of 
the beam and the length of it on one side of the given point, less the square of the 
length on one side of the given point. 

2. When the Breadth is uniform, The depth will be as the square root of the prod- 
uct of the length of the beam and the length of it on one side of the given point, less 
the square of the length on one side of the given point. 

3. When the Section at every point is similar, as a Circle, Ellipse, Square, and 
Rectangle, The section at every point being a regular figure, the cube of the depth 
will be as the product of the length of the beam and the length of it on one side of 
the given point, less the square of the length on one side of the given point. 

General Deductions from the Experiments of Stephenson, Fairbairn, Cubitt, 
Hughes, etc. 

Fairbairn shows in his experiments that with a stress of about 12320 lbs. per 
square inch on cast iron, and 28000 lbs. on wrought iron, the sets and elongations 
are nearly equal to each other. 

A cast-iron beam will be bent to one third of its breaking weight if the load is 
laid on gradually ; and one sixth of it, if laid on at once, will produce the same ef- 
fect, if the weight of the beam is small compared with the weight laid on. Hence 
beams of cast iron should be made capable of bearing more than 6 times the greatest 
weight which will be laid upon them. 

In beams of cast or wrought iron the flanges should be proportionate to the rela- 
tive crushing and tensile resistances of the material. 

The breaking weights in similar beams are to each other as the squares of their 
like linear dimensions ; that is, the breaking weights of beams are computed by 
multiplying together the area of their section, their depth, and a constant, determ- 
ined from experiments on beams of the particular form under investigation, and di- 
viding the product by the distance between the supports. 

Cast and wrought iron beams, having similar resistances, have weights nearly as 
2.44 to 1. 

The range of the comparative strength of girders of the same depth, having a top 
and bottom flange, and those having bottom flange alone, is from having but a little 
area of bottom flange to a large proportion of it, from % to X greater strength. 

A box beam or girder, constructed of plates of wrought iron, compared to a single 
rib and flanged beam I, of equal weights, has a resistance as 100 to 93. 

The resistance of beams or girders, where the depth is greater than their breadth, 
when supported at top, is much increased. In some cases the difference is fully one 
third. 

When a beam is of equal thickness throughout its depth, the curve should be an 
ellipse to enable it to support a uniform load with equal resistance in every part ; 
and if the beam is an open one, the curve of equilibrium, for a uniform load, should 
be that of a parabola. Hence, when the middle portion is not wholly removed, the 
curve should be a compound of an ellipse and a parabola, approaching nearer to the 
latter as the middle part is decreased. 

Girders of cast iron, up to a span of 40 feet, involve a less cost than of wrought 
iron. 

Cast iron beams and girders should not be loaded to exceed one fifth of their break- 
ing weight; and when the strain is attended with concussion and vibration, this pro- 
portion must be increased. 

Simple cast-iron girders may be made 50 feet in length, and the best form is that 
of Hodgkinson : when subjected to a fixed load, the flange should be as 1 to 6, aud 
when to a concussion, etc., as 1 to 4. 

The forms of girders for spaces exceeding the limit of those of simple cast iron are 
various ; the principal ones adopted are those of the straight or arched cast-iron gird- 
ers in separate pieces, and bolted together — the Trussed, the Bow-string, and the 
wrought-iron Box and Tubular. 

A Straight or Arched Girder is formed of separate castings, and is entirely depend- 
ent upon the bolts of connection for its strength. 



46-4 STRENGTH OF MATERIALS. 

A Trussed or Bow-string Girder is made of one or more castings to a single piece, 
and its strength depends, other than upon the depth or area of it, upon the proper 
adjustment of the tension, or the initial strain, upon the wrought-iron truss. 

A Box or Tubular Girder is made of wrought iron, and is best constructed with 
cast-iron tops, in order to resist compression: this form of girder is best adapted to 
afford lateral stiffness. 

Floor Beams, Grirders, etc. 

The condition of the stress borne by a floor beam is that of a beam supported at 
both ends and unifonnly loaded ; but from the irregularity in its loading and un- 
loading, and from the necessity of its possessing great rigidity, it is impracticable to 
e-timate its capacity other than as a beam having the weight borne upon the middle 
of its length. 

To Compute tlie Depth, of* a Floor Beam, 

When the Length and Breadth are given, and the Distance between the 
Centres of the Beam is One Foot. Rule. — Divide the product of the square 
of the length in feet and the weight to be borne in pounds per square foot 
of floor, by the product of 4 times the breadth and the value of the mate- 
rial from the Table (page 455), and the square root of the quotient will 
give the depth of the beam in inches. 

Example.— A white pine beam is 2 ins. wide, and 12 feet in length between the 
supports ; what should be the depth of it to support a weight of 175 lbs. per square 
foot? 

192y 1TK 

~^ ■= 105, and V1C5 = 10.25 ins. 

When the Distance between the Centres of the Beam is greater or less than 
one Foot. Rulk. — Divide the product of the square of the depth for a 
beam, when the distance between the centres is one foot, by the distance given 
in inches by 12, and the square root of the quotient will give the depth of 
the beam in inches. 

Example. — Assume the beam in the preceding case to be set 15 ins. from the cen- 
tres of its adjoining beams ; what should be its depth ? 

10 25 2 y 15 
*~ == 131.25, and ^131.25 = 11.45 ins. 

Header and. Trimmer Beams. 

The conditions of the stress borne or to be provided for bj~ them arc as 
follows : 

Header or Trimmer beams support 3^ of the weight of and upon the tail 
beams inserted into or attached to them. 

Trimmer Beams support, in addition to that borne by them directly as a 
floor beam, each 3^ the weight on the headers. 

The stress, therefore, upon a header is due directly to its length, or the 
number of tail beams it supports ; and the stress upon the trimmer beams 
is that of their own stress as a floor beam, and % of the weight upon the 
header supported by them. 

Note — The distance between the support of the trimmer beams and the point of 
connection with the header does not in anywise affect the stress upon the trimmer 
beams; for in just proportion as this distance is increased, and the stress upon them 
consequently increased, by the suspension of the header from them nearer to the mid- 
dle of their length, so is the area of the surface supported by the header reduced, and, 
consequently, the load to be borne by it. 

G-irder. 

The condition of the stress borne by a Girder* is that of a beam fixed or 
supported at both ends, as the case ma}* be, supporting the weight borne 

* When a girder has four or more supports, its condition as regards a stress upon its middle is that 
of a beam fixed at both ends. 



STRENGTH OF MATERIALS. 465 

by all of the beams resting thereon, at the points at which they rest ; and 
its dimensions must be proportionate to the stress upon it, and the distance 
between its points of insertion or support. 

Illustration — It is required to determine the dimensions of a pitch-pine girder, 
15 feet between its several points of supports, to support the ends of two lengths of 
beams each 20 feet in length, having a superincumbent weight, including that of the 
beams, of 200 lbs. per square foot. 

The condition of the stress upon such a girder would be that of a number of beams, 
40 feet in length (20x2), supported at both ends, and loaded uniformly along their 
length, with 200 lbs. upon every superficial foot of their area. 

Hence the amount of the weight to be borne is determined by 20x2x15x200 = 
120 000 lbs. ■— the product of twice the length of a beam, the distance between the 
supports of the girder and the weight borne per square foot of area ; and the resist- 
ance to be provided for is that to be borne by a beam, 15 feet in length, fixed at both 
ends, and supporting 120000 lbs. uniformly laid along its length, equal to 60 000 lbs. 
supported at its centre. 

Consequently, — = 3000 = quotient of the product of the length and weight 

-f- the product of 6 times the Value of the material; and assuming the girder to be 12 

inches wide, then / ^— =. 15.8 ins. 

Formulas to Compute trie Values arid, the Dimensions of 
Beams, Bars, etc., of various Sections.— [Tredgold.] 

For a Square, Rectangle, Rectangle the diagonal being vertical, and Cylinder, 
they are alike to those already given, substituting in the rectangles for bd 2 , S 3 . 
For a Grooved or Double-flanged, Open, and Single-flanged Beam they are as fol- 



lows : 



Grooved. Open. 



ZW _ 

bd'Hl— y 3 )~ Y ' 

IW 

. y 

6tZ 2 (l— J/ 3 ) 

ZW 



bd 2 (1— 2/ 3 ) 
mnW 



= V. 



bd 2 m-\-n (1 — y 3 ) 
mnW 



- = V. 



1. Fixed at one End, Weight\ ZW _ 
suspended from the other, j ^ ^2 m q »3) — " 

2. Fixed at both Ewds,Weight) ZW _ 
suspended from the middle, / b d2 ( i qy- s )~~ 

3. Supported at both Ends, "i ^ -yy 

Weight suspended from the> =V. 

middle, ) bd2 (1 — ?2/ 3) 

4. Supported at both Ends A mnW 

Weight suspended at any v — - = V. 

other point than the middle, J bd 2 m-\-n(l — q y 3 ) 

5. Fixed at both Ends, Weigh U mnW 

suspended at any other point V z= = V. 

than the middle, ) pdfim \ n (1 — qy 3 ) bd 2 m-\-n{\—y 3 ) 

XC ZW 2."} For the other conditions of 

Single- ( . \ b d 2 m _ q y 3) (1—q) 3 * (. a Beam, Bar, etc., use the 

flanged\ ' j ' = V. 4. ("same formula as the above, 

(.(•/l — qy 3 -\-Vl — q) 2 5. J multiplying the Value ob- 
tained above by G, 4, 1, and 1.5 respectively, y and q representing , , — ^-f- 

whole depth of beam 
. whole breadth of beam — width of web 

r= y, and — ■ = q. 

whole breadth of beam 

Transverse [Resistance from End. Pressure applied Hori- 
zontally. 

L Wrought Iron.— 1% feet in length ; flanges, 6x3>£ ins. X % depth ; area, 5^ 
square ins. ; 50000 lbs. produced no set ; 5S240 lbs. produced a set of 1% ins. 
White Oak.— Rectangle 10 feet in length, 11 X4^ ins. ; 33600 lbs. gave a deflec- 
tion of .% in. ; 50400 lbs. gave a deflection of .5 in. ; 67200 lbs. gave a deflection of 
.%, and with '.8400 it broke. 

Rr 



466 



STRENGTH OF MATERIALS. 



Transverse Strength of Cast-iron Grirders and. Beams, 
deduced from tlie Experiments of Barlow, Hodgkin- 
son, Hoaghes, Tredgold, etc. 

Reduced to a uniform Measure of One Inch in Depth, one Foot in Length, supported 
at both Ends ; the Stress or Weight applied in the Middle. 





Flar 


ges. 


"3 




* 


a 


A a 




•^ .. 








:3 
® . 


9 

3 


•S 
E 


a 
.2 
'« 2 


too 


to J 




Section of 

GlBDER OB 






Beam. 


o 


3 

o 
"o 

a 


9 




a 
2 
a 


a 

2 
< 




33 « 


3-5 




Sq. Ins. 


Sq. Ins. 


Inch. 


Inch. 


Inch. Sq. In. 


Lbs. 


Lbs. 


Lbs. 


Eq. area") 










1 








&W* of flange ( 


1.75X.42 


1.77X.39 














rJL. at top & C 


= .735 


= .69 


.29 


5.125 


1.77 2.S2 


30150 


rotes 


2100 


bottom, ) 


















\\ do. { 


2.02X.515 


2.02 X- 515 














= 1.045 


= 1.045 


At* 


2.02 


2.02 2.53 


10276 


3052 


IE 00 


Area "\ 




















W of sec. / 

I of top > 

«L & bot. \ 


2.23X.31 


6.67X.66 
















= .72 


= 4.4 


.206 5.:- 


6.G7 6.23 


117450 


1SS52 


3650 


1 to 6j 




















JL { 




5 X.3 
















— 


= 1.5 


.3C5 


1.56 


5. 


1.96 


7280 


3714 


2350 


T ! 


5 X.3 
= 1.5 


23.9x3.12 


.365 


1.56 


5 


1.96 


2366 


1213 


750 


^_ L [ 


— 


= 74.56 


3.3 


36.1 


23.9 


183.5 


8066240 


4395S 


1200 


i { 


.5 X5 


1.5x5 
















= .25 


= .75 


.5 


4.t 


1.5 


It 


199S0 


19980 


5000 


T { 


1.5 X.5 


.5X.5 
















= .75 


= .25 


.5 


4.t 


1.5 


1. 


7252 


7252 


1S0O 


+ { 


4 X 2 


















= 8 


— 


2. 


4. 


4. 


12. 


33600 


2S00 


700 


1 


5.1 X2.33 


12.1x2.07 
















JL i 


= 11.88 


= 25.04 


2.08 
.994 


30.5 
2.012 


11.1 

.994 


C0.S 
2.0-5 


4793800 
9440 


52795 
4662 


1700 


| Rectangu-) 
| lar Prism, / 


2350 


1.005X-9S 


1.005X.99 


1.005 


2.51 


1.005 1.98 


12340 


6232 


2450 


G S? \ 


.995x1.01 


.9f5x 1 


.995 3.01 


.995 2. 


15420 7710 


£550 


n s 1 


1.005X.98 


1.005X-99 


1.005 4. 


1.005 1.9S 


21765 10902 


2700 


y a ) 


.771X1.51 


.771x1-5 .771 4.04 


.771 2.322 


2570511070 


2750 


« a / 


1.507X.74 


1. 507 X. 74 1.507 4.04 


1.507 2.23 


25735 11540 


2S50 


o ^ 


1. 5-25 X. 78 


1.525X-78 


1.525 4.07 


1.525 2.35 


30000 


126S9 


3100 


Square ") 










1 








|l|j Prism, f 
fPtH Stress at ( 


— 


— 


1.02 


1.01 


1.02 1.03 


2635 


2552 


2500 


Side, ) 


















Mm Cylinder, 


— 


— 


1.122 1.122 


1.122 .933 


2370 


2396 


2150 


jk Square ) 








1 








Mjk> Prism, V 


■ — 


— 


.4431 1.443 1.-143 1.041 


2269 


2182 


1500 


\s angle up) 





















* Horizontal web. f Depth of openinp, 3 inches. 

t A repretenting area oftection, d the depth in inchet, I the length in feet, and VV the breaking weight 
in pound*. 



STRENGTH OF MATERIALS. 467 

Comparative Resistance or Strength, of GJ-irders, Beams, 
etc., of J^CLnal Sectional Areas and Depths. 



Description of Girder or Beam. 



Comparative 

Strength. 



Rectangular beam 

Grooved beam, top and bottom flanges of equal areas, of uniform 
thickness of metal throughout, and the depth three times the 
breadth (Tredgold) 

Single-flanged beam; width of flange five twelfths of height; width 
of rib half the depth of flange (Watt and Fairbairn) 

Open beam, the space half the depth , 



Double-flanged beam ; area of top flange one sixth of that of bot- 
tom ; depth of top flange half that of bottom ; width of bottom 
2 flange 1% the depth of the beam (Hodgkinson) . 



1.16 

1.27 
1.5 

l.CC 



To Compute the Transverse Strength, or the Breaking 
"Weight of* Cast-iron Grirders or Beams, of various Fig- 
ures and. Sections, when Supported, at tooth Ends, the 
"Weight applied in the Middle. 

When the Section of the Girder or Beam is that of a Rectangle, a Grooved, 
Open, Single or Double Flanged Beam, and is alike to any of the Exam- 
ples given in the preceding Table. 

Rule 1.* — Divide the product of the area of the section, the depth, and 
the Value for the girder, etc., from the Table, by the length between the 
supports in feet, and the quotient will give the breaking weight in pounds. 

Example. — The dimensions of a Hodgkiuson beam, having top and bottom flanges 
in the proportion of 1 to 6, give an area of section of 25.6 square inches, a depth of 
20.5 inches, and a length between its supports of 18 feet ; what is its breaking weight? 

Note. — In consequence of the increased area of the metal over the example given 
in the Table, the unit of Value of 3650 is (page 465) reduced to 3500. 

_. 25.6X20.5X3500 w 1836800 Q 

Then — X — rs — = 102044.4 lbs. 

lo lo 

2. From the product of the breadth and square of the depth in inches of 
the rectangular solid, the dimensions of which are the depth and the great- 
est breadth of the beam in its centre, subtract the product of the breadths 
and the square of the depths of that part of the beam which is wanting to 
make it a uniform solid, and then proceed to determine its resistance by 
the rule for the particular case as to its being supported or fixed, etc. 

Note 1.— These rules are applicable to all cases where the flange of the beam is set, 
as shown in the Table, when the beam rests upon two supports, or contrariwise, as to 
position of flange, when the beam is fixed at one end only. 

2.— When the case under consideration is alike in its general characters to one in 
the Table, but differs in some one or more points, an increase or decrease of the metal 
is obtained by a reduction or increase of the value, according as the differences may 
affect the resistance of the beam. 

Example — What is the load that will break a Hodgkinson beam of the following 
dimensions, 10 feet in length between its supports, the load applied in its middle ? 

* The utility of these rules in preference to those of Hodgkinson, Fairbairn, Tredgold, Hughes, 
and Barlow, is manifest, as in the one case the Value of the metal is considered, and in the other 
cases the metal is assumed to be of a uniform value or strength; and when the range in this ele- 
ment, both in weight and cost, render it imperative that, in a structure of iron of the highest trans- 
verse strength, the weight due to the requirements of dimensions of the lowest transverse strength 
should not be increased, and contrariwise. 

The only variable element not embraced in this rule is that consequent upon any peculiarity of 
form of section ; as, for instance, in that of the Hodgkinson, or like beams, when the area of one 
flange greatly exceeds the rest of the section, and this flange is other than below when the beam 
rests upon two supports or is fixed at both ends, or than above when the beam is fixed at one end. 



46 S STRENGTH OF MATERIALS. 



Top flange 7x1 inch. 

Bottom flange 21x2 " 

Width of web S " 



"Whole depth of beam .... 21 ins. 

Area of whole section .... 63.4 u 

Dimensions of rectangle .. 21x21 " 



Hence 21x212 = 9261 ins. 
7 — . 8— 6 .2 ins. =a width of space between both extremities of top flange and rib. 
21 — 2-f-l =1S= depth of space between top and bottom flanges. 

Hence 6.2x182= 200S 8. 
21 — 7 = 14= width of space between both extremities of top and bottom flanges. 
21 — 2 = 19 = depth of space above bottom and outside of top flange. 

Hence 14X19 2 =5054. Sum, 7062.8. 

And 9261 — 7062.8x4x460* = 4044688 == difference of products of the breadth 
and of the depth of the circumscribing solid, and the breadth and square of the depth 
of the parts wanting to complete the rectangle, multiplied by four times the value 
of the metal, which -J- 10/or the length = 4044GS.S lbs. 

In the example given above, the formulae of various authors give the fol- 
lowing results : 

2 

Hodgkinson. — — X (b d 3 —(b — b') d rj ) = W d representing depth of 

beam, d' depth to bottom flange, b breadth of bottom flange, b' thickness of vertical 
web, all in inches, I length infect, and W weight in tons. 
2 



-X (21X213 — (21 - .S)X19 3 ) = 177.55, which X2240 = 397722 lbs. 



3X21X10/ 

_ . , . 2.166arf I" ' ■'■'■ 2.166X42X21 
Fairbairn. — — = W, a represent g area of bottom flange ; :. — 

= 191.1, which X 2240 = 42S 064 lbs. 

ad 2v4^v21 

Hughes. — = W ; :. " ■ = 176.4, which X2240 = 395 136 lbs. 

Barlow.- ^ = W ; /. 1{] ] = 150.4, which X2240 = 336 S36 lbs. 

Experiments upon the breaking weight of girders of English cast iroii 
have given the following results : 

Dimensions of Girders. 1 and 2. 3. 

Top flange... 3^X1% ins. &AX\)4ins. 

Web IK " 1^ " 

Bottom flange 9 Xl% u 15 x2^ " 

Whole depth 22 " 24^ " 

Area of bottom flange 11^ u 33% t; 

Whole area 39.69 sq. " 70,69 sq. " 

Length between supports 19 ft. BOX ft. 

Breaking weight j 2 - 1253SD '"*' 3-145 20S lbs. 

Breaking "Weights computed "by various Formulse, 

1 and 2. 3. 1 and 2. 3. 

By Hughes 58 352 lbs. 119 240 lbs. I By Hodgkinson 94 99S lbs. 139 0S2 lbs. 

" Fairbairn... 63 213 " 129272 " | " Barlow.... 116323 " 141120 " 

Formula of Table, page 466, using 3300 and 2000t 105 701 " 144 820 " 

Comparative "Valxies of Cast-iron Bars, Hollow Girders, 
or Tubes of various Figures (English. Iron). 



Square bar, small 1. 

Square bar, large 75 

Kound bar, small 675 

Square tubes, uniform thickness . . 1.075 



Rectangular tubes, uniform thickness. .85 
Circular tubes, uniform thickness .... ,9 
Elliptic tubes, uniform thickness 95 



* Assumed breaking -weight of the metal, if of American iron. In connection with this, it is to be 
borne in mind that, the greater the area of the section of the metal, the less its strength, and the 
longer the beam, the greater the risk of deflection from a flaw in its structure. 

f This is an interesting case, as it exhibits the great reduction of the value consequent upon an in- 
crease of dimensions, as the proportionate valve for a girder of the proportion of flanges, but of small 
dimensions, would be 3200. whereas it is but 2600. 



STRENGTH OF MATERIALS. 



469 



Transverse Strength of "Wrought - iroix G-irders and 
Beams, deduced from the Experiments of Barlow, 
ITairhairn, Hughes, etc. 

Reduced to a uniform Measure of One Inch in Depth, one Foot in Length, supported 
at both Ends ; the Stress or Weight applied in the Middle. 





Flan 


ores. 


"flS 




u 


a 


"S ® 


. 


Ji .. 






•H 


•§ 


i 


a 

.2 


tt o 


<^c 1 « II 


Section of 






co J IgJI . 


Girder 
or Beam. 




£ 

o 
W 


1 


3 

o 

Q 


i 

8 


to a 
oO 

c5 

< 


^2 

klj'hD 

£ a 
PQ 


puv 
03 ►"-" 


£- 5 




Sq. Ins. 


Sq. Ins. 


Inch. 


Inch. 


Inch. 


Sq.In. 


Lbs. 


Lbs. 


Lbs. 


I fl 


2.5 X 1 


4 X .33 
















= 2.5 


= 1.52 


.325 


8.38 


4. 


6.295 


132000 


20952 


2500 


T i 


2.85X .38 


















L=1.08 


— 


.31 


2.5 


2.85 


1.73 


12560 


7260 


2900 


A "{ 




2.S5X .3S 
















— 


= 1.08 


.31 


2.5 


2.S5 


1.73 


12032 


6955 


2750 


X i 




35 X .6 
















— 


= 2.1 


.8 


3.5 


3.5 


6.25 


49280 


7822 


2200 


_a ^| 


2.86X .33 


2.86X .S3 
















A-!. 


= .944 

5 X .25 

= 1.25 


= .944 


.66 


3.7 


2.86 


3.8S 


56000 


14433 


3800 


i " ) 


2of2.25x 
2.25X .3 


















V 


= 2.82 


— 


.54 


2.6 


5. 


4.07 


234S5 


5770 


2250 


ta w ( 


2 of 3.5 X 


2 of 3.5 X 
















B \ 


3.5 X .5 


3.5 X .5 
















l " j 


= 7 
2 of 2. 125 X 


= 7 
2of2.125x 


.37 


16. 


7.37 


19.92 


768000 


38593 


2400 




.28 = 1.19 


.30=1.27 


.25 
Thick. 

of 
Plates. 


7. 


4.5 


4.26 


170660 


40345 


5S00 




— 


— 


.065 


5.8 


3.S 


1.24 


23670 


140S9 


3200 







— 


.061 


3. 


1.95 


.6 


9450 


15750 


5200 


n 


— 


— 


.1325 


6. 


4. 


2.62 


75600 


28855 


4700 


1 1 " * 


— 


— 


.124 


24. 


15. 


9.6 


375000 


39063 


1600 


LsJ 


— 


— 


.272 


23.75 


15.5 


21.2 


1536000 


72452 


3000 




— 





.525 


24. 


16. 


41.45 


3864000 


93221 


3900 


k 


— 





.75 


36. 


24. 


87.75 


4310400 


149333 


4900 




9.6 X.252 


9.6X.075 


















= 2.419 


= .72 


.074 


9.5 


9.5 


4.36 


146528 


33607 


3450 


ngapsa 


9.25X.149 


9. 25 X. 269 
















1 1 u 


= 1.378 


= 2.488 


.059 


18.25 


9.25 


6.03 


119210 


19768 


1050 


1 1 


2.25X.26 


2.25X.26 
















Iftmr-rffl 


= .585 


= .585 


.131 


15. 


2.25 


5.1 


452400 


88706 


5500 




1 X -2S2 


1 X .116 
















, 


= .2S2 


= .116 


.067 


8. 


1. 


1.47 


123794 


81214 


10300 




24.* 


12S 




54. 


2.92 


45.82 


9443400 


Mote 


3800 


^^ +■ ( 




( 


.375t. 


) 












11 & ' 


— 


- < 


.25 b. 


V24. 


16. 


12.94 


1SS160 


14540 


6050 


ll 5 "j 




I 


.125 s. 


f 












vsa^ H ( 






.143 


15. 


9.75 


5.56 


27S250 


50045 


3300 


Oli 








.0408 


12. 


12. 


1.4 


44200 


31571 


2600 


v*/J1 


— 


— 


.095 


24. 


24. 


7.13 


21)8629 


41743 


1725 



* Thickness of plates, bottom, .156; top, .147 ; sides, .0' 
f The lateral strength of this was ascertained to be 380i 
iimate deflection was 1\ ins. 

Rr* 



I. Area of bottom, 8.8 ins. 

, or .613 of its vertical strength. The ul 



470 STRENGTH OF MATERIALS. 

The above and many of the preceding results are deduced from girders of the 
length of from 20 to 30 feet ; hence, when the length is less, the breaking weight 
may be increased, in consequence of the increased stability of the girder. 

Ad V 

These results are very conclusive of the correctness of the formula used, viz., — - — ' 

as will be s?en in the cases here given, in the 10th and 15th cases, where the rela- 
tions between breadth, depth, and thickness are nearly identical; and in the i'th 
and 15th cases, where the relations between breadth are the same, but the thickness 
and consequent area differ. 

To Compute the Transverse Strength., or the Loads that 
zuajr "be borne Id y "VTrought-iron Girders, Beams, or 
Tubes, of various Figxxres and Sections, when Support- 
ed at "both Ends, the Load applied in the IXIiddle. 

When the Section of the Girder or Beam is that of any of the Figures in 
the preceding Table. Rllk. — Divide the product of the area of the sec- 
tion, the depth, and the Value for the girder, etc., from the Table, by the 
length between the supports in feet, and the quotient will give the de- 
structive weight in pounds. 

Note 1. — The Rule given on page 467 for cast-iron girders, etc., will also apply 
here, when the metal is of such thickness as to give the girder, etc., full resistance to 
lateral flexure, and when the construction is sucti as to bring the stress upon the ten- 
sion and compression of the metals, and not upon the rivets. 

For Note 2, see page 467. 

3.— The Values here given are based altogether upon experiments with English iron. 

Example. — What is the load that will destroy a wrought-iron solid grooved beam 
of the following dimensions, and 10 feet in length between the supports ? 

Top flanges 3x1^ ins. I Width of web 4 inch. 

Bottom flange 4X .5 w | Depth of beam 9. 

3X1.25 4-4X. 5 = 3. 75 + 2 = 5. 75 ins., which + 9 — 1.25 + .5X.4 = 2.90 = S.e5ins. 
, . _. 8.65X9X3000 on _ K Jx 

= area of section. Then : = 233o5 lbs. 

J 10 

Formulas for 13 earns and Tubes of Wrought Iron. 

Faikbaie.n.* 

.V, SSOOAd T TO "' • 2912 A d _ 
Solid beams, = L. Plate beams, = L. 

tt* a • wu 1792to2S00Ad T w " 1A . 16S0to5510Ad _ 
Cylindrical tubes, = L. Elliptical tubes, ■ = L. 

HODGKINSON. 

iL . -•■... 60000 to 90000 (&<£» — V d'^ _ 
Rectangular beams, — — = L. 

' ' , t 3.1416X22500 to S5500 . 

Cvlmdncal tubas, — (r* — r 4 ) = L. 

A I 

^„. ,' : ; 4 -; 3.1416x20000 to37000(ct3_ c'f3) 
Elliptical tubes, — = L ; 

&, b'. and d. d' representing the external and internal breadths and depths, r and r* 
the external and internal radii, and c c' and t t' semi-conjugate and semi-transverse 
diameters, and I the length in inches. 

Comparative Value of "Wrought-iron 33ars, Hollow CrircL- 
ers, or Tubes of various Figures {English Iron). 

Square bar 25 ) I Round bar 1^5 

Rectangular tubes, plates at top and bottom thick, at sides thin 425 

Welded Tubes without Rivets. 

Rectangular, uniform thickness 37o Rectangular tubes, riveted 2S0 

Circular, uniform thickness 325 Elliptic tubes, riveted 2^0 

Elliptic, uniform thickness 350 Flanged beams 240 

Circular tubes, riveted 190 Plate beams 320 

* See Report of Commissioners on Railway Structures, 1849. 



STRENGTH OF MATERIALS. 



471 



CRUSHING STRENGTH. 

The Crushing Strength of any body is in proportion to the area of its 
section, and inversely as its height. 
In tapered columns, the strength is determined by the least diameter. 

Crushing Strength of* various Materials, clecLnced. from 
tlie Experiments of ]VLaj. Wade, Hodgkin^on, and. Capt. 
JVIeigs, TJ.S.^l. 

Reduced to a uniform Measure of One Square Inch. 

Figures and Material. WeVht Figures and Material. 



Cast Ikon. 

American, gun-metal. 

44 mean 

English, Low Moor, No. 1 

" No. 2 

" Clyde, No. 3 

44 Stirling, mean of all . . 

u 4 * extreme 

Weougut IpwOn. 

American 

W mean 



English j 

Various Metals. 

Fine brass 

Cast copper 

Cast steel 

Cast tin 

Lead 

Woods. 

Ash... 

Beech 

Birch 

Box 

Cedar, red 

Chestnut > 

Elm 

Hickory, white 

Locust '. 

Mahogany, Spanish 

Maple 

Oak, American white 

u Canadian white ; . . . 

44 44 live 

" English | 

Pine, pitch 

u white 

u yellow 

Spruce, white 

Sycamore 

Teak 

Walnut 

Stones, Cements, eto. 
Brick, machine-pressed j 



Lbs. 

174S03 
129000 
02450 
92330 
106039 
122395 
134400 

12TT20 
83500 
05200 
40000 

1C4800 

117000 

295000 

15500 

7730 

6663 
6963 
796) 

10513 
5968 
5350 
6831 
8925 
9113 
8198 
8150 
6100 
59S2 
6850 
T500 
6484 
8947 
5775 
8 .00 
5 50 
7082 

12100 
6645 

6222 

1421G 

4000 

800 



Clay, fine, baked 

44 44 rolled and baked . . 

Common brick masonry 

Crown glass 

Craigleith Limestone, English < 

Aberdeen granite . . u i 

Arbroath u 

Caithness " 

Limestone " 

Portland M -j 

Portland cement ... u 
u mean u . 

Portland oolite u 

Fire-brick, Stourbridge 

Freestone, Belleville 

44 Caen 

u Connecticut 

44 Dorchester 

" Little Falls 

Gneiss 

Granite, Patapsco 

44 Quincy 

Marble, Baltimore, large 

u ' 44 i mull 

" East Chester! 

41 Hastings, N. Y 

" Italian 

44 Lee, Mass 

44 Montgomery co., Pa. . . 

" StockbridgeQ 

44 Symington, large 

44 44 fine crystal 

44 44 strata horizontal 

44 u strata vertical . . 

Mortar, good 

44 common 

Normandy Caen 

Portland cement 1, sand 1 

Roman u 

Sandstone, Adelaide 

44 Acquia Creek* 

44 Senecat 

Stock brick 

Sydney 4t 



Crushing 

Weight. 



* Same as that of the Capitol, Treasury Department, and Patent. Office, Washington, T). C. 

+ Same as that of the Smithsonian Institute. J Same ns that of the General 1'ost-offioe, Wash'n. 

5 Same as that of the City Hall, New York. || Same as that of the Nat. Wash. Monument. 



472 



STRENGTH OF MATERIALS. 



When the height of a prism or column is not 5 times its side or diame< 
ter, the crushing strength is at its maximum. 

Experiments upon cast-iron bars give a crushing stress of 5000 lbs. per 
square inch of section as just sufficient to overcome the elasticity of the 
metal ; and when the height exceeds 3 times the diameter, the iron 3'ields 
by bending. 

When it is 10 times, it is reduced as 1 to 1.75 ; when it is 15 times, it is 
reduced as 1 to 2; when it is 20 times, it is reduced as 1 to 3; when it is 
30 times, it is reduced as 1 to 4 ; and when it is 40 times, it is reduced as 
1 to 6. 

The experiments of Mr. Hodgkinson have determined that an increase 
of strength of about one eighth of the breaking weight is obtained by en- 
larging the diameter of a column in its middle. 

In cast-iron columns of the same thickness, the strength is inversely 
proportional to the 1,? power of the length nearly. Thus, in solid col- 
umns, the ends being flat, the strength is as j-^, I representing the length, 
and d the diameter. 

Hollow columns, having a greater diameter at one end than the other, 
have not any additional strength over that of uniform cylindrical columns. 

Experiments upon wrought iron give a mean crushing stress of 74250 
lbs. per square inch. Cast iron is decreased in length nearly double what 
wrought iron is by the same weight ; but wrought iron will sink to any 
degree with little more than 26680 lbs. per square inch, while cast iron will 
bear 97500 lbs. to produce the same effect. 

A wrought bar will bear a compression of 3^3 of its length, without its 
utility being destroyed. 

With cast iron, a pressure beyond 26680 lbs. per square inch is of little, 
if any, use in practice. 

For equal decrements of length, wrought iron will sustain double the 
pressure of cast iron. 

Glass and the hardest stones have a crushing strength from 7 to 9 times 
greater than tensile; hence an approximate value of their crushing strength 
may be obtained from their tensile, and contrariwise. 

Various experiments show that the power of stones, etc., to resist the 
effects of freezing is a fair exponent of that to resist compression. 

"Wronglvt-irorL IPlates, Cylindrical Tubes. 



Length. 


Width. 


Thickness. 


Area. 


Crush. Weight 


Plates. 


Ins. 


Ins. 


Ins. 


Lbs. 


10 feet 


• 2.98 

3.01 

External. 


.497 

.766 

Internal. 


1.48 

2.3 


815 


10 " 


3379 


Hollow Cylinders. 




10 feet 


1.495 

2.49 
6.366 


1.292 
2.275 
6.106 


.444 

.804 
2.547 


14661 


10 " 


29779 


10 " 


35886 


Rectangular Tubes. 




81 \ 


4.1 


4.1 


.504 


10980 


4.1 


4.1 


1.02 


19261 


10 y lap-riveted -{ 


4.25 


4.25 


2.395 


21585 


101 1 

10J I 


8.4 


4.25 


6.89 


29981 


8.1 


8.1 


2.07 


13276 


( lap-riveted, and ) 










10 < two internal di- > 


8.1 


8.1 


3.551 


19800 


( aphragm plates ) 











STRENGTH OF MATERIALS. 



473 



Comparative Resistance of* Cast and. "Wrought Iron Bars 
to "bear Compression in the Direction of their Length, 
set Vertical, and. inclosed in a Frame to maintain them 
in that ^Position. 

One Inch Square, and Ten Feet in Length. 

DECREASE IN LENGTH. 

Wrought Iron. 



Weight. 


Cast Iron. 


Wrought Iron. 


1 Weight. 


Cast Iron. 


Lbs. 


Ins. 


Ins. 


Lbs. 


Ins. 


5054 


.054 


.028 


| 27498 


.3 


9578 


.102 


.052 


i 31978 


.357 


14058 


.151 


.073 


1 40938 


.503 



Ins. 

.143 
.174 



Ultimate Practical Resistance. 

Cast Iron. — Mean weight, 12800 lbs. ; mean decrease, .135 ins. 

Hence, the length of the bars being 10 feet = 120 ins., . Jjf § — 888. a cast- 
iron bar will bear a compression of 8 Jg of its length without its utility 
being destro} r ed, although its elasticit} r will be materially injured. 

Wrought Iron. — Mean weight, 26650 lbs. ; mean compression, .139 ins. 

Hence, the length of the bars being 10 feet = 120 ins , ,ff§ == 863, a 
wrought-iron bar will bear a compression of 303 of its length without its 
utility being destroyed. 

To Compute the Crushing Strength of a Solid Cylindrical 
Column of Cast Iron. 

^3.6 

jY^j X 100000 = W, d representing the diameter of the column in inches, I its 

length in feet, and W the crushing weight in pounds. 

Example. — What is the resistance to crushing of a solid cylinder, 2 inches in di- 
ameter and 5 feet in length ? 

2 3>6 12.125 

L _ ^ 1 ^ xl()00C0 _ TS601 lhs 

5t; 7 15.-i2o 
For Rectangular Columns put 160000 for the multiplier. 

To Compute the ^iltimate Crushing Strengtli of a Hollow- 
Cylindrical Column of Cast Iron. 

J)3.6_ J3.6 

j-^j X 100000 = W, D representing the greater diameter. 

Example What is the resistance to crushing of a hollow cylindrical column hav- 
ing diameters of 2 and 1.25 inches, and a length of 7 feet? 

23 6_i.253 6 12.125 — 2.233 ,^ AAA n „^ 

-Xl000C0 = 361C0 lbs. 



71.7 



27,332 



For Wrought Iron and Oak. 



Hollow cylinder, pine 8200 

Rectangular column, wrought iron 2T0000 

Rectangular column, oak 17400 

Rectangular column, pine 11280 



Solid cylinder, wrought iron 170D00 

Solid cylinder, oak 10SS0 

Solid cylinder, pine 8200 

Hollow cylinder, wrought iron . . . 170000 
Hollow cylinder, oak 108S0 

The above formulae are those of Hodgkinson for the breaking or crush- 
ing weight. The formulae of Euler, which are for the incipient breaking 



for solid cylinders, and 



D* - d* 



I* 



X 100000 =3 W, for hollow cylinders. 



474 



STRENGTH OF MATERIALS. 



The safe load that may be borne bj' a column of cast iron, independent 
of any considerations, regarding the operation of its ends as to their be- 
ing flat or rounded, etc., is from 5000 to 8000 lbs. per square inch for short 
or stable bodies, or about % of the result by the above rule. 

Note. — The preceding formulae apply to all columns where the length is not less 
than about 30 times the external diameter; for columns shorter than this, a modifi- 
cation of the formulae is necessary, as in shorter columns the breaking weight is a 
large portion of that necessary to crush the column. 

For Columns, the Length of which exceed 5 Diameters and is less than 30. 

w c 
— —TV, iv representing the weight as obtained from the preceding formula, 

and c the crushing resistance of the material in pounds. 

Weight that can he borne with safety- by Cast-iron. Col- 
umns in lOOO Lbs.- (Trenton Iron Co.) 



Lenjrth. 


2 


3 


4 


5 


6 


7 


8 


9 


10 


ii 


12 


13 


14 


15 


Feet. 


Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


5 


12.4 


44 


102 


ist 


2S8 


414 


560 


72S 


916 


11-26 


1354 


_ 


_ 




6 


9.4 


86 


88 


164 


261 


3S6 


532 


69S 


SSI 


10S2 


1320 


1570 








7 


7.2 


30 


76 


U6 


212 


360 


502 


660 


850 


1056 


12S2 


1530 


179S 


2086 


8 


— 


24 


6G 


130 


218 


332 


470 


630 


812 


1016 


1240 


14S6 


1754 


2040 


9 





20 


56 


114 


198 


306 


440 


596 


774 


974 


1196 


1440 


1700 


1992 


10 


— 


18 


48 


102 


180 


282 


410 


560 


739 


932 


1152 


1392 


1656 


1940 


12 





— 


3S 


80 


136 


238 


354 


494 


65S 


846 


1056 


1292 


1550 


1S2S 


14 





— 


2S 


64 


122 


200 


304 


432 


5S6 


774 


966 


1192 


1440 


1712 


16 








— 


52 


mo 


170 


262 


378 


520 


6S6 


S7S 


1094 


1332 


15?6 


18 








— 


44 


S4 


144 


226 


332 


462 


616 


796 


1000 


1228 


14S2 


20 


— 


— 


— 


— 


72 


124 


196 


292 


410 


552 


720 


912 


1130 


1372 



For Tubes or Hollow Columns, subtract the weight that may be borne by a col- 
umn of the diameter of the internal diameter of the tube. The thickness of metal 
should not be less than one twelfth their diameter. 

RelativeValue ofvarious Woods, their Crushing Strength 
and. Stiffness being combined. 



Ash 


3571 


Beech 


3079 


Cedar 


700 


Elm 


346S 



English oak 4074 

Mahogany 2571 

Quebec oak 2927 

Spruce 2522 



Sycamore 1833 

Teak 6555 

Walnut 237S 

Yellow pine 2193 



Comparative Value of Long Columns of* various IVIate- 

rials. 
Cast Iron 1000 ] Oak 108.8 | Pine 78.5 

Bridges. 

Iron bridges with a circular arc should have a rise of .1 of the chord 
line, and a width of pier of .1 of span. 

Grirders combined with Suspension Chains.— [P. W. Barlow.] 
In a suspended girder, the stress is resisted by back chains or wire rope. 
The economy of metal in a suspension-bridge, under the average cir- 
cumstances of its attainable depth, is from ){ to % of that in a tubular or 
simple girder-bridge of equal strength and rigidity. 

Comparison between the Two largest Itailway X3ridges. 

Niagara — Wire. Having a roadway, and a single railway of three 
gauges in a span of 820 feet ; weighs 1000 tons. 

Britannia — Tubular. Having a double line of railway in a span of 460 
feet ; weighs 3000 tons. 



STRENGTH OF MATERIALS. 475 

Trussed Beams or GrircLers. 

Wrought and cast iron possess differeut powers of resistance to tension and com- 
pression ; and when a beam is so constructed that these two materials act in unison 
with each other at the stress due to the load required to be borne, their combination 
will effect an essential saving of material. In consequence of the difficulty of adjust- 
ing a tension-rod to the strain required to be resisted, it is held to be impracticable to 
construct a perfect truss beam. 

Fairbairn declares that it is better for the tension of the truss rod to be low than 
high, which position is fully supported by the following elements of the two metals : 

Wrought Iron has great tensile strength, and, having great ductility, it undergoes 
much elongation when acted upon by a tensile force. On the contrary, Cast Iron has 
great crushing strength, and, having but little ductility, it undergoes but little elon- 
gation when acted upon by a tensile force ; and, when these metals are released from 
the action of a high tensile force, the set of the one differs widely from that of the 
other, that of wrought iron being the greatest. Under the same increase of tempera- 
ture, the expansion of wrought is considerably greater than that of cast iron; 1.81* 
tons per square inch is required to produce in wrought iron the same extension as in 
cast iron by 1 ton. 

Fairbairn, in his experiments upon English metals, deduced that within the limits 
of strain of 13440 lbs. per square inch for cast iron, and 30240 lbs. per square inch 
for wrought iron, the tensile force applied to wrought iron must be 2.25 times the 
tensile force applied to cast iron, to produce equal elongations. 

The relative tensile strengths of cast and wrought iron being as 1 to 1.85, and their 
resistance to extension as 1 to 2.25, therefore, where no initial tension is applied to a 
truss rod, the cast iron must be ruptured before the wrought iron is sensibly extended. 

The resistance of cast iron in a trussed beam is not wholly that of tensile strength, 
but it is a combination of both tensile and crushing strengths, or a transverse strength ; 
hence, in estimating the resistance of a girder, the transverse strength of it is to be 
Used in connection with the tensile strength of the truss. 

The mean transverse strength of a cast-iron bar, one inch square and one foot in 
length, supported at both ends, the stress applied in the middle, is about 2500 lbs. ; 
and as the mean tensile strength of wrought iron is about 40000 lbs. per square inch, 
the ratio between the sections of the beams and of the truss should be in the ratio 
of the transverse strength per square inch of the beam and of the tensile strength of 
the truss. 

The girders under consideration are those alone in which the truss is attached to 
the beam at its lower flange, in which case it presents the following conditions: 

1. When the truss runs parallel to the lower flange. 2. When the truss runs at 
an inclination to the lower flange, being depressed below its centre. 3. When the 
beam is arched upward, and the truss runs as a chord to the curve. 

Consequently, in all these cases the section of the beam is that of an open one with 
a cast-iron upper flange and web, and a wrought-iron lower flange, increased in its 
resistance over a wholly cast-iron beam in proportion to the increased tensile strength 
of wrought iron over cast iron for equal sections of metals. 

As the deductions of Fairbairn as to the initial strain proper to be given to the 
truss are based upon a cast-iron beam with the truss inserted into the upper flange 
of the beam, whereby it was submitted almost wholly to a tensile strain, they will 
not apply to the two constructions of trussed beams under consideration. 

As each construction of trussed beam will produce a strain upon the truss in ac- 
cordance with the position of the neutral axis of the section of the whole beam, and 
as the extension of the truss will vary according as it is more or less ductile, it is im- 
practicable, in the absence of the necessary elements, to give an amount of initial 
strain that would be applicable as a rule. 

From the various experiments made upon trussed beams, it is shown : 
1. That their rigidity far exceeds that of simple beams ; in some cases it was from 
7 to 8 times greater. 2. That when the truss resists rupture, the upper flange of the 
beam being broken by compression, there is a great gain in strength. 3. That their 
strength is greatly increased by the upper flange being made larger than the lower 
one. 4. That their strength is greater than that of a wrought-iron tubular beam 
containing the same area of metal. 

* The elongation of cast and wrought iron being 5500 and 10000, hence 10000-^-5500 =• 1.81. 



476 



STRENGTH OF MATERIALS. 



Flesxilts of Experiments upon tne Deflection of Bars, 
Beams, etc., of various Sections, etc. ; "by XT. S. Ordnance 
Corps, Barlow, Fairbairn, Hodgkinson, Stephenson,etc. 

Bar, Beams, etc., supported at both Ends ; Stress or Weight applied in the Middle. 

Value for De. 



Material and Section. 



•& 


JZ 




*o tc 












T3 


A 




= 5 


c3 


P< 


% Z 


^K 








« 


P 


cc 



53 flection given. 



Woods. 

Fir. Rectangle 

u Square 

Ash. Cylinder 

" '.* hollow .... 

" Square 

Yellow pine. Square 

Oak. Square 

Pine. Rectangle 

Metals. 

Cast iron, English 

u dry sand, square 
" green sand, u 

Flange, 5X. 3.... / 
Jj^ Flange, 5x. 3.... | 



Ft. 


Ins. 


3 









3 


10 


3 


10 


7 




5 




3 




40 




2 


10 


1 


s 


1 


8 


6 


6 



Flange, 1.5X. 5... 3 1 



Flange, 23.9x3.125 
6.5x1). 

area IS/ 

* 4 5x .S75) 
8 Xl.25 / 

Rectangular, ) 
area 1.965 J 

Open beam, ) 

area 2 J 



Wrought iron. Square . . . 
u Rectangle. 

T Flange, 4.5 X. 5 \ 
Rib, 3.25 diameter,/ 



Flanges, 2 of 



2 of ) 

2.25X.28V 
2.25X.3 ) 



I Tubes, thickness 

.03 in.) 
.525 U J 

i Tubes, thickness ) 
.037 in. J 

Corrugated plates 

Tubes, thickness 

.0410 in.) 
.143 » / 

Steel, cast, soft 

Brass, cast 



17 



luch. 

1.5 

1.5 

2. 

2. 

2. 

.75 
1. 
7.5 

1. 

2. 

2. 
.36 
.36 
.36 
.36 

.5 

3.29 
.91 

1.12 
.975 

1. 



9 2. 
9 1.5 



31 6 

17 

17 
3 2 

1 



| Inch. 
2. 
2. 



.75 
1. 

9.25 

1. 

2. 

2. 

1.55 

1.55 

1.55 
1.55 



14. 

36. 
2.015 

2.5 

2. 
3. 

10. 



.25 



1.9 3. 
15.5 24. 



12. 

3.1 
9.25 



12. 



14.62 



9.75 16. 
.23 .52 
.7 I .45 



- 



11.925 



13.535 



Lbs. 
120 
ISO 
715 
657 
225 
16 
15S 

1700 

800 
10S00 
5000 
112 
336 
112 
336 

2016 

60000 
44S0 



22400 
712 

712 

2240 
2240 

3136 
164S0 

44S 
5GS5 

2755 

44S0 

2262 



14.714 16S00 



Inch. 

.09 
1. 

2.7* 
2.5* 
1.2661 
1.5 
2.95 
5.25 

.161 
.11 
.045f 
.273 
1.03 
.27 
.895 

.079 

.1 
.3 



.004 



.132 



.0GS 
.074 



.375* 



.C5 



.1 
.12 



.65* 



.62 



.62* 
1.39* 
22 I .331 
60 I .04t 



4107 
14SS 



* Breaking weight 



f Elasticity perfect. 



$ Permanent set. 



STRENGTH OF MATERIALS. 477 

The Value in the preceding Table is for the Deflection given; hence, 
when the deflection that a bar, beam, etc., may be permitted to bear is 
given, the weight it may bear with that deflection will be determined by 
the formula, substituting for D the deflection it may bear. 

Deflection of Bars, Beams, Grirders, etc. 

The expedients of Barlow upon the deflection of wood battens determ- 
ined that the deflection of a beam from a transverse strain varied as the 
breadth directly, and as the cubes of both the depth and length, and that 
with like beanis and within the limits of elasticity it was directly as the 
weight. 

In bars, beams, etc., of an elastic material, and having great length 
compared to their depth, the deductions of Barlow will apply with suffi- 
cient accuracy for all practical purposes ; but in consequence of the va- 
ried proportions of depth to length of the varied character of materials, 
of the irregular resistance of beams constructed with scarfs, trusses, or 
riveted plates, and of the unequal deflection at initial and ultimate strains, 
it is impracticable to give any positive laws regarding the degrees of de- 
flection of different and dissimilar bars, beams, etc. 

In the experiments of Hodgkinson, it was farther shown that the sets 
from deflections was ver} T nearly as the squares of the deflections. 

In a rectangular bar, beam, etc., the position of the neutral axis is in its 
centre, and it is not sensibly altered by Variations in the amount of strain 
applied. In bars, beams, etc., of cast and wrought iron, the position of the 
neutral axis varies in the same beam, and is only fixed while the elastici- 
ty of the beam is perfect. When a bar, beam, etc., is bent so as to injure 
its elasticity, the neutral line changes, and continues to change during the ■ 
loading of the beam, until it breaks. 

When bars, beams, etc., are of the same length, the deflection of one, 
the weight being suspended from one end, compared with that of a beam 
uniform!}' loaded, is as 8 to 3 ; and when a beam is supported at both ends, 
the deflection in like cases is as 5 to 8. Whence, if a bar, etc., is in the 
first case supported in the middle, and the ends permitted to deflect ; and 
in the second, the ends supported, and the middle permitted to descend, 
the deflection in the two cases is as 3 to 5. 

Of three equal and similar bars or beams, one inclined upward, one 
downward at the same angle, and the- other horizontal, that which has 
its angle upward is the weakest, the one which declines is the strongest, 
and the one horizontal is a mean between the two. 

When a bar, beam, etc., is Uniformly Loaded, the deflection is as the 
weight, and approximately as the cube of the length or as the square of 
the length; and the element of deflection and the strain upon the beam, 
the weight being the same, will be but half of that when the weight is 
suspended from one end. 

The deflection of a bar, beam, etc., Fixed at oneEnd, and Loaded at the 
other, compared to that of a beam of twice the length, Supported at both 
ends, and Loaded in the Middle, the strain being the same, is as 2 to 1 ; and 
when the length and the loads are the same, the deflection will be as 16 
to 1, for the strain will be four times greater on the beam fixed at one end 
than on the one supported at both ends ; therefore, all other things being 
the same, the element of deflection will be four times greater ; also, as the 
deflection is as the element of deflection into the square of the length, 
then, as the lengths at which the weights are borne in their cases are as 
1 to 2, the deflection is as 1 : 2 2 x4=l to 16. 

The deflection of a bar, beam, etc., having the section of a triangle, and 
supported at its ends, is X greater when the edge of the angle is up than 
when it is down. 

Ss 



478 



STRENGTH OF MATERIALS. 



When the Length is uniform, with the same weight, the deflection is in- 
versely as the breadth and square of the depth into the element of de- 
flection, which is inversely as the depth. Hence, other things being equal, 
the deflection will vary inversely as the breadth and cube of the depth. 

Illustration. — The deflections of two pine battens, of uniform breadth and depth, 
and equally loaded, but of the lengths of 3 and 6 feet, were as 1 to 7.8. 

If a bar or beam is cylindrical, the deflection is 1.7 times that of a square 
beam, other things being equal. 

1 3 W 

Bars, Beams, etc. — When Fixed at one End, and Loaded at the other, ■= V. 

' bd 3 D * 
a constant quantity. 

• When Fixed at one End, and uniformly Loaded, 
When Fixed at both Ends, and Loaded in the Middle, 



3Z3W 



= V. 



I'W 



When Supported at both Ends, and Loaded in the Middle 
When Supported at both Ends, and uniformly Loaded, 



24 ud a D 

Z3W 



= V. 



It) b d a D ' 
5Z 3 W 



= V. 



8X10 ud 
When Supported in the Middle, and the Ends uniformly Loaded, 



3/3W 



5xl06d3D 



= V. 



When Supported at both Ends, and the Weight suspended from any other Point 
m2 W 2W 



than the Middle, ■ 



- z=z V, I representing the length in feet, b its breadth, d its 



lbd*D 

depth, W the weight or stress with which it is loaded, m n the distances of the weight 
. from the supports, and D the deflection in inches. 

Hence, in order to preserve the same stiffness in bars, beams, etc., the 
depth must be increased in the same proportion as the length, the breadth 
remaining constant. 

The deflection of different bars, beams, etc., arising from their own 
weight, having their several dimensions proportional, will be as the 
square of either of their like dimensions. 

Note— In the construction of models on a scale intended to be executed in full di- 
mensions, this result should be kept in view. 

Trtesixlts of Experiments upon the Transverse Strengtli 
and. Deflection of* Wrought-iron Rails.— [Barlow.] 





Rails. 


Weight 

per 
Yard. 


Length 

of 
Bearing. 


Depth. 


Area. 


Weight. 


Defle 


ction. 




For 

Weight. 


For each 
Ton. 






Lbs. 


Feet. 


Inch. 


Inch. 


Lbs. 


Inch. 


Inch. 




( 


60 


2.75. 


4.5 


6.166 


£240 


.027 


— 




Flanges, 2 25 ) 


60 


2.75 


4.5 


6.166 


41S0 


.031 


.004 


c? 


rib, .65. . ^ 


60 


2.75 


4.5 


6.166 


17020 


.057 


.005 


i/l 


I 


60 


2.75 


4.5 


6.166 


26680 


.0S7 


.01 


C_) 


" 2.6X1. 25ins.\ 
rib, .85 / 


75 


4.5 


5. 


7.5 


4480 


.05 


— 




75 


4.5 


5. 


7.5 


20160* 


J65 


.023 


JL 


" 3.5x.6in. ..) 
rib, 8 / 


57 


2.75 


3.5 


5.S5 


4480 


.05 





57 


2.75 


3.5 


5.85 


17920* 


.152 


.039 


» 


Head, 2.25x1 in. i 

rib, .75 ... > 

Flange, 3.5 X. 8 . ) 


60 


2.75 


4. 


6.7 


4480 


.034t 





JL 


60 


2.75 


4. 


6.7 


17920 


.064 


.082 


tamp. 


Head,2.5x.O ...) 
















|| 


rib, .6 .... > 


51 


3. 


4.5 


5.55 


6720* 


.024 





& 


Bottom, 1.25X-SS) 

















* Destructive weights. 

f The deflection between this and a like bar to this, reversed, wns, for between 5 and 10 tons 
weight, as .0074 and .0059. J Destructive weight 7 tons. 



STRENGTH OF MATERIALS. 



479 



As it is impracticable to give an}' general rule for the deflection of bars, 
beams, etc., of different lengths and sections, reference must be had to the 
results of previous experiments upon bars, beams, etc., of a like character 
to that of those for which the deflection is required. 

Thus, in the preceding Tables, page 476 to 478, are given the deflections 
ascertained in very many cases,, added to which is given a value or con- 

.I 3 W 
stant, obtained by the formula Jq^Tq- 

In the first and second examples of the Table are results of two experi- 
ments with a like material, but of differing dimensions. 

In order, then, to determine the relative values of the Constants, the va- 
rying elements of the case must be reduced to a uniform measure. 

In the examples referred to, the Values or Constants are as 187 and 202, 
their sections (bd 3 ) as 12 and 12, the weights applied as 120 and 180, and 
the lengths as 3 3 and 6 3 . 

If the deflections were in conformance with the formula, the Values here 
deduced would be equal, instead of 187 and 202 ; the proportion of which 

187 
is obtained by 5x5 == .92 of the deflection given by the formula. The deflec- 
tion as furnished by the Table for the second experiment is 1 ; hence, as 
.92 : 1 : : 1 : 1.09 — the calculated deflection of it. 

When it is required to estimate the Deflection for Differing Weights, Lengths, 
and Sections, and contrariwise, to estimate Weights, Lengths, and Sections 
for a given Deflection. 

Rule. — Divide the deflections by the cubes of the lengths- and by the 
weights ; or, multiply the deflections by the sections (b d 3 ). 

Thus, if deflections are as .15 and 1.2 inches, weights as 125 and 250 
lbs., lengths as 1 and 2 feet, and sections as lx^ 3 and 2 x 2 3 inches. 

.15 l 3 .15 <dt 
Then '^—-^--~ = -r£~ quotient of the deflection -4- the cubes of the lengths, 

which, being equal, shows the deflections to be as the cubes of the lengths. 
.15 125 .0012 2 . jl i J i 7/7 . 

IK • 9^0 ~ ooOfi ~ T = < l uotim t °f " ie reduced deflection -^ the weights ; 

hence the deflections are but one half of that due to the weights. 

2 lx2 3 16 1 

T X 9 — ~ = — = - = product of the preceding quotient and the sections 

(bd 3 ); hence the reduced deflections are as the sections. 



Relative Elasticity of varions Materials.— [Trumbull.] 



Ash 2.9 

Beech 2.1 

Cast iron 1. 



Elm 2.9 

Oak 2.8 

Pine, white 2.4 



Pine, yellow 2.C 

u pitch 2.9 

Wrought iron. . . .86 



Comparative Strength and Deflection of Cast-iron 
Flanged. 33 earns. 



Description of Beam. 


Comp 

Strength. 


Description of Beam. 


Comp. 

Strength, 


Beam of equal flanges. 

Beam with only bottom flange. 

Beam with flanges as 1 to 2 

Beam with flanges as 1 to 4 ... . 


.58 
.72 

.73 


Beam with flanges as 1 to 4.5. . . 
Beam with flanges as 1 to 5.5. .. 

Beam with flanges as 1 to 6 

Beam with flanges as 1 to C.73.. 


.78 
.82 
1. 
.92 S 



480 STRENGTH OF MATERIALS. 

GJ-eneral Deductions. 

In Cast Iron, the permanent deflection is from )^ to ^ of its breaking 
weight, and the deflection should never exceed % of the ultimate deflec- 
tion. 

All rectangular bars of Wrought Iron, having the same bearing length, 
and loaded in their centre to the full extent of their elastic power, will be 
so deflected that their deflection, being multiplied by their depth, the prod- 
uct will be a constant quantity, whatever may be" their breadth or other 
dimensions, provided their lengths are the same. 

The heaviest running weight that a bridge is subjected to is that of a 
locomotive and tender, which is equal to 1.5 tons per lineal foot. 

Girders should not be deflected to exceed the ^ °f an mcn to a f° ot m 
length. 

In cast iron, the 7^7 to ife of the breaking weight will give a visible set. 

When a load on a girder is supported by the bottom flange of it alone, 
it produces a torsional strain. 

A continuous weight, equal to that a beam, etc., is suited to sustain, 
will not cause the deflection of it to increase unless it is subjected to con- 
siderable changes of temperature. 

The heaviest load on a railway girder should not exceed % of that of 
the breaking weight of the girder when laid on at rest. 

Deflection consequent upon Velocity of the Load. — Deflection is very much 
increased by instantaneous loading ; by some authorities it is estimated to 
be doubled. 

The momentum of a railway train in deflecting girders, etc., is greater 
than the effect from the dead weight of it, and the deflection increases with 
the velocit3 T . 

Experiments made by the Commissioners of Railway Structures of 1849 
showed that a passingload produced a greater effect on a beam than a 
load at rest. 

A carriage was moved at a velocity of 10 miles per hour ; the deflection 
was .8 inch, and when at a velocity of 30 miles the deflection was 1)4 
inches. 

In this case, 4150 lbs. would have been the breaking weight of the bars 
if applied in their middle, but 1778 lbs. would have broken them if passed 
over them with a velocity of 30 miles per hour. 

Cast iron will bend to X °f its ultimate deflection with less than ^ of 
its breaking weight if it is laid on gradually, and but % if laid on rapidly. 

When motion is given to the load on a beam, etc., the point of greatest 
deflection does not remain in the centre of the beam, etc., as beams broken 
by a traveling load are always fractured at points bej-ond their centres, 
and often into several pieces. 

Chilled bars of cast iron deflect more readily than unchilled. 

liesnlts of* Experiments on tlie Subjection of Iron Bars 

to COntiimal Strains.— [Rep. Comm. on Raihca>/ Structures.] 

Cast-iron bars subjected to a regular depression, equal to the deflection 
due to a load of X °f their statical breaking weight, bore 10 000 success- 
ive depressions, and when broken by statical weight gave as great a re- 
sistance as like bars subjected to a like deflection by statical weight. 

Of two bars subjected to a deflection equal to that carried by half of 
their statical breaking weight, one broke with 28 G02 depressions, and the 
other bore 30 000, and did not appear weakened to resist statical pressure. 

Hence Cast-iron bars will not bear the continual applications of ^ of 
their breaking weight. 



STRENGTH OF MATERIALS. 



481 



A bar of Wrought Iron, 2 inches square and 9 feet in length between its 
supports, was subjected to 100 000 vibratory depression's, each equal to the 
deflection due to a load of f. of that which permanently injured a similar 
bar, and their depressions only produced a permanent set of .015 inch. 

The greatest deflection which did not produce any permanent set was 
due to rather more than J£ the statical weight, which. permanently injured 
it. . 

A wrought-iron box girder, 6x6 inches and 9 feet in length, was sub- 
jected to vibratory depressions, and a strain corresponding to 3762 lbs., 
repeated 43 370 times, did not produce any appreciable effect on the rivets. 

Mr. Tredgold, in his experiments upon Cast Iron, has shown that a load 
of 300 lbs., suspended from the middle of a bar 1 inch square and 34 inch- 
es between its supports, gave a deflection of .16 of an inch, while the elas- 
ticity of the metal remained unimpaired. Hence a bar 1 inch square and 
1 foot in length will sustain 650 lbs., and retain its elasticity. 

TORSIONAL STRENGTH. 

The Torsional Strength of any square bar or beam is as the cube of its 
side, and of a cylinder as the cube of its diameter. Hollow cylinders or 
shafts have greater torsional strength than solid ones containing the same 
volume of material. 

The Torsional Angle of a bar, etc., under equal pressures will vary as the 
length of the bar, etc. Hence the torsional strength of bars of like diam- 
eters is inversely as their lengths. 

The strength of a cylindrical prism compared to a square is as 1 to .85. 

When a bar, beam, etc., having a length greater than its diameter, is 
subjected to a torsional strain, the direction of the greatest strain is in the 
line of the diagonal of a square, and if a square be drawn on the surface 
of the bar, etc., in its primitive form, it will become a rhombus by the ac- 
tion of the strain. 

Torsional Stx*engtlx of Cast Iron, deduced from the Ex- 
periments ofHVIajor Wade, TU. S. -A.. 
Reduced to a uniform Measure of One Inch Square or in Diameter ; Weight or Stress 
applied at One Foot from Centre of Axis of the Material, and at the Face of the 



Axis or Jo\ 


irnal. 




Breaking Weight of Figures 






Area of Cross- 
section. 


Square, b d 2 . 


Cylinder, d 2 . 


Area of 

Section. 


Hollow Cylin- 
der, rf 3 -<*' 3 . 


Area of 
Section. 


Hollow Cylin- 
der, d 2 — >3. 


Inch. 
1 

2 
3 


Lbs. 

730 
677 
840 


Lbs. 
013 
098 
636 


Inch. 

.905 

1.931 

2.72S 


Lbs. 
d' — .h 563 
d' = .5 597 
d' = .5 540 


Inch. 

1.012 

1.967 
2.966 


Lbs. 

d' = .6 5S5 
d' — .l 579 

d' = .S 476 


Mean 


749 


649 




5G7 




547 



Summary of Results. 

All of the bars were from the same mixture of common foundr}- iron, 
of a mean torsional strength of 644 lbs. per square inch of section. 

From these results it appears that solid square shafts have about J less 
strength than solid C3 r linders of equal areas. 

The stress which will give a bar a permanent set of }4° * s about jfe of 
that which will break it, and this proportion is quite uniform, even when 
the strength of the material may vary essentially. 

The strongest bars give the longest fractures. 

Wrought Iron, compared with Cast Iron, has equal strength under a 
stress which does not produce a permanent set, but this set commences 

Ss* 



482 



STRENGTH OF MATERIALS. 



under a less force in wrought iron than cast, and progresses more rapicH? 
thereafter. The strongest bar of wrought iron acquired a permanent s^t 
under a less strain than a cast-iron bar of the lowest grade. The mean 
Values of cast and wrought iron and bronze, for bars of small diameters 
for a permanent set of ^£°, are as 1, .6, and .33. 

The torsional strength and rigidity of Puddled Steel does not differ es- 
sentially from that of cast iron. 

The coefficients for the torsional breaking stress of iron and bronze, as 
determined by Major Wade, are : Wrought Iron, 640 ; Cast Iron, 5G0 ; 
Bronze, 460. 

Torsional Strength of Cast and Wrought Iron arid Bronze, 
-with their Values for different Diameters. 

Reduced to a uniform Measure of One Inch Square or in Diameter ,- Weight or Stress 
applied at One Foot from Centre of Axis of the Material, and at the Face of the 
Axis or Journal 

Length of Journal, or of the Bar or Beam submitted to Stress, for which the Values 
are given, three times the Diameter or Side of the Shaft 



FlGUKES 


Specific 
Gravity. 


Length oi 
Journal 
or Side. 


Breaking 
Weight. 


Va 
2 Ins 


ue tor 
5 Ins 


Diamet 
10 Ins. 


r of 
15 Ins. 


. Cylinder (Cast Iron). 

Good common castings 

" cold blast, mean 
of S trials 


T.1S 

7.32 
7.724 

7.S55 

— 

8.71 
7.2~ 
.7.S55 


Inch 

a 

8. 
8. 
S. 

8. 

8. 

S. 

s. 

3. 

4.S 

3. 


Lbs 

5S3 

705 
750 
833 

300> 
642) 

192> 
458 j 

730) 
840/ 


Lbs 

170 
175 

l'.>0 

200 
130 

55 

220 
170 


Lbs 

115 

120 

130 
1.5 

128 

45 

150 
105 


Lbs 

105 

110 
120 

125 

125 

35 

140 

1G0 


Lbs 

100 

105 
115 


u greatest extreme 

Cylinder (Wrought Iron). 
Begins to yield, permaneut set . 

Bends without breaking 

Cylinder (Bronze). 
Begins to yield, permanent set . 

Bends without bre iking 

Square (Cast Iron) 


120 
123 

33 
134 


" ( Wrought Iron) 


152 



The Torsional Strength of Cast Steel is about double that of Cast Iron. 

The experiments above given were made with bars not exceeding 2 
inches in diameter; the relations given, therefore, do not hold, as the'di- 
ameters are increased, in consequence of the shrinking of the cast metals 
in cooling, which, by cooling at the outer surface first, draws the metal 
from the centre, and in effect gives to a bar or shaft the properties of a 
hollow cylinder. In shafts of 10 inches in diameter, the torsional strength 
of wrought iron is fully equal to that of cast iron ; and with larger diam- 
eters it would be much greater, but that it suffers deterioration as its di- 
ameter increases, from the increased difficulty in effecting welding and the 
reduction of the metal to a fibrous texture. 

The following rules are purposed to apply in all instances to the diam 
eters of the journals of shafts, or to the diameter or side of the bearings of 
the beams, etc., where the length of the journal or the distance upon which 
the strain bears does not greatly exceed the diameter of the journal or 
side of beam, etc. : hence, when the length or distance is greatly increased. 
the diameter or sidj must be correspondingly increased. 

To Compute tlie Torsional Strength of Square or Roxmd 
Sliafts, etc. 

^ Rule. — Multiply the Value in the preceding Table 03- the cube of the 
side or of the diameter of the shaft, etc., and divide the product by the 
distance from the axis at which the stress is applied in feet ; the quotient 
will give the resistance in pounds. 



STRENGTH OF MATERIALS. 483 

Example. — What torsional stress may be borne by a cast-iron shaft of the best 
material, 2 inches in diameter, the power being applied at two feet from its axis ? 

200X2 3 = 1G00, and — = S00 lbs. 

To Compiite the Diameter of a Sqxxare or Round Shaft, 
etc., to resist Torsion. 

Rule. — Multiply the extreme of pressure upon the crank-pin, or at the 
pitch-line of the pinion, or at the centre of effect upon the blades of the 
wheel, etc., that the shaft may at any time be subjected to, by the length 
of the crank or radius of the wheel, etc., in feet; divide their product by 
the Value in the preceding Table, and the cube root of the quotient will 
give the diameter of the shaft or its journal in inches. 

Example. — What should be the diameter for the journal of a wrought-iron water- 
wheel shaft, the extreme pressure upon the crank-pin being 59 400 lbs., and the crank 
5 feet in length ? 

5940^0x5 rj_ 23TG ^ and ^ 22T6 ,__ 13>34 inche ^ 

When two Shafts are used, as in Steam-vessels with one Engine, etc. 

Rule. — Divide three times the cube of the diameter for one shaft by 
four, and the cube root of the quotient will give the diameter of the shaft 
in inches. 

Example.— The area of the journal of a shaft is 113 inches; what should be the 1 
diameter, two shafts being used ? 

Diameter for area of 113 = 12. 

Qy193 

Then —j— = 1296, and ^1296 = 10.9 inches. 

Note. — The examples here given are deduced from instances of successful prac- 
tice; where the diameter has been less, fracture has almost universally taken place, 
the strain being increased beyond the ordinary limit. 

2. — When the work to be performed is of a regular character, and the stress is 
consequently uniform, the proportion of % may be reduced to %. 

Ttelative Values of* Diameters. 

When shafts of less diameter than 12 inches are required, the Values here given 
may be slightly reduced or increased, according to the quality of the iron and the 
diameter of the shaft to be used; but when they exceed this diameter, the Values 
may not be increased, as the strength of a cast or wrought iron shaft decreases very 
materially as its diameter increases. 

To Compute tlie Torsional Strength, of Hollow Sliafts 
and. Cylinders. 

Rule. — From the fourth power of the exterior diameter subtract the 
fourth power of the interior diameter, and multiply the remainder by the 
Value of the material ; divide this product by the product of the exterior 
diameter and the length or distance from the axis at which the stress is 
applied in feet ; the quotient will give the resistance in pounds. 

Example. — What torsional stress may be borne by a cast-iron hollow shaft, hav- 
ing diameters of 3 and 2 inches, the power being applied at 1 foot from its axis ? 

34 _ 2*X105 = S1 -1GX105^0S25, which -=- 3x1 = ^5 = 2275 lbs. 

The order of shafts, with reference to the degree of torsional stress to 
which they are subjected, is as follows : 

1. Fly-wheel. 3. Secondary. 

2. Water-wheel. 4. Tertiary, etc. 

Hence the diameters of their journals may be reduced in this order. 



484 



STRENGTH OF MATERIALS. 



Relative "Valixe of different T^igxires to Resist Torsion, 
having eciual Sectional Areas. 



Solid 
Cylinder. 



Solid 
Square. 



.875 



Hollow Cylinders, the interior and exterior Diameters of which 

are in the Proportion of 
4 to 10 I 5 to 10. I 6 to 10. I 7 to 10. I 8 to 10 



1.2653 



1.4433 



1.7 



2.0S64 



2.7377 



Variovis Qualities of different ]Vletals, 
As determined by the Experiments of Major Wade for the U. S. Ordnance Corps. 



Bronze . 



Cast iron 



Cast steel . . . 
Wrought iron 



(Least. . . 
"^Greatest 

(Least... 
'••"'• (Greatest 
Mean . . 

(Least. . . 

(Greatest 

(Least. 



(Greatest 






7.978 

8.953 

6.9 

7.4 

7.225 

7.729 

8 953 

7.704 

7.853 







Torsiona 


Strength 








g.3 

* SB 

« a 


lor Diam. 1 Inch, 


Is 


OS 


•ss 


and Length of 1 
Foot. 


a 


« » 








H 

Ota 


"2 




At h De- 


Ulti- 


5 
35 






gree. 


mate. 






PerSq. In. 


PerSq In 


Lbs. 


Lbs. 


PerSq In. 




17 093 


. — 


620 


SS2 





4.57 


50 7S6 


— 


833 


1147 





5.94 


9 000 


416 


1006 


2 732 


S4 529 


4.57 


45 970 


958 


2500 


5012 


174 1-20 


33.51 


31 829 


680 


19:0 


3 785 


144 910 


22.34 





— 








198 944 





123 000 


1916 


— 


13 600 


391 985 


- 


38 027 


542 


970 


2175 


40 000 


10.45 


74 592 


— 


1320 


2 608 


127 720 


12.14 



DETRUSIVE STRENGTH. 

The Detrusive Strength of any body is directly as its strength, or thick- 
ness, or area. 



R-esi 



Its of Experiments upon 
of JVtetals with. 



the Detrusive 
a. Punch. 



Strength 



Metals. 


Diameter 

of 

Punch. 


Thickness 

of 

Metal. 


Power 
exerted 


Power required for a 

Surface of Metal of One 

Square Inch. 


Brass 

Cast iron 


Inch 
1. 

.5 

.5 

1. 

.5 
.5 
.5 
.5 

1. 

2. 


Inch. 
.045 

.08 
.17 
.3 
.25 
.OS 
.17 
.24 
.615 
1.06 


Lbs 

5 44S 

3 9S3 

7S23 

21250 

34 720 

6 025 
11950 
17 000 
82 S70 

297 400 


Lbs. 
37 000 
30 000 

| 30 000 

22 300 
90 COO - 

1 45 000 

43 000 

44 300 


"2 o 55 




2 '* s 


Steel 

Wrought iron < 


o s L 
3 ™ ® 
IF? 

is: 

^ 2. 3 



To Compute the Power necessary to Punch Iron, 
Brass, or Copper Plates. 

Rulk. — Multiply the product of the diameter of the punch and the 
thickness of the metal by 150 000 if for wrought iron, by 128 000 if for 
brass, and b} r 9!) 000 if for copper, and the product will give the power re^ 
quired in pounds. 

Comparison between Detrusive and Transverse 
Strengths. 

Assuming the compression and abrasion of the metal in the application 
of a punch of one inch in diameter to extend to % of an inch beyond the 
diameter of the punch, the comparative resistance of wrought iron to de- 
trusive and transverse strain, the latter estimated at 600 lbs. per square 
inch, for a bar one foot in length, is as 2.5 to 1. 



STRENGTH OF MATERIALS. 



485 



Results of Experiments upon the Detrusive Strength. 

of* IMetals -with. Shears. 

Made by Parallel Cutters-. 

. Wrought Iron Thickness from .5 to 1 inch, 50 000 Ihs. per square inch. 

Made by Inclined Cutters, angle 1 in 8 = 7°. 



Sheet Metals. 




Thickness. 


Power. 


Bolts. 




Diameter. 


Power. 




Ins. 
.05 
.291 
.24 
.51 
1. 


Lbs. 
540 
11196 
14 930 

39 150 

44 800 


Brass 


Ins. 
1.11 

.775 

.775 

1.142 

.32 


Lbs. 
29 700 


Copper 

Steel - 


Copper 

Steel 


11310 

23 720 


Wrought iron . 


{ 


Wrought iron . 


•{ 


35 410 . 
3 093 



The resistance of wrought iron to shearing is about 75 per cent, of its 
resistance to tensile stress. 

The resistance to shearing of plates and bolts is not in a direct ratio. It 
approximates to that of the square of the depth of the former, and to the 
square of the diameter of the latter. 

Character of* Strains to -which. Connecting TiocLs, Straps, 
GriDs, and. Keys are snhjected.. 

Heads of Rods— At sides of keyholes, tensile and crushing ; at front of 
keyholes, detrusive. 

Straps. — At crown and at the sides of keyhole, tensile; at back of key 
holes, detrusive. 

Gib. — Transverse, uniformly loaded along its length, fixed at both ends. 

Key. — With single gib, transverse, uniformly loaded along its length, 
fixed at both ends. 

Key. — With double gib, transverse, uniformly loaded along its length, 
fixed at both ends. 

.Woods. 

When a beam or any piece of wood is let in (not mortised) at an incli- 
nation to another piece, so that the thrust will bear in the direction of the 
fibres of the beam that is cut, the depth of the cut at right angles to the 
fibres should not be more than .2 of the length of the piece, the fibres of 
which, by their cohesion, resist the thrust. 

Shafts and. Gudgeons. 

Shafts are divided into Shafts and Spindles, according to their magni- 
tude. 
A Gudgeon is the metal journal or arbor upon which a wooden shaft 

revolves. 

Shafts are subjected to Torsion and Lateral Stress combined, or to Lat- 
eral Stress alone. 

Lateral Stiffness and Strength. — Shafts of equal length have lateral stiff- 
ness as their breadth and the cube of their depth, and have lateral strength 
as their breadth and the square of their depths. Hence, in shafts of equal 
lengths, their stiffness b}' any increase of depth increases in a greater pro- 
portion than their strength. 

Shafts of different lengths have lateral stiffness, directty as their breadth 
and the cube of their depth, and inversely as the cube of their length ; and 
have lateral strength directly as their breadth and as the square of their 
depth, and inversely as their length. Hence, in shafts of different lengths, 
their stiffness by any increase of their length decreases in a greater pro. 
portion than their strength. 



486 STRENGTH OF MATERIALS. 

Hollow shafts having equal lengths and equal quantities of material 
have lateral stiffness as the square of their diameter, and have lateral 
strength as their diameters. Hence, in hollow shafts, one having twice 
the diameter of another will have four times the stiffness, and but double 
the strength ; and when having equal lengths, by an increase in diameter 
they increase in stiffness in a greater proportion than in strength. 

The stress upon a shaft from a weight upon it is proportional to the 
product of the parts of the shaft multiplied into each other. Thus, If a 
shaft is 10 feet in length, and a weight upon the centre of gravity of the 
stress is at a point 2 feet from one end, the parts 2 and 8, multiplied to- 
gether, are equal to 16 ; but if the weight or stress were applied in the mid- 
dle of the shaft, the parts 5 and 5, multiplied together, would produce 25. 

The ends of a shaft having to support the whole weight, the end which 
is nearest the weight has to support the greatest proportion of it, in the 
inverse proportion of the distance of the weight from the end. Hence, 
when a shaft is loaded in the middle, each of the journals or gudgeons has 
half the weight or stress to support. 

When the load upon a shaft is uniformly distributed over any part of it, 
it is considered as united in the middle of that part ; and if the load is not 
uniformly distributed, it is considered as united at its centre of gravity. 

When the transverse section of a shaft is a regular figure, as a square, 
circle, etc., and the load is applied in one point, in order to give it equal 
resistance throughout its length, the curve of the sides becomes a cubic 
parabola ; but when the load is uniformly distributed over the shaft, the 
curve of the sides becomes a semi-cubical parabola. 

The deflection of a shaft produced by a load which is uniformly distrib- 
uted over its length is the same as when % of the load is applied at the 
middle of its length. 

The resistance of the body of a shaft to lateral stress is as its breadth 
and the square of its depth ;" hence the diameter will be as the product of 
the length of it and the length of it on one side of a given point, less the square 
of that length. 

Itj/ustration — The length of a shaft between the centres of its journals is 10 fret; 
what should be the relative cubes of its diameters when the load is applied at 1, 2, 
and 5 feet from one end ? and what when the load is uniformly distributed over the 
length of it ? 

IX l l — I 3 = rf 3 ; and when uniformly distributed, rf 3 -4- 2 = d l . 
10x1 = 10 — 12 — 9 = cube of diameter at 1 foot ; 10x2 = 20 — 22 — 1G = cube of 

diameter at 2 feet ; 10x5 = 50 — 5 2 = 25= cube of diameter at 5 feet. 

When a load is uniformly distributed, the stress is greatest at the middle of the 
length, and is equal to half of it ; 25-=- 2 = 12.5 = cube of diameter at 5 feet. 

CYLINDRICAL OR SOLID SHAFTS. 

To Compute the Diameter of a Shaft of Cast Iron, to re- 
sist Lateral Stress alone. 

When the Stress is in or near the Middle. Rule. — Multiply the weight 
b}' the length of the shaft in feet ; divide the product by '500, and the 
cube root of the quotient will give the diameter in inches. 

Example.— The weight of a water-wheel upon a shaft is 50000 lbs , its length 30 
feet, and the centre of stress of the wheel 7 feet from one end; what should be the 
diameter of its body ? 

\/\ — nw — ) ~ ^-4-^ ms i if the weight was in the middle of its length. 

Bene? the diameter at 7 feet from one end Mill he, as by preceding Rule, 30x7 — 
7 2 = \^\ — relative cube of diameter at 7 feet ; 33x15 — 15 2 = 225 = relative cube 
of diameter at 15 feet. 

Then, as $/2:5 : 14.42 : : \/\<Sl : V2.S0 ins., the diameter of the shaft at 7 feet from 
one e?id. 



STRENGTH OF MATERIALS. 487 

When the Stress is uniformly laid along the Length of the Shaft. Rulk.^— 
Divide the cube root of the product of the weight and the length by 9.3, 
and the quotient will give the diameter in inches. 
Example. — Apply the rule to the preceding case. 
V50 000X3 Q . 

— o — = 12 - 31 ^- 

Or, When the Diameter f on the Stress applied in the Middle is given. 
Rulic. — Take the cube root of % of the cube of the diameter, and this root 
will give the diameter required. 

Example.— The diameter of a shaft when the stress is uniformly applied along its 
length is 14.422 ins. ; what should be its diameter, the stress being applied in the 
middle ? 

V%X14.4J2 3 — £/%Xo000 = 12.33 ins. 

HOLLOW SHAFTS OF CAST IRON. 

When the Stress is in or near the Middle. Rule. — Divide the continued 
product of .012 times the cube of the length, and the number of times the 
weight of the shaft in pounds Iry the square of the internal diameter added 
to 1, and twice the square root of the quotient added to the internal di- 
ameter, will give the whole diameter in inches. 

Example. —The weight of a water-wheel upon a hollow shaft 30 feet in length is 
2 5 times its own weight, and the internal diameter is 9 ins. ; what should be the 
whole diameter of the shaft ? 

'.012X303X2.5 /BIO .... 

82 =3 * 14 mS ' 



// .012XSQ3X2.5 _ n 

V\ 1 + 92 — V i 

Then 3.14x2 -j- 9 = 15.28 ins, the diameter. 



To Compute the Diameter of a Solid Sliaft of Cast Iron 
to resist its o^vxi "Weight alone. 

Rule. — Multiply the cube of its length by .007, and the square root of 
the product will give the diameter in inches. 

Example. — The length of a sliaft is 30 feet; what should be its diameter in the 
body ? v/(303x.C07) — 1S9, and V1S9 = 13.75 ins. 

To Compute the Diameter of* a Sliaft, tlie Stress "being 
applied, in tlie IVtiddle, when it has to resist l)oth Tor- 
sional and Lateral Stress combined. 

Rulh. — Ascertain the diameter for each stress, and the cube root of the 
sum of their cubes will give the diameter required. 

Example. — The diameter of the journal of a shaft to resist torsional stress is as- 
certained to be 17 ins., and the diameter of its body in the. centre to resist lateral 
stress has also been ascertained to be 14.422 ins. ; what should be the diameter of the 

body? VH73 + 14 .4223) =7913, and tymS = 19.927 ins. 

The strength of a cylindrical shaft compared to a square one, the diameter of the 
one being equal to the side o.f the other, is as 1 to 1.2, and of a square shaft to a cy- 
lindrical as 1 to .85. 

To Compute tlie Deflection of a Cylindrical Sliaft. 

Rule. — Divide the square of three times the length in feet by the prod- 
uct of the following Constants and the square of the diameter in inches, 
and the quotient will give the deflection. 

Cast iron, cylindrical 1500 [ Wrought iron, cylindrical . . 1980 

Cast iron, square 25G0 | Wrought iron, square 3360 

Example.— Tlie length of a cast-iron cylindrical shaft is 30 feet, and its diameter 
in the centre 15 ins. ; what is its deflection ? 

•> 

30x3 81 HO 



1500X152 337500 



= .0£4 ms. 



488 STKEXGTH OF MATERIALS. 

To Compute the Diameter of Shafts of "Wrought Iron., 
Oak, and. [Pine. 

Multiply the diameter ascertained for Cast Iron as follows : Wrought 
Iron by .935, Oak by 1.83, Yellow Pine by 1.716. 

To Compute tlie Length, of a Cylindrical Shaft. 

Bulk. — Multiply the preceding Constant by the deflection, and the square 
of the diameter and % of the square root of the product will give the length 
in feet. 

Example.— The diameter of a cast-iron cylindrical shaft is 15 ins., and the de- 
flection assigned to it is .024; what should be its length? 
V1500X. 024X15= 90 OA _ m 

- g — ~s~ do f eet - 

GUDGEONS. 

To Compute the Diameter of a Single G-udgeon of Cast 
Iron, to Support a given "Weight or Stress. 

Rulk. — Divide the square root of the weight in pounds by 25 for cast 
iron, and 26 for wrought iron, and the quotient will give the diameter in 
inches. 

Example. — The -weight upon a gudgeon of a cast-iron water-wheel shaft is 62 500 
lbs. ; what should be its diameter? 

V02 500 250 &•; 
-^- = -=10zns. 

To Compute tlie Diameter ofTwo G-udgeons of Cast Iron, 
to Support a given Stress or "Weight. 

Rule. — Multiply the square root of the weight of half the wheel by 
.048, and the product will give the diameter in inches. 



The flexure of a spring is proportional to its load and to the cube of its 
length. 

Deflection of a Carriage Spring. 

A railway-carriage spring, consisting of 10 plates %, thick and 2 of 
% inch, length 2 feet 8 ins., w^idth 3 ins., and camber or spring 6 ins., 
deflected as follows, without any permanent set : 

%i ton, % inch. I 1% ton, ll{ inch. 3 tons, 3 inches. 

1 " 1 ".', 2 « 2 I ./« 4 " 

Compression of an India-rubber Buffer of 3 ins. Stroke. 

1 ton, 1.3 inch. 2 tons, 2 inches. I 5 tons, 2% inches, 

1% " \% " 3 " 2% " | 10 " 3 

TUBES AND FLUES. 

Resistance of "Wrought-iron Tubes to External and 
Internal Pressixre.- [W. Fairbaibs.] 

It has been considered a rule that a cylindrical tube, such as a boiler- 
flue, when subjected to a uniform external pressure, was equally strong in 
every part, and that the length did not affect the strength of a tube so 
placed. Although this rule may be true when applied to tubes of indefi- 
nite lengths, it is very far from true where the lengths are restricted with- 
in certain apparently constant limits, and where the ends are securely 
faetened, as in heads or tube sheets, which prevent their yielding to an 



STRENGTH OF MATERIALS. 



489 



external force, or where, as in flues constructed in courses, the laps pre- 
sent a ring which greatly increases their resistance. 

In some experimental tests to prove the efficiency of large boilers, it 
was ascertained that flues 35 feet long were distorted with considerable 
less force than others of a similar construction 25 feet long. 

Results ofExperiments upon tlie Resistance of "Wrought-* 
iron Tubes and Flues to External Pressure or Collapse. 



N 




S 


• 8 

S3 a • 










£ 


to 


r «''o « 


Scq a 




OJ 


* 


{£ aj 1 " 1 


P 


J 


H 


P, 


Ins. 


Ins. 


Ins. 


Lbs. 


4 


19 


.043 


1T0 


4 


40 


.043 


65 


4 


60 


.043 


43 


6 


30 1 


.043 


4S 



Welded Tubes, and Ends secured to Head Plates. 



Ins. 




30 
30 



Ins. 
.043 
.043 
.043 
.043 



Lbs. 
32 
65 
39 

32 



Ins. 
10 
10 
12 
12 



Ins. 

50 
30 
60 
30 



Ins. 
.043 
.043 
.043 
.043 



See a 
P- 1 2 



Lbs. 
19 
33. 
12.5 



Tubes and Flues, Lap and Abut 
Joints. 


Riveted Flues, Over-lap Joints, Ends 
closed. 


Diameter. 


Length. 


Thickness 

of 

Plates. 


Pressure 

per Square 

Inch. 


Diameter. 


Length. 


Thickness 

of 

Plates. 


Pressure 

per Square 

Inch. 


Ins. 
Tube 1S% 
Lap 9 
Abut 9 


Ins. 
61 
37 
37 


Ins. 
.25 
.14 
.14 


Lbs. 
420 
262 

378 


Ins. 

by 

14X 


Ins. 

V 60 


Ins. 

.125 


Lbs. 
125 



Rivets }£ in. , and 1^ ins. apart. 





Cylindrical and Elliptical \ 


Riveted Flues. Abut Joints 








. 




m 


£8 




^ 




$ 


Si 


Efl 


1 

a 


be 

a 
3 


"Sort 

2 S 


Sec c 


a 


S 

S 


to 

a 

OJ 


Eh 






Ins. 


Ins. 


Ins. 


Lbs. 




Ins. 


Ins. 


Ins. 


Lbs. 


Cylindrical 


18.X 


61 


.25 


420 


Elliptical 


20^x15^ 


61 


.25 


127.5 


u 


12. 


61 


.043 


12,5 


" 


14. X10X 


62 


.043 


6.5 



To Compute Collapsing Pressure upon a Tube or Flue. 

The total external pressure upon a tube or flue varies directly as its 
longitudinal section, that is, as product of the length and the diameter. 

P' /dC = P; P' representing the pressure to which the tube is subjected in 
pounds per square inch, I the length of the tube in feet, d diameter in inches, 
and C a constant to be determined. 

It has been ascertained by experiment that the resistance of thin metal 
plates to a force tending to crush or to crumple them, varies directly as a 
certain power (x) of their thickness. 

P 

Hence the Value of a tube, etc., to resist collapse is as—, t representing 

the thickness of the metal in inches. 

The mean of the product of P Id in the several experiments here given, 
where the metal was of a uniform thickness of .043 inches is 850. for a 
thickness of % mcn 9140, etc.; and the mean of the value of x for all 
thicknesses is 2 - 19 . 

Tt 



490 



STRENGTH OF MATERIALS. 



By taking ».19 for the index off, this formula becomes Vx — =P' the 

Id ' 

collapsing pressure, which is the general formula for calculating the 
strength of wrought-iron tubes and short flues subjected to external 
pressure— that is, provided their length is not less than 1.5 feet, and not 
greater than 10 feet. 

V varies somewhat with the length of the flues and tubes, and is taken 
by Fairbairn at 806 300. 



£•2.19 



Hence, -_ X 806 300=P', and ^1/^^=4 
Id ' V 806 300 

Illustration.— 1. If the flue of an iron boiler is .25 in. thick, 16 ins. in diameter, 
and 20 feet in length, Avhat is its resistance to collapsing in lbs. per square inch ? 

X806I- 

being 



.25*. 19 log. 2.6S15 04S 

— — - X S06 300= a — X S06 300=*—- X 806 300=120. 9 lbs. 

fi 'A io o'zO 0*40 



2. Its pressure being ascertained, what-should be its thickness? 

2.19 /'20X16X1--0 9 2 - 19 / 

vf .04S. L-g. of .04S=2.G312. 



SOG 300 



Then, 2.6812 
2.19 



.ST 12^2. 19= . 3979=1. 3979=log. of .25=. i5 in. 
Power of Thicknesses of Tubes and. Flues. 



hickness. 


ower c 
2.19 


)I J.X1J.O 

Thickness. 


Kiiesses 


.049 


.00135 


.134 


.01226 


.05S 


.00196 


.148 


.01524 


.065 


.00251 


.165 


.01933 


.072 


.00314 


.18 


.02339 


.0S3 


.0'">429 


.1875 


.02557 


.095 


.00577 


.203 


.03044 


.109 


.007S 


.22 


.0363 


.12 


.00953 


.23S 


.04313 









Thickness. 




Thickness. 




.25 


.04S02 


.425 


.1535 


.259 


.0519 


.4375 


.1636 


.284 


.06349 


.454 


.7775 


.3 


.07159 


.5 


.2192 


.3165 


.07S29 


.5625 


.2S35 


.34 


.09429 


.625 


.3573 


.375 


.12672 


.6S75 


.4402 


.38 


.1202 


.75 


.5327 



.Resvilt 
iron 1 

Diameter. 


s ofEx; 
ITulDes o 

Length. 


perimei 
r Flues 

Thickness. 


its upoi 

i to Illt< 

Pressure 

per Square 

Inch. 


a the K.€ 
jrnal Jr* 

Diameter. 


>sistanc 
ressure 

Length. 


'e ofWrought- 
or 13xxrsting. 

1 Pressure 
Thickness. [ per Square 
Inch. 


Ins. 
6 
6 


Ins. 

12 
24 


Ins. 
.043 
.043 


Lbs. 

475 
235 


Ins. 

6 

12 


Ins. 

30 
60 


Ins. Lbs. 
.043 230 
.043 110 



Formulae of Resistance of Cylindrical Tubes or Flues 
to Internal Pressure, omitting Element of Length. 

Tx°^ _ Pxd 2txT "^ . 

— —— ==P; -^-=-=£; and — — =rf } I representing tensile resistance of 

material per square inch in pounds, d diameter of tube or flue, t thickness, 
both in inches, and P pressure requisite to produce rupture of tube or flue in 
pounds per square inch. 

Illustration. — If diameter of an iron flue is 12 inches, and its thickness .04 in.. 
whut will be its bursting pressure ? and having ascertained its pressure, deduce a 
diameter and thickness to correspond. 

Assume T=45000 lbs., hence 45000X.5 for single riveting =22 500 lbs. 

22500X2X.04 1800 «„__ 2x. 04x22500 1800 >*"\ , 

= =150 lbs.: = =12 ins.: and 

12 12 ' 150 150 ' 



150X12 1^00 



2X22500 45000 



=.04 ins. 



STRENGTH OF MATERIALS. 491 

MEAN OF THE RESULTS OF EXPERIMENTS UPON THE RESISTANCE OF 
WROUGHT-IRON CYLINDRICAL TUBES TO INTERNAL PRESSURE. 

To Compete tlie Thickness of a "Wronglit-iron riveted 
Tube or Fine. 

When the Diameter of the Tube and the Pressure in Pounds per Square 
Inch are given. Rulk. — Multiplj' the pressure in pounds per square inch 
by the diameter of the tube in inches, and divide the product by twice 
the tensile resistance of the metal in pounds per square inch. 

P2xample. — The diameter of a wrought-iron tube is 6 ins., and the pressure to 
which it is to he submitted is 425 lbs. per square inch ; what should be the thickness 
of the metal ? 

Assume the tensile strength to be 29 651 lbs. 
425x6 2550 

2^G5T^2 = 5-9302 :=r - 043mS - 

The tenacity or tensile resistance of wrought-iron boiler plates ranges from 42 000 
to 62 000 lbs.* per square inch. 

Tubes or flues subjected to internal pressure or bursting have much greater re- 
sistance than when subjected to external pressure or collapsing; in some cases, 
where the lengths of the collapsed tubes were 25 feet, the differenca was about 6.2 
times. 

The difference, however, between these strains can not be determined as a rule, for 
the reason that the resistance to internal pressure is inversely as the diameter of the 
tube or flue alone, without regard to its length ; whereas, with the resistance to col- 
lapse, the stress is inversely as the product of the diameter and the length. 

Application, to Constimction of* tlie iResnlts of the 

Experiments. 

With drawn or brazed tubes, when there are no courses and laps, their length is 
an essential element in an estimate of their resistance to collapse ; but with riveted 
flues, constructed in courses, the objection to length is removed, as the addition of 
the laps is a source of great resistance to collapse, rendering the flue alike to a scries 
of lengths, each, equal to the distanca between the centres of the courses. 

In a boiler of the ordinary construction, of 30 feet in length and 3)^ feet 
in diameter, with two flues 16 ins. in diameter, the cylindrical external 
shell has 2.8 times resistance to the force tending to burst it that the flues 
have to resist the same force to collapse them. 

To Compute tlie "Ultimate Collapsing Resistance ofa Klne. 

Rulk.— Take the square of the thickness of the metal in decimals of an 
inch, or that due to the number of it, if given by a wire gauge, and multi- 
ply it by its proportional unit or multiplier from the Table (page 490), the 
thickness and length being duly considered, and divide the product by the 
product of the diameter of the flue in inches and the. length of it in feet. 

Example.— The diameter of a flne is IS ins., the thickness of the metal No. 3 U. S. 
wire gauge (.23 in.), and the length of it 30 feet; what is its ultimate resistance to 
collapse per square inch ? 

Multipliers for thicknesses from % to % in., and for a length of 30 feet, are 810 000 
to 920000, the difference of which is 920 000 — 810 800 = 110 000, and the difference 
in thickness .25 — 125=. 125. Then, as .125 : 110 000 : : .105 (.23 — .U5) : 92 400. 

Difference in length, 35 — 25 = 10. Then, as 10 : 110 000 :: 5 (35 — GO) : 55 000. 

• 92400 + 55 000 
Consequently, 1 = 73700, a mean multiplier of thickness and length,, 

which, added to S10 000, the multiplier for % in. in thickness and 25 feet in length, 

23 2 ( 5->9 

*7 S83 700. Hence -^ — — X 8S3 700 = — — X 833 700 = 103. SS lbs. 
nO X 15 450 

* Including English plates. 



492 



STRENGTH OF MATERIALS. 



The following Table exhibits the collapsing pressure of flues, and burst* 
ing pressure of boilers of different diameters and thickness of metal : 

Resistance of Wrought-iron !FTu.es to an External or Col- 
lapsing Pressure, and. of the Shells of Boilers to an In- 
ternal or Bursting Pressure. 

Tensile Resistance of the Plates without Riveting is taken at a Mean of 55 000 

pounds per Square Inch. 

Flues. Shells. 



' 












Bursting 


Pressure 


Diameter. 


. Length. 


Thickness. 


j Collapsing 
Pressure 


Diameter. 


i 

Thickness. 


per Squ 


are Inch. 


per Square 


Single 


Doub'e 








Inch. 






Riveted. 


Riveted. 


Ins. 


Feet. 


Ins. 


Lbs. 


Feet. 


Ins. 


Lbs. 


Lbs. 


6 


10 


2 


417 


2 


•U 


573 


745 


6.5 


10 


.2 


385 


2.6 


•X 


458 


596 


7 


10 


.2 


357 


3 


'X 


382 


496 




10 


-X 


580 


3.4 


•X 


318 


414 


7.5 


10 


.2 


333 




•9i 


398 


518 




10 


•% 


542 


• 3.6 


-X 


327 


426 


8 


10 


.2 


312 




•% 


409 


532 




10 


•X 


508 


4 


} X 


286 


372 


8.5 


10 


.2 


294 




-X, 


358 


465 




10 


,X 


478 


4.6 


.% 


254 


331 


9 


10 


.2 


278 




•E 


318 


413 




10 


•X 


451 


5 


■% 


229 


298 


9.5 


10 


.2 


263 




»x 


286 


372 




10 


-X 


427 


o.6 


•X 


208 


270 


10 


12 


.2 


227 




A 


260 


338 




12 


3% 


354 




•% 


312 


406 




12 


•Ye 


612 


6 


-X 


191 


248 


10.5 


12 


.2 


216 




•& 


239 


311 




12 


.& 


337 




>% 


286 


372 




12 


V 

*/16 


583 


6.6 


.% 


220 


287 


11 


12 


.2 


206 




-% 


264 


344 




12 


•U 


322 


7 


•36 


204 


266 




12 


•X 


557 




<% 


245 


319 


11.5 


12 


.2 


197 


7.6 


.% 


191 


248 




12 


M 


308 




•% 


229 


298 




12 


.n 


532 


8 


•% 


179 


233 


12 


15 


.2 


153 




•% 


215 


279 




15 


•K 


239 


S.6 


X 


168 


219 




15 


.% 


415 




m 


202 


263 


12.5 


15 


% 


229 


9 


\ 


159 


207 




15 


-M 


398 




•% 


191 


248 


13 


15 


-X 


220 


9.6 


.* 


150 


196 




15 


.%, 


384 




.% 


181 


235 


13.5 


15 


• K 


212 


10 


• X 


143 


186 




15 


.& 


369 




-% 


172 


224 


14 


18 


-% 


176 




-H 


229 


298 




18 


•a; 


305 


10.0 


.& 


136 


177 


14.5 


18 


-x 


168 




•% 


163 


212 




18 


JK 


294 




■S 


218 


284 


15 


20 


-X 


157 


11 


•% 


156 


203 




20 


-% 


276 




% 


208 


271 


15.5 


20 


«X 


152 


11.6 


.% 


149 


194 




20 


s 


267 




• l A 


199 


259 


16 1 


20 


-« 


148 


12 


• % 


143 


166 


20 


-An I 


231 


, 


•X 


191 


248 



STRENGTH OF MATERIALS. 



493 



Note.— The single-riveted are estimated at .5 the resistance of the plates, and the 
Btaggered riveted at .65; this reduction from .50 and .7, as determined by Fairbairn, 
is to meet defects of rivets, cracks of plates from the pinning of rivet holes, etc., his 
deductions being taken from experiments made with rivets and plates in a normal 
condition. 

From the results given in the Table and deduced from the rules, such allowances 
for the resistance and wear of the plates , oxydation, etc., are to be made, as the char- 
acter of the metal, the nature of the service, and the circumstance of using fresh or 
salt water, etc., will render necessary. 

In riveted plates, it is customary in practice to estimate the safe tensile resistance 
of the metal of a boiler or tube, when exposed to salt-water, at one fifth of its ulti- 
mate resistance or bursting pressure ; and, when exposed to fresh-water alone, at one 
fourth of it. 



Resistance of tlie 



To Compute the Ultimate Bursting 
Shell of a Boiler. 

Rule. — Double the thickness given or ascertained by a wire gauge ; 
multiply the sum by the tensile resistance of the material as it may be 
constructed, and divide the product b} T the diameter of the boiler or flue in 
inches. 

Example.— The diameter of the shell of a wrought-iron boiler, single riveted, U 
5 feet, and the thickness of the metal is .28 in. ; what is the ultimate resistance to a 
bursting pressure ? 

.28-J-.2SX55000, which X-5 for reduction of resistance of the plates for single 
riveting = 15400, and ^i!^— 256.6 lbs. 

Deductions. — 1. The resistance of Tubes or Flues to an External or Internal Press- 
ure varies directly and inversely as their diameters. 2. The resistance of a Tube or 
Flue to External Pressure, up to the lengths experimented upon, is inversely as its 
length. Consequently, the resistance of tubes or flues to external pressure, of differ- 
ent diameters but of equal lengths, varies inversely as their diameter, and contrari- 
wise. 3. The Tubes or Flues, witli lap-joints, have one third less res stance to ex- 
ternal pressure than when their joints are abutted. 4. A Cylindrical Tube or Flue 
has three times the resistance to external pressure of an Elliptical Tube or Flue, of 
the proportionate diameter given in the experiments noticed. 5. The lengih of Tubes 
or Flues, to resist Internal pressure, has no essential effect. 6. With Tubes or 1 lues 
of like thickness, their resistance varies inversely as the product of their lengths by 
their diameters. 

liesnlts of Experiments upon tlie Resistance of Elliptical 
Fines to External IPressixre or Collapse. 

By comparing the results of Experiments upon Elliptical tube ■ with those upon 
Cylindrical tubes, it appears the preceding general formula will apply approximately 
to elliptical tubes, by substituting for d in that formula the diameter of the circle of 
curvature touching the extremity of the minor axis. Thus : 

Diameter of the circle of curvature (page 4S9, Ex. 4, 19th line from bottom) = 

ZL = ^- = 19.12 ins. 

r 5. 1 25 

The pressure upon this tube was 6.5 lbs., which, reduced to unity of length and di- 
ameters 641.3 (19.12X5.16X6.5). 

Comparison between the Resistance to External and In- 
ternal Pressure in 'W'roiight Iron Single-riveted. Elnes 
of different Diameters and Lengths. 









External 


Internal 




Diameter. 


Thickness. 


Length. 


Pressure per 
Square Inch. 


Pressure per 
Square Inch. 


Ratio. 


Ins. 


Ins. 


Feet. 


Lbs. 


Lbs. 




6 


.15 


10 


205 


1875 


1 to 0.7 


13 


.2 


15 


1(53 


017 


1 to 5.6 


18 


•X 


20 


lo5 


764 


1 to 5.G 



Tt» 



494 



STRENGTH OF MATERIALS. 



Resistance of* Lead. Tubes to Internal Pressure. 



Diameter. 


Length- 


Thickness. 


Pressure of Rupture 
per Square Inch. 


Ins. 

3 
3 


Ins. 

31 


Ins 


Lbs. 

374 

364 



Assume 370 as the mean of the pressure of rupture of lbs. per square 
inch. 

To Compute the Thickness of a Lead Tf*ipe when the Di- 
ameter and. the Pressure in Pounds per Square Inch is 
given . 

Rclm. — Multiply the pressure in pounds per square inch by the diam- 
eter of the pipe in inches, and divide the product by twice the tensile re- 
sistance of the metal in pounds per square inch. 

Example. — The diameter of a lead pipe is 3 inches, and the pressure to which it 
is to be submitted is 370 lbs. per square inch; what should be the thickness of the 



metal ? 



370x3 
2220X2 : 



1110 
"4440" 



..25 in. 



Resistance of* Grlass GJ-lohes and Cylinders to Internal 
Pressure and Collapse. 

Globes (Flint Glass). Cylinder. 
Bursting Pressure. 



Diameter. 


Thickness. 


Per ) 
Square Inch. 


Diameter. 


Length. 


Thickness. 


Per 
Square Inch. 


Ins. 


Ins. 


Lbs. 


Ins, Ins. 


Ins. 


Lbs. 


4 


.024 


84 


4 1 7 


.079 


282 


4 


.038 


150 


Elliptical (C 


rown Glass) 




5 


.022 


90 


4.1 ■ I 7 


.019 


109 


6 


.059 


152 | 


1 










Collapsing Pressure. 






5 


.014 


292 


3 


14 


.014 


85 


4 


.025 


1000* 


4 


7 


.034 


202 


6 


.059 


900* 


4 


14 


.004 


297 



MEMORANDA. 

Repetition of Stress. — A piece of cast iron submitted to transverse stress 
broke at the 1956th strain, with a stress three fourths of that of its orig- 
inal ultimate resistance. 

Resistance to Bursting of Thick Cylinders. — The mean resistance to 
bursting of the chambers of cast-iron guns, from experiments of Major 
Rodman, is as follows : 

Thickness of metal — 1 calibre, length — 3 calibres, 52 217 lbs. per sq. in. 

Thickness of metal = % calibre, length = 3 calibres, 49 100 lbs. per sq. in. 

The tensile strength of the iron being 18 820 lbs. 

Diam. of cylinder 2 ins., length 12 ins., metal 2 ins., 80 229 lbs. per sq. in. 

Diam. of cylinder 3 ins., length 12 ins., metal 3 ins., 93 702 lbs. per sq. in. 

The tensile strength of the iron being 20 866 lbs. 

Average Tensile strength of Gun-metal (cast iron), 37 774 lbs. 

Wire Ttopes. 
The ultimate strength of iron wire ropes is 4480 lbs. for each pound in 
weight per fathom, and for galvanized steel ropes 6720 lbs. 



iron. 495 



IRON. 



The foreign substances which iron contains modify its essential proper- 
ties. Carbon adds to its hardness, but destroys some of its qualities, and 
produces Cast Iron or Steel according to the proportion it contains. Sul- 
phur renders it fusible, difficult to weld, and brittle when heated or "hot 
short." Phosphorus renders it "cold short" but may be present in the 
proportion of Tppo to ic?o o without affecting injuriously its tenacity. An- 
timony, Arsenic, and Copper have the same effect as sulphur, the last in a 
greater degree. 

Cast Iron. 

The process of making cast iron depends much upon the description of 
fuel used; whether charcoal, coke, bituminous or anthracite coals. A 
larger yield from the same furnace, and a great economy in fuel, are ef- 
fected D3 T the use of a hot blast. The greater heat thus produced causes 
the iron to combine with a larger per-centage of foreign substances. 

Cast iron for purposes requiring great strength should be smelted with 
a cold blast. Pig-iron, according to the proportion of carbon which it con- 
tains, is divided into Foundry^ Iron and Forge Iron, the latter adapted only 
to conversion into malleable iron ; while the former, containing the largest 
proportion of carbon, can be used either for castings or bars. 

There are many varieties of cast iron, differing by almost insensible 
shades ; the two principal divisions are gray and white, so termed from 
the color of their fracture. Their properties are very different. 

Gray Iron is softer and less brittle than white iron ; it is in a slight de- 
gree malleable and flexible, and is not sonorous ; it can be easily drilled 
or turned in a lathe, and does not resist the file. It has a brilliant frac- 
ture, of a gray, or sometimes a bluish-gray, color ; the color is lighter as 
the grain becomes closer, and its hardness increases at the same time. It 
melts at a lower heat than white iron, and preserves its fluidity longer. 
The color of the fluid metal is red, and deeper in proportion as the heat is 
lower ; it does not adhere to the ladle ; it fills the molds well, contracts 
less, and contains fewer cavities than white iron ; the edges of its cast- 
ings are sharp, and the surfaces smooth and convex. A medium-sized 
grain, bright gray color, fracture sharp to the touch, and a close, compact 
texture, indicate a good quality of iron. A grain either very large or very 
small, a dull, earthy aspect, loose texture, dissimilar crystals mixed to- 
gether, indicate an inferior quality. 

^Gray iron is used for machinery and ordnance purposes where the 
pieces are to be bored or fitted. Its tenacity and specific gravity are di- 
minished by annealing. Its mean specific gravity is 7.2. 

White Iron is very brittle and sonorous ; it resists the file and the chisel, 
and is susceptible of high polish ; the surface of its castings is concave ; 
the fracture presents a silvery appearance, generally fine-grained and 
compact, sometimes radiating or lamellar. When melted it is white, and 
throws off a great number of sparks, and its qualities are the reverse of 
those of gray iron ; it is, therefore, unsuitable for machinery purposes. Its 
tenacity is increased, and its specific gravity diminished by annealing. Its 
mean specific gravity is 7.5. 

Mottled Iron is a mixture of white and gray ; it has a spotted appear- 
ance ; it flows well, and with few sparks ; its castings have a plane sur- 
face, with edges slightly rounded. It is suitable for shot, shells, etc. 

A fine mottled iron is the only kind suitable for castings which require 
great strength, such as beam centres, cylinders, and cannon. The kind 
of mottle will depend much upon the size of the casting. 

Besides these general divisions, the different varieties of pig-iron are more 
particularly distinguished by numbers, according to their relative hardness. 



496 IRON. 

No. 1 is the softest iron, possessing in the highest degree the qualities 
belonging to gray iron ; it has not much strength, but on account of its 
fluidity when melted, and of its mixing advantageously with old or scrap 
iron and with the harder kinds of cast iron, it is of great use to the found- 
er, and commands the highest price. 

No. 2 is harder, closer grained, and stronger than No. 1 ; it has a grav 
color and considerable lustre. It is the character of iron most suitable 
for shot and shells. 

No. 3 is still harder than No. 2. Its color is gray, but inclining to 
white ; it has considerable strength, but it is principally used for mixing 
with other kinds of iron. 

No. 4 is bright iron ; No. 5, mottled; and No. G, white, which is unfit for 
general use by itself. 

The qualities of these various descriptions depend upon the proportion 
of carbon, and upon the state in which it exists in the metal ; in the dark- 
er kinds of iron, where the proportion is sometimes 7 per cent., it exists 
partly in the state of graphite or plumbago, which makes the iron soft. 
hi white iron, the carbon is thorough^ combined with the metal, as in 
steel. 

Cast iron frequenti}* retains a portion of foreign ingredients from the 
ore, such as earths or oxides of other metals, and sometimes sulphur and 
phosphorus, which are all injurious to its quality. Sulphur hardens the 
iron, and, unless in a very small proportion, destroys its tenacity. 

These foreign substances, and also a portion of the carbon, are separa- 
ted by melting the iron in contact with air, and soft iron is thus rendered 
harder and stronger. The effect of remelting varies with the nature of 
the iron and the character of ore from which it has been extracted ; that 
from the hard ores, such as the magnetic oxides, undergoes less alteration 
than that from the hematites, the latter being sometimes changed from 
No. 1 to white b} T a single remelting in an air furnace. 

The color and texture of cast iron depend greatly upon the volume of 
the casting and the rapidity of its cooling; a small casting, which cools 
quickly, is almost always white, and the surface of large castings par- 
takes more of the qualities of white metal than the inferior. 

All cast iron expands at the moment of becoming solid, and contracts 
in cooling ; gray iron expands more and contracts less than other iron. 

The contraction is about T Jo for £ ra 3 T and strongly-mottled iron, or % 
of an inch per foot. 

Remelting iron improves its tenacity ; thus, a mean of 14 cases for two 
fusions gave, for 1st fusion, a tenacity of 29 284 lbs. ; for 2d fusion, 33 790 
lbs. For 2 cases— for 1st fusion, 15 129 lbs. ; for 2d fusion, 35 786 lbs. 

"Wrcmglit Iron. 

Wrought iron is made from the pig-iron in a Bloomery Fire or in a Pud- 
dliruj Furnace— generally in the latter. The process consists in melting it 
and keeping it exposed to a great heat, constantly stirring the mass, bring- 
ing every part of it under the action of the flame until it loses its remain- 
ing carbon, when it becomes malleable iron. When, however, it is de- 
sired to obtain iron of the best quality, the pig-iron should be refined. 

Refining.— This operation deprives the iron of a considerable portion of 
its carbon ; it is effected in a Blast Furnace, where the iron is melted by 
means of charcoal or coke, and exposed for some time to the action of a 
great beat; the metal is then run into a cast-iron mold, by which it is 
formed into a large broad plate,. As soon as the surface of the plate is 
chilled, cold water is poured on to render it brittle. 

The Bloomery resembles a large forge lire, where charcoal and a strong 



STEEL. 497 

blast are used ; and the refined metal or the pig-iron, after being broken 
into pieces of the proper size, is placed before the blast, directly in contact 
with charcoal; as the metal fuses, it falls into a cavity left for that pur- 
pose below the blast, where the bloomer works it into the shape of a bally 
which he places again before the blast, with fresh charcoal ; this operation 
is general!}' again repeated, when the ball is ready for the Shingler. 

The Puddling Furnace is a reverberator}- furnace, where the flame of bi- 
tuminous coal is brought to act directly upon the metal. The metal is first 
melted ; the puddler then stirs it, exposing each portion in turn to the ac- 
tion of the flame, and continues this as long as he is able to work it. 
When it has lost its fluidit}-, he forms it into balls, weighing from 80 to 
100 lbs., which are next passed to the shingler. 

Shingling is performed in a strong squeezer or under the trip-hammer. 
Its object is to press out as perfectly as practicable the liquid cinder 
which the ball still contains ; it also forms the ball into shape for the 
puddle rolls. A heavy hammer, weighing from 6 to .7 tons, effects this 
object most thoroughly, but not so cheaply as the squeezer. The ball re- 
ceives from 15 to 20 blows of a hammer, being turned from time to time 
as required : it is now termed a Bloom, and is" read j* to be rolled or ham- 
mered ; or the ball is passed once through the squeezer, and is still hot 
enough to be passed through the puddle rolls. 

Puddle Rolls. — By passing through different grooves in these rolls, the 
bloom is reduced to a rough bar from three to four feet in length, its name 
conveying an idea of its condition, which is rough and imperfect. 

Piling. — To prepare rough bars for this operation, they are cut, by a 
pair of shears, into such lengths as are best adapted to the size of the fin- 
ished bar required ; the sheared bars are then piled one over the other, 
according to the volume required, when the pile is ready for balling. 

Balling. — This operation is performed in the balling furnace, which is 
similar to the puddling furnace, except that its bottom or hearth is made 
up, from time to time, with sand ; it is used to give a welding-heat to the 
piles to prepare them for rolling. 

Finishing Rolls.— The balls are passed successively between rollers of 
various forms and dimensions, according to the shape of the finished bar 
required. 

The quality of the iron depends upon the description of pig-iron used, 
the skill of the puddler, and the absence of deleterious substances in the 
furnace. 

The strongest cast irons do not produce the strongest malleable iron. 

For many purposes, such as sheets for tinning, best boiler-plates, and 
bars for converting into steel, charcoal iron is used exclusively ; and, gen- 
erally, this kind of iron is to be relied upon, for strength and toughness, 
with greater confidence than any other, though iron of superior quality is 
made from pigs made with other fuel, and with a hot blast. Iron for gun- 
barrels has been lately made from anthracite hot-blast pigs. 

Iron is improved in quality by judicious working, reheating it, and 
hammering or rolling: other things being equal, the best iron is that 
which has been wrought the most. 



STEEL. 

Steel is a compound of Iron and Carbon, in which the proportion of the 
latter is from 1 to 5 per cent., and even less in some kinds. Steel is dis- 
tinguished from iron by its fine grain, and by the action of diluted nitric 
acid, which leaves a black spot upon steel, and upon iron a spot which is 
lighter colored in proportion to the carbon it contains. 



49 S STEEL. 

There are many varieties of steel, the principal of which are : 

Natural Steel, obtained by reducing rich and pure descriptions of iron 
ore with charcoal, and refining the cast iron, so as to deprive it of a suf- 
ficient portion of carbon to bring it to a malleable state. It is used for 
files and other tools. 

Indian steel, termed Wootz, is said to be a natural steel, containing a 
small portion of other metals. 

Blistered Steel, or Steel of Cementation, is prepared by the direct corn- 
bination of iron and carbon. For this purpose, the iron in bars is put in 
layers, alternating with powdered charcoal, in a close furnace, and ex- 
posed for seven or eight days to a heat of about 9000°, and then put to cool 
for a like period. The bars, on being taken out, are covered with blisters, 
have acquired a brittle quality, and exhibit in the fracture a uniform crys- 
talline appearance. The degree of carbonization is varied according* to 
the purposes for which the steel is intended, and the best qualities of iron 
(Russian and Swedish) are used for the finest kinds of steel. 

Tilted Steel is made from blistered steel moderately heated, and subject* 
ed to the action of a tilt hammer, by which means its tenacity and density 
are increased. 

Shear Steel is made from blistered or natural steel, refined by piling thin 
bars into fagots, which are brought lo a welding heat in a reverberatory 
furnace, and hammered or rolled again into bars ; this operation is repeat- 
ed several times to produce the finest kinds of shear steel, which are dis- 
tinguished by the names of half shear, single shear, and double shear, or 
steel of 1, 2, or 3 marks, etc., according to the number of times it has been 
piled. 

Cast Steel is made by breaking blistered steel into small pieces and melt- 
ing it in close crucibles, from which it is poured into iron molds ; the ingot 
is then reduced to a bar bj^ hammering or rolling. Cast steel is the best 
kind of steel, and best adapted for most purposes ; it is known by a very 
fine, even, and close grain, and a silvery, homogeneous fracture; it w 
very brittle, and acquires extreme hardness, but is difficult to weld with- 
out the use of a flux. The other kinds of steel have a similar appearance 
to cast steel, but the grain is coarser and less homogeneous ; they are 
softer and less brittle, and weld more readily. A fibrous or lamellar ap- 
pearance in the fracture indicates an imperfect steel. A material of great 
toughness and elasticity, as well as hardness, is made by forging together 
steel and iron, forming the celebrated damasked Steel, which is used for 
sword-blades, springs, etc. ; the damask appearance of which is produced 
by a diluted acid, which gives a black tint to the steel, while the iron re- 
mains white. 

VariousyVmc?/ steels, or alloys of steel with silver, platinum, rhodium, and 
aluminum, have been made with a view to imitating the Damascus steel, 
wootz, etc., and improving the fabrication of some of the finer kinds of 
surgical and other instruments. 

Properties of Steel. — After being tempered it is not easily broken; it 
Melds readily ; it does not crack or split ; it bears a very high heat, and 
preserves the capability of hardening after repeated working. 

Hardening and Tempering. — Upon these operations the quality of man. 
ufactured steel in a great measure depends. 

Hardening is effected by heating the steel to a cherry-red, or until the 
scales of oxide are loosened on the surface, and plunging it into a liquid, 
or placing it in contact with some cooling substance ; the degree of hard- 
ness depends upon the heat and the rapidity of cooling. Steel is thus ren- 
dered so hard as to resist the hardest files, and it becomes at the same time 
extremely brittle. The degree of heat, and the temperature and nature 
of the cooling medium, must be chosen with reference to the quality of the 



LIMES, CEMENTS, MORTARS, AND CONCRETES. 499 

steel and the purpose for which it is intended. Cold water gives a greater 
hardness than oils or other'fatty substances, sand, wet-iron scales, or cin- 
ders, but an inferior degree of hardness to that given by acids. Oil, tal- 
low, etc., prevent the cracks which are caused by too rapid cooling. The 
lower the heat at which the steel becomes, hard, the better. 

Tempering. — Steel in its hardest state being too brittle for most pur- 

Eoses, the requisite strength and elasticity are obtained b} T tempering — or 
Ming doicn the temper, as it is termed — which is performed by heating the 
hardened steel to a certain degree and cooling it quickly. The requisite 
heat is usually ascertained by the color which the surface of the steel as- 
sumes from the film of oxide thus formed. The degrees of heat to which 
these several colors correspond are as follows : 

At 430°, a very faint yellow (Suitable for hard instruments ; as hammer- faces. 
At 45;)°, a pale straw color. \ drills, etc. 

At 470°, a full yellow (For instruments requiring hard edges without elastici- 

At 490°, a brown color ( ty ; as shears, scissors, turning tools, etc. 

At 510°, brown, with purple (^ ^ for cutting wQod and ^ metalg; ^ ag 

Ats^p^pie.;:;;;;;;;;;) piane-inm* knives, etc. 

At 550°, dark blue (For tools requiring strong edges without extreme 

At 500°, full blue ( hardness ; as cold-chisels, axes, cutlery, etc. 

At 000°, grayish-blue, verg- (For spring-temper, which will bend before break- 
ing on black \ ing ; as saws, sword-blades, etc. 

If the steel is heated higher than this, the effect of the hardening proc- 
ess is destroj*ed. 

Case-riarderLirig. 

This operation consists in converting the surface of wrought iron into 
steel, by cementation, for the purpose of adapting it to receive a polish or 
to bear friction, etc. ; this is effected by heating iron to a cherry-red, in a 
close vessel, in contact with carbonaceous materials, and then plunging it 
into cold water. Bones, leather, hoofs and horns of animals are general- 
ly used for this purpose, after having been burned or roasted so that they 
can be pulverized. Soot is also frequently used. 



LIMES, CEMENTS, MORTARS, AND CONCRETES. 

Limestones. 

The calcination of marble or any pure limestone produces lime (quick- 
lime). The pure limestones burn white, and give the richest limes. 

The finest calcareous minerals are the rhombohedral prisms of calca- 
reous spar, the transparent double-reflecting Iceland spar, and white or 
statuary marble. 

The property of hardening under water, or when excluded from air, 
conferred upon a paste of lime, is effected by the presence of foreign sub- 
stances — as silicum, alumina, iron, etc. — when their aggregate presence 
amounts to ^o 0I * tne whole. 

Limes are classed: 1. The common or fat limes. 2. The poor or mea- 
gre. 3. The hydraulic. 4. The hydraulic cements. 5. The natural puz- 
zuolanas, including puzzuolana properly so called, trass or terras, the 
arenes, ochreous earths, basaltic sands, and a variety of similar substances. 

Rich Limes are fnlly dissolved in water frequently renewed, and they 
remain a long time without hardening; they also increase greatly in vol- 
ume, from 2 to 3% times their original bulks, and will not harden with- 
out the action of the air. They are rendered hydraulic by the admixture 
of puzzuolana or trass. 



500 LIMES, CEMENTS, MORTARS, AND CONCRETES. 

Rich, fat, or common Limes usually contain less than 10 per cent, of inu 
purities. 

Hydraulic Limestones are those which contain iron and clay, so as to 
enable them to produce cements which become solid when under water. 

The pastes of fat limes shrink, in hardening, to such a degree that they 
can not be used as mortar without a large dose of sand. 

Poor Limes have all the defects of rich limes, and increase but slightly 
in bulk. 

The poorer limes are invariably the basis of the most rapidly-setting 
and most durable cements and mortars, and they are also the only limes 
which have the property, when in combination with silica, etc., of indu- 
rating under water, and are therefore applicable for the admixture of hy- 
draulic cements or mortars. Alike to rich limes, they will not harden if 
in a state of paste under water or in wet soil, or if excluded from contact 
with the atmosphere or carbonic acid gas. They should be employed for 
mortar only when it is impracticable to procure common or hydraulic lime 
or cement, in which case it is recommended to reduce them to powder by 
grinding. 

Lime absorbs, in slaking, a mean of 2^ times its volume, and 2% timea 
its weight of water. 

Hydraulic Limes are those which readily harden under water. The 
most valuable or eminently hydraulic set from the 2d to the 4th dav after 
immersion ; at the end of a month they become hard and insoluble, and 
at the end of 6 months they are capable of being worked like the hard, 
natural limestones. They absorb less water than the pure limes, and only 
increase in bulk from 1% to 2)^ times their original volume. 

The inferior grades, or moderately hydraulic, require a longer period, say from 15 
to 20 days' immersion, and continue to harden for a period of 6 months. 

The resistance of hydraulic limes increase if sand is mixed in the proportion of 50 
to ISO per cent, of the part in volume ; from thence it decreases. 

Slaked Lime is a hydrate of lime. 

M. Vicat declares that lime is rendered hydraulic by the admixture with it of frcm 
33 to 40 per cent, of clay and silica, and that a lime is obtained which does not slake, 
and which quickly sets under water. 

Artificial Hydraulic Limes do not attain, even under favorable circum- 
stances, the same degree of hardness and power of resistance to compres- 
sion as the natural limes of the same class. 

The close-grained and densest limestones furnish the best limes. 

Hydraulic limes lose or depreciate in value b}- exposure to the air. 

Arenes is a species of ochreous sand. It is found in France. On ac- 
count of the large proportion of clay it contains, sometimes as great as 
&, it can be made into a paste with water without any addition of lime ; 
hence it is sometimes used in that state for walls constructed en pise, as 
well as for mortar. Mixed with rich lime, it gives excellent mortar, which 
attains great hardness under water, and possesses great hydraulic energy. 

Puzzuolana is of volcanic origin. It comprises trass or terras, the ardnes, 
some of the ochreous earths, and the sand of certain gravwackes, granites, 
schists, and basalts ; their principal elements are silica' and alumina, the 
former preponderating. None contain more than 10 per cent, of lime. 

When finely pulverized, without previous calcination, and combined with the paste 
of fat lime in proportions suitable to supply its deficiency in that element, it pos- 
sesses hydraulic energy to a valuable degree. It is used in combination with rich 
lime, and may be made by slightly calcining clay and driving off the water of com. 
bination at a temperature of 1200°. 

Brick or Tile Dust combined with rich lime possesses hydraulic energy, 



LIMES, CEMENTS, MORTARS, AND CONCRETES. 501 

Trass or Terras is a blue-black trap, and is also of volcanic origin. It 
requires to be pulverized and combined with rich lime to render it fit for 
use, and to develop any of its hydraulic properties. 

General Gillmore* designates the varieties of hydraulic limes as follows : 
If, after being slaked, they harden under water in periods varying from 15 
to 20 daj r s after immersion, slightly hydraulic ; if from six to eight days, 
hydraulic ; and if from one to four days, eminently hydraulic. 

Pulverized silica burned with rich lime produces hydraulic lime of ex- 
cellent qualit} T . Hydraulic limes are injured by air-slaking in a ratio 
varying directly with their hydra ulicity, and they deteriorate by age. 

For foundations in a damp soil or exposure, hydraulic limes must be 
exclusively employed. 

Cements. 

Hydraulic Cements contain a larger proportion of silica, alumina, mag- 
nesia, etc., than any of the preceding varieties of lime ; they do not slake 
after calcination, and are superior to the very best of hydraulic limes, as 
some of them set under water at a moderate temperature (65°) in from 3 
to 4 minutes ; others require as many hours. They do not shrink in hard- 
ening, and make an excellent mortar without any admixture of sand. 

Roman Cement is made from a lime of a peculiar character, found in En- 
gland and France, derived from argillo-calcareous kidney-shaped stones 
termed " Septaria." 

Rosendale Cement is from Rosendale, New York. 

Portland Cement is made in England and France. It requires less wa- 
ter than the Roman cement, sets slowly, and can be remixed with addi- 
tional water after an interval of 12 or even 24 hours from its first mixture. 

The property of setting slow may be an obstacle to the use of some designations of 
this cement, as the Boulogne, when required for localities having to contend against 
immediate causes of destruction, as in sea constructions having to be executed un- 
der water and between tides. On the other hand, a quick-setting cement is always 
difficult of use ; it requires special workmen and an active supervision. A slow-set- 
ting cement, however, like the natural Portland, possesses the advantage of being 
managed by ordinary workmen, and it can be remixed with additional water after 12 
or even 24 hours. 

Artificial Cement is made by a combination of slaked lime with un- 
burned clay in suitable proportions. 
Artificial Puzzuolana is made by subjecting clay to a slight calcination. 
Salt-water has a tendency to decompose cements of all kinds. 

IVIortars. 

Lime or Cement paste is the cementing substance in mortar, and its pro- 
portion should be determined by the rule that the volume of the cementing 
substance should be somewhat in excess of the volume of voids or spaces in the 
sand or coarse material to be united, the excess being added to meet imper- 
fect manipulation of the mass. 

Hydraulic Mortar, if re-pulverized and formed into a paste after hav- 
ing once set, immediately loses a great portion of its hydraulicity, and de- 
scends to the level of the moderate hydraulic limes. 

All mortars are much improved by being worked or manipulated ; and as rich limes 
gain somewhat by exposure to the air, it is advisable to work mortar in large quan- 
tities, and then render it fit for use by a second manipulation. 

For an Analysis of Limestones, etc., etc., see Gen. Gilmore's Treatise, p. 22, 125. 

White lime will take a larger proportion of sand than brown lime. 

The use of salt-water in the composition of mortar injures the adhesion of it. 

* See hie Treatises on Limes, Hydraulic Cements, and Mortars, of Papers on Practical Engineer 
ing, Engineer Department, U. S. A. 

Uu 



502 LIMES, CEMENTS, MORTARS, AND CONCRETES. 

Mortar. — When a small quantity' of water is mixed with slaked lime, a 
stiff paste is made, which, upon becoming dry or hard, has but very little 
tenacity, but, by being mixed with sand or like substances, it acquires 
the properties of a cement or mortar. 

The proportion of sand that can be incorporated with mortar depends 
partly upon the degree of fineness of the sand itself, and partly upon the 
character of the lime. For the rich limes, the resistance is increased if 
the sand is in proportions vailing from 50 to 240 per cent, of the paste in 
volume ; be3'ond this proportion the resistance decreases. , 

Stone Mortar. — 8 parts cement, 3 parts lime, and 31 parts of sand. 

Brick Mortar. — 8 parts cement, 3 parts lime, and 27 parts of sand. 

Brown Mortar. — Lime 1 part, sand 2 parts, and a small quantity of hair; 

Lime and sand, and cement and sand, lessen about % in volume when mixed to- 
gether. 

Calcareous Mortar, being composed of one or more of the varieties of 
lime or cement, natural or artificial, mixed with sand, will vary in its 
properties with the quality of the lime or cement used, the nature and 
quality of sand, and the method of manipulation. 

Mortar. — Lime, 1 ; clean sharp sand, 2)^. An excess of water in slak- 
ing the lime swells the mortar, which remains light and porous, or shrinks 
in drying; an excess of sand destroys the cohesive properties of the mass. 

It is indispensable that the sand should be sharp and clean. 
Turkish Plaster, or Hydraulic Cement. — 100 lbs. fresh lime reduced to 
powder, 10 quarts linseed-oil, and 1 to 2 ounces cotton. Manipulate the 
lime, gradually mixing the oil and cotton, in a wooden vessel, until the 
mixture becomes of the consistenc}^ of bread-dough. 

Dry, and, when required for use, mix with linseed-oil to the consistency of paste, 
and then lay on in coats. Water-pipes of clay or metal, joined or coated with it, re- 
sist the effect of humidity for very long periods. 

Exterior Plaster or Stucco. — 1 volume of cement powder to 2 volumes 
of dn- sand. 

In India, to the water for mixing the plaster is added 1 lb. of sugar, or 
molasses, to 8 Imperial gallons of water, for the first coat; and for the 
second or finishing, 1 lb. sugar to 2 gallons water. 

Powdered slaked lime and Smith's forge scales, mixed with blood in 
suitable proportions, make a moderate hydraulic mortar, which adheres 
well to masonry previously coated with boiled oil. 

The plaster should be applied in two coats laid on in one operation, the first coat 
being thinner than the second. The second coat is applied upon the first while the 
latter is yet soft. 

The two coats should form one of about 1)4 inches in thickness, and when finished 
it should be kept moist for several days. 

This process may be modified by substituting for the first coat a wash of thick 
cream of pure cement, applied with a stiff brush just before the plaster is laid on. 

When the cement is of too dark a color for the desired shade, it may be mixed 
with white sand in whole or in part, or lime paste may be added until its volume 
equals that of the cement paste. 

Khorassar, or Turkish Mortar, used for the construction of buildings 
requiring great solidity, % powdered brick and tiles, % fine sifted lime. 
Mix with water to the" required consistency, and lay on layers of 5 and 6 
inches in thickness between the courses of brick or stones. 

Interior Plastering. — The mortars used for inside plastering are termed 
Coarse, Fine, Gauge or hard finish, and Stucco. 

Coarse, Stuff.— Common lime mortar, as made for brick masonry, with 
a small quantity of hair ; or by volumes, lime paste (30 lbs. lime) 1 part, 
sand 2 to 2J£ parts, hair % P art - 



LIMES, CEMENTS, MORTALS, AND CONCRETES. 503 

When full time for hardening can riot be allowed, substitute from 15 to 20 pc? 
cent, of the lime by an equal proportion of hydraulic cement. 

For the second or brown coat the proportion of hair may be slightly diminished. 
' Fine Stuff" (lime putty). — Lump lime slaked to a paste with a moderate 
volume of water, and afterward diluted to the consistency of cream, and 
then to harden by evaporation to the required consistency for working. 

In this state it is used for a slipped coat, and when mixed with sand or plaster 
of Paris, it is used for the finishing coat. 

Gauge Stuff, or Hard finish, is composed of from 3 to 4 volumes fine stuff 
and 1 volume plaster of Paris, in proportions regulated by the degree of 
rapidity required in hardening; for cornices, etc., the proportions are 
equal volumes of each, fine stuff and plaster. 

Stucco is composed of from 3 to 4 volumes of white sand, to 1 volume of 
fine stuff, or lime putty. 

. Scratch Coat. — The first of three coats when laid upon laths, and is 
from }£ to % of an inch in thickness. 

r One-coat Work. — Plastering in one coat without finish, either on mason- 
ry or laths — that is, rendered or laid. 

Two-coat Work. — Plastering in two coats is done either in a laying coat 
and set, or in a screed coat and set. 

The Screed coat is also termed a Floated coat. Laying the first coat in 
two-coat work is resorted to in common work instead of screeding, when the 
finished surface is not required to be exact to a straight-edge. It is laid 
in a coat of about % an inch in thickness. 

The laying coat, except for very common work, should be hand-floated. 

The firmness and tenacity of plastering is very much increased by hand-floating. 

Screeds are strips of mortar 6 to 8 inches in width, and of the required 
thickness of the first coat, applied to the angles of a room, or edge of a 
wall and parallelly, at intervals of 3 to 5 feet over the surface to be cover- 
ed. When these have become sufficiently hard to withstand the pressure 
of a straight-edge, the inter-spaces between the screeds should be filled out 
flush with them, so as to produce a continuous and straight, even surface. 

Slipped Coat is the smoothing off of a brown coat with a small quantity 
of lime putty, mixed with 3 per cent, of white sand, so as to make a com- 
paratively even surface. 

This finish answers when the surface is to be finished in distemper, or paper. 

Hard Finish. — Fine stuff applied with a trowel to the depth of about y% 
of an inch. 

Estimate of ]M!aterials and. H<al3or for lOO Square Yards 
of I^ath and IP 1 aster. 



Materials 
and Labor. 


Three Coats 
Hard Finish. 


Two Coats 
Slipped. 


Materials 
and Labor. 


Three Coats 
Hard Finish. 


Two Coats 
Slipped. 


Lime 


4 casks. 

% " 

X " 

'2000. 

4 bushels. 

7 loads. 


%% casks. 

200:). 

3 bushels. 
6 loads. 


White sand.. . 


2U bushels 
13 lbs. 
4 days. 
H " 
1 " 




Lump lime 

Plaster of Paris 


13 lbs. 


Masons 

Laborer 

Cartage 


oX days. 

2 l - 


Hair 


% " 


Sand 





Concrete or Beton 
Is a mixture of mortar (generally h3'draulic) with coarse materials, as 
gravel, pebbles, stones, shells, broken bricks, etc. Two or more of these 
materials, or all of them, may be used together. As lime or cement paste 
is the cementing substance in mortar, so is mortar the cementing sub- 
stance in concrete or beton. The original distinction between cement and 
beton was, that the latter possessed hydraulic energy, while the former did 
not. 



504 LIMES, CEMENTS, MORTARS, AND CONCRETES. 



Hydraulic. — \% parts unslacked hydraulic lime, 1% parts sand, 1 part 
gravel, and 2 parts of a hard broken limestone. 

This mass contracts one fifth in volume. Fat lime may be mixed with concrete, 
without serious prejudice to its hydraulic energy. 

"Varioias Compositions of Concrete. -Forts P&ichmond 
and Tompkins, TJ. S. 

Hydraulic— 308 lbs. cement = 3.65 to 3.7 cubic feet of stiff paste. 12 
cubic feet of loose sand — 9.75 cubic feet of dense. 

For Superstructure. — 11.75 cubic feet of mortar as above, and 16 cubic 
feet of stone fragments. 

In the foundations of Fort Tompkins, about ^ of its volume was com- 
posed of stones from ~% to % of a cubic foot in volume, rammed into the 
wall as the concrete was laid. 

Sea Wall. — Boston Harbor. — Hydraulic. — 308 lbs. cement, 8 cubic feet 
of sand, and 30 cubic feet of gravel. The whole producing 32.3 cubic feet. 

Superstructure. — 308 lbs. cement, 80 lbs. lime, and 14.6 cubic feet dense 
sands. The whole producing 12.825 cubic feet. 

Cost of labor and materials expended in laying concrete foundation at 
Fort Tompkins, during the year 1849, per cubic yard as laid, $2.26. 

Transverse Strengtli of Concretes, Cements, Mortars, 
Puzznolana, and. Trass, dednced from the Experiments 
of GJ-enerals Totten and Grillmore, TJ. S. -A-., G-eneral 
Treussart, and 3V£,"Voisin. 

Reduced to a uniform Measure of One Inch Square and One Foot in Length. Sup- 
ported at both Ends. 

^ -T-j- t 2 = y per square inch of section, representing value f on* general use, 
being % of ultimate breaking strain. 

Experiments op Voisin, 1857. 



Mortar. 






Concrete. 




Mortar. 






Concrete. 




One Volume 


s 


One Volume 


Value. 


One Volume 


1 


One Volume 


Value. 


of Sand. 


1 


of Pebbles. 


of Sand. 


t3 
o 

m 

s 


of Pebbles. 


c 






T3 
S 3 


1 


a 


c 


1 
i 


Cj 


-a 




8J 


S 

6 


as 


a 
> 


o 


313 
o g 


Q 

o 




o 


1 

6 


> 


o 
5 


o £ 
> & 


Q 

o 


Q 

o 
to 












Lbs. 


Lbs. 












Lbs. 


Lbs. 


1 


.62 


1.69 


1 


1.56 


2.3 


2.9 


% 


.38 


1.12 


Y> 


1.03 


.58 


1.2 








% 


1.03 


1.7 


3.2 


X 


.35 


1.05 


1 


1.4 


.48 


1. 








H 


1. 


1.8 


3.1 








% 


1.01 


.35 


.85 








M 


1. 


1. 


1. 


i 


.34 


1. 


1 


1.45 


.3 


.83 






M 


.43 


1.24 


i 


1.45 


1.6 


2.7 








% 


1.03 


.44 


.65 








V? 


11 


1. 


1.9 


V 


.32 


.96 


1 


1.45 


.41 


.81 


'A 


.38 


1.12 


i 


1.4 


.86 


.91 








% 


1.03 


.36 


.79 



Experiments of General Totten, 1837. 







Mortar. 




Concrete.* 




Mortar. 




Concrete.* 


Cement 1. 


Cement 1. 
Sand .5 


Cement 1. 
Sand 1. 


Cement 1. 


Cement 1. 
Sand .5. 


Cement 1, 
Sand 1. 


Granite . \\ 
Mortar . . if 
Gravel . . 1\ 
Mortar . . 2j 


Lbs. 
2.9 

1.4 


Lbs. 
2.4 

2.4 


Lbs. 
2.3 

.7 


Stone ) 

Gravel f 

Hrick \ 

Gravel | 


Lbs. 
1.9 

.9 


Lbs. 
1. 

1.4 


Lbs. 
.6 

1.6 



* The granite, bricks, etc., were broken into fragments or spalls of the required size, 



LIMES, CEMENTS, MORTARS, AND CONCRETES. 505 



Tensile Strength ofvarious Cements, Mortars, and ]VIa- 
sonry, deduced from the Experiments of A^icat and 
Chatoney at Cherbourg, Gren.Grillmore,U. S. A.., Crys- 
tal Palace, London, etc. 



Weight or Power required to Tear asunder One Square Inch. 

Ultimate 



Materials and Mixtures. 



Resist- 
ance. 



Materials and Mixtures. 



Ultimate 
Resist- 
ance. 



Boulogne, 100 parts, water 50 . . . 

90 days, 100 parts, water 50 

Boulogne, 1 year, Portland (nat- 
ural) 

English, 1 year, Portland (arti- 
ficial) 

Portland, 42 days, cement 1, sand 1 

44 15 " 

44 135 " 

u English, 320 days, pure. 

fct " " cement 1, 

sand 1, 



Lbs. 
112 
52 

■ 675 

• 462 

142 

134 

233 

1152 

■ 948 



Portland, English, 320 days, ce- 
ment 1, sand 2 

u 45 days, pure and mixed, 

stiff 

u English, pure, 1 month.. 

" u 44 6 mos. . . . 

Roman, 1 year, from Septaria. . . 

" 42 days, cement 1. sand 1 

" " " 1, " 2 

U « u 1? u 3 

Stone masonry, Roman cement, 

5 mos 



Brick and Granite Masonry, 320 Days. 



Cement, Delafield and Baxter . 



" Lawrence Co 



f Pure 

Cement 4) 

Sand 1/ 

Cement 5) 

Siftings 1/ 

Cement 1 ) 

Siftings . .. 2/ 



James River 






j Pure. 
1 Pure. 



Cement 4) 

Sand l] 

( Pure 

" Newark Lime and Cement Co •< Cement 1 ) 

(Sand 2/ 

u Brighton and Ro3endale Pure 

u Newark and Rosendale Pure 

u Pure upon bricks 

" 1, sand 1 pure upon bricks 

" 1, " 3 " u 

44 Pure upon granite. . . . . 

u 1, water .5 

u 1, 4t .42.. ,,. 

44 Pure upon bricks, without mortar, mean 

Common lime 1 ) u 

sand <ly 2 ) 



siS , d pa . 3 . te :;.::::::::'.'J}«p- t -'-- 
1} 



Lime paste 
Sand 

Lime paste 1) 

Sand 3 V 

Cement pa^te 5) 



Lbs. 

. 713 

• 203 

303 
424 

191 

284 
199 

1GG 

77 



} * 



Lbs 

6S.5G 

63.5 

79.S7 ' 

74.5 

87.37 
53.08 

62. 

C3.25 

39.62 

8025 
75.81 
31. 
16. 
7, 
27. 
20. 
27. 
45. 



4.13 



11.41 



Crushing Strength ofCements, Stone, etc.— {Crystal Tatace, London.] 
Reduced to a uniform Measure of One Square Inch. 



Material. 


Ultimate 
Pressure. 


Material. 


Ultimate 
Pressure. 


Portland cement, area 1, height 1 
44 cement i 
44 sand j 


Lbs. 
16S0 

1244 


Portland cement 1 ) 

44 Hand 4 J 

Roman cement, pure 


Lbs. 

1244 

342 



Uu* 



506 LIMES, CEMENTS, MORTARS, AND CONCRETES. 



Experiments of General Gillmore, 



Cements 
and Mixtures. 



[Value. : 



Cements 
and Mixtures. 



Pure. 



D ^ e ",... and j- Stiff paste. 

High Falls (NJ 
V.), 270 days .j 

James River.... ^ n 7 nt :;;:: ;l 

( Cement ... 4. \ 

James River, 59 \ Water 2.6 J 

days ) Cement. . . 4. \ 

I Water. ... 1.4) 

Portland (Eng.), ( Purecement 



320 days . 



7 < Cement 
* ( Sand . . . 



m 



Lbs. 
6.* 

11.3 

5.9 

1.9 

3.4* 

10.6 

6.6 



Portland Pure 
( Eng. ), 100 
days . . 



Cement 

Sand 

Cement 

Sand 

Roman (Eng.), /Cement 

100 days (Sand 

!Pure 
Cement....' 
Lime 

Rosendale (Hoff- (Stiff paste . . 
man), 320 days (Thin •' . . . 



Lbs. 
12.5 
13. 

S.5 

4. 
7. 
6.7 

3.9 

4.4* 

4.S* 





Cement. 




Value 




Cement. 




Value 






A 

3 


ll 


C S« 

s s 


£ 


a -- 
E a 
Cx. 


w«3 


Akron, ! 
Brightor 
Cumber 
James P 
Newark 
Portlanc 
Remingt 


\ T ew York 

l and Rosendale 

hnd,Md 

iver, Va 

and Rosendale. 

1, English 

on, Conn 


5.2 
4.9 
6.5 

5.8 
10.5 
6.5 


Lbs. 
4.4 

3.8 
6.3 
4.2 
3.S 
8.6 
4.8 


Lbs. 
4.1 
3.4 
3.8 
4.4 
3.4 
6.5 
3.4 


Round Top, Md 

Rosendale, Hoffman. . . 
u Lawrence . . 

Sandusky, Ohio 

Shepherdstown, Va 

Utica,Ill 


5.S 
5.3 

3.S 
51 
5.1 


Lbs. 
4.1 
4.1 

3.2 
4.2 
1.2 


Lbs. 

3Tl 
3.8 



Note. — When the paste is not subjected to compression during setting, a thin 
paste produces as strong a mortar as a stiff one. 

Experiments of General Treussart. 

Puzzuolana and Trass — Mortar. 



Strasburgh 



'Puzzuolana 1 
k Sand 
) Trass 

\Lime 1 1 

rSand 1 >4 

^Puzzuolana l) 



'\\ 



5 days 



Value. 
Lbs. 

2.S 
3.4 


Puzzuolana and Trass — Mortar. 


Value. 


("Lime paste. 1 \ ^ , 
Stras- ] Puzzuolana 2^) ° ys 
burgh. . 1 Lime paste. 1 ) Q 4 

f^Trass 2 j b 

^ ite isand 6 1 U i 

Marble. ^ a r n s d g ;;;;;; 1 1 | 5 


Lbs. 
3.S 

3.1 
2.1 



Cement paste, 95 days 13.8 

u 1, lime paste % 13 6 

" 1, H X H.3 

" 1, " 1 7.9 



Cement paste X* lime paste 1 4.2 

Fire-brick beamt 2.1 

Portland cement, 4 mos 21.3 

Roman u 4 " 14.8 



Deductions. — 1. Particles of unground cement exceeding ^L. of an inch in diam- 
eter may be allowed in cement paste without sand, to the extent of 50 per cent, of 
the whole, without detriment to its properties, while a corresponding proportion of 
sand injures the strength of mortar about 40 per cent. 

2. When these unground particles exist in cement paste to the extent of 66 per 
cent, of the whole, the adhesive strength is diminished about 28 per cent. For a 
corresponding proportion of sand the diminution is 6S per cent. 

3. The addition of sittings exercises a less injurious effect upon the cohesive than 
upon the adhesive property of cement. The converse is true when sand, instead of 
sittings, is used. 

* All except the first %vere submitted to a pressure of 32 lbs. per square inch. 
f Loaded partly along the bricks, and broke through them. 



LIMES, CEMENTS, MORTARS, AND CONCRETES. 507 

4. In all the mixtures with sittings, even when the latter amounted to 66 per cent, 
of the whole, the cohesive strength of the mortars exceeded its adhesion to the bricks. 
The same results appear to exist when the sittings are replaced by sand, until the 
volume of the latter exceeds 20 per cent, of the whole, after which the adhesion ex- 
ceeds the cohesion. 

5. At the age of 320 days (and perhaps considerably within that period) the co- 
hesive strength of pure cement mortar exceeds that of Croton front bricks. The 
converse is true when the mortar contains 50 per cent, or more of sand. 

6. When cement is to be used without sand, as may be the case when grouting is 
resorted to, or when old walls are to be repaired by injections of thin paste, there is 
no advantage in having it ground to an impalpable powder. 

7. For economy it is customary to add lime to cement mortars, and this may be 
done to a considerable extent when in positions where hydraulic activity and 
strength are not required in an eminent degree. 

Slaking. — The volume of water required to slake lime will vary with limes from 
2.5 to 3 times the volume of the lime (quicklime), and it is important that all the wa- 
ter required to reduce the lime to a proper consistency should be given to it before the 
temperature of the water first given becomes sensibly elevated. 

Immediately upon the limu being provided with the requisite volume of water, it 
should be covered, in order to confine the heat, and it should not be stirred while 
slaking. When the paste is required for grouting or whitewashing, the water re- 
quired should be given at once, and in larger volume than when the paste is required 
for mortar, and when slaked the mass should be transferred to tight casks to prevent 
the loss of water. When the character of the limes, as with those of hydraulic en- 
ergy, will not readily reduce, their reduction, which is an indispensable condition, 
must be aided by mechanical means, as a mortar mill. 

The process here given is termed drowning. When the lime is retained in a bar- 
rel, or like instrument, immersed in water, and then withdrawn before reduction oc- 
curs, it is termed immersion, and when it is reduced by being exposed to the atmos- 
phere, and gradually absorbing moisture therefrom, it is termed air-slaked. 

Bricks should be well wetted before use. Sea sand should not be used in the 
composition of mortar, as it contains salt and its grains are round, being worn by at- 
trition, and consequently having less tenacity than sharp-edged grains. 

Fine Clay. — The fusibility of clay arises from the presence of impurities, such as 
lime, iron, and manganese. These may be removed by steeping the clay in hot mu- 
riatic acid, then washing it with water. Crucibles from common clay may be made 
in this manner. 

Pise is made of clay or earth rammed in layers of from 3 to 4 inches in depth. In 
moist climates, it is necessary to protect the external surface of a wall constructed 
in this manner with a coat of mortar. 

Asphalt Composition. — Mineral pitch 1 part, bitumen 11, powdered stone, or wood 
ashes, 7 parts. 

2. Ashes 2 parts, clay 3 parts, and sand 1 part, mixed with a little oil, makes a 
very fine and durable cement, suitable for external use. 

Mastic. — Pulverized burnt clay 93 parts, litharge ground very fine 7 parts, mixed 
with a sufficient quantity of pure linseed oil. 

3. Silicious sand 14, pulverized calcareous stone 14, litharge 2, and linseed oil 4 
parts by weight. 

The powders to be well dried in an oven, and the surface upon which it is to be 
applied must be saturated with oil. 

4. For Roads. — Bitumen 16.875 parts, asphaltum 225 parts, oil of resin 6.25 parts, 
and sand 135 parts. Thickness, from \\ *>° 1% inches. 

Asphaltum 55 lbs. and gravel 28. 7 lbs. will cover an area of 10.75 square feet. 

Notes by General Gillmore, U. S. A. — All the lime necessary for any required 
quantity or batch of mortar should be slaked at least one day before it is mixed with 
ihe sand. 

All the water required to slake the lime should be poured on at one time, the lime 
should be submerged, and the mass should then be covered with a tarpaulin or can- 
vas, and allowed to remain undisturbed for a period of 24 hours. 

The ingredients should be thoroughly mixed, and then heaped for use as required. 

Recent experiments have developed that most American cements will sustain, 
without any great loss of strength, a dose of lime paste equal to that of the cement 
paste, while a dose equal to y z to % the volume of cement paste may be safely add- 



508 LIMES, CEMENTS, MORTARS, AND CONCRETES. 



ed to any Eosendale cement without producing any essential deterioration of the 
quality of the mortar. Neither is the hydraulic activity of the mortars so far ini- 
paired hy this limited addition of lime paste as to render them unsuited for concrete 
under water, or other submarine masonry. By the use of lime is secured the double 
advantages of slow setting and economy. 

■Pointing Mortar is composed of a paste of finely-ground cement and clean sharp 
silicious sand, in such proportions that the volume of cement paste is slightly in ex- 
cess of the volume of voids or spaces in the sand. The volume of sand varies from 
2% to 2% that of the cemant paste, or by weight, 1 of cement powder to 3 to 3% of 
sand. The mixture should be made under shelter, and in quantities not exceeding 
from 2 to 3 pints at a time. 

Before pointing, the joints should be reamed, and in close masonry they mutt be 
open to i of an inch, then thoroughly saturated with water, and maintained in a 
condition that they will neither absorb water from the mortar or impart any to it. 
Masonry should not be allowed to dry rapidly after pointing, but it should be well 
driven in by the aid of a caulking iron and hammer. 

In the pointing of rubble masonry the same general directions are to be observed. 

Notes by General Totten, U. S. A. — 240 lbs. lime = 1 cask, will make from 7.8 to 
8.15 cubic feet of stiff paste. 

303* lbs. of finely-ground cement will make from 3.7 to 3.S cubic feet of stiff paste ; 
79 to 83 lbs. of cement powder will mak3 1 cubic foot of stiff pa^te. 

1 cubic foot of dry cement powder, measured when loose, will measure .78 to .8 
cubic foot when packed, as at a manufactory. 

100 yards of lath and plaster work, with wr.ges of masons at $1.75 per day, and 
Rockland lime at $1 per cask, cost, respectively : 

3 Coats hard finish work $25.50 | 2 Coats slipped work $19. f 5 

Mural Efflorescences. — White alkaline efflorescences upon the surface of brick walls- 
laid in mortar, of which natural hydraulic lime or cement is the basis. 

The crystallization of these salts within the pores of bricks, into which they have 
been absorbed from the mortar, causes disintegration. 
. Asphalte Flooring. — S lbs. of composition will cover 1 sup. foot, % inch thick. 

Plastering.— \ bushel, or 1^ cubic foot of cement, mortar, etc., will cover 1% 
square yards % in. thick. 75 volumes are required upon brick work for 70 upon laths! 

Cost of IVLasonry, ofvarious Kinds, per Cubic Yard, and 
the Volume of jVEortar required, for eacli.— [Gen. Gilmokk, U.S. A.] 







■c 


tS 


1 S . 

2i| 


Cost. 




£ 










O a> 


c £ o 


« 




Mortar. 




o"s 


* S 






J« 




S 

3 


C 19 

fa 




c ^ a 

&£3 


1 

9 

■6 


If 




> 


tJ 


o 


a 


3 


u 




Cu. Ft. 


Bbls. 


Bbls. 


$ Cts. 


$ Cts. 


$ Cts. 


Rough, in rubble or gravel, from % 














to . 1 cubic foot in volume 


10.3 


.5:5 


1.22 


90 


4.10 


5. 


Blocks, large and small, not in 














courses; joints hammer-dressed. . . 


8.1 


.423 


.92 


62 


7. 


7.63 


Large masses ; headers and stretchers 












: 


dovetailed; hammer-dressed; beds 














and joints laid close 


1. 


.05 


.11 


08 


9. 


9.08 


Ordinary ; courses 20 to 32 in rise . . . 


1.5 


.08 


.17 


12 


5.70 




Ordinary; courses 12 to 20 in rise . . . 


2. 


1.05 


.22 


1G 


2.19 




Brick 


8. 


.42 


.9 


66 


5.70 


6.10 


Concrete, good 


11. 


.54 


1.75 


1.91 


2.19 


3.20 


" medium 


9. 


.41 


LOG 


65 


1.56 


2.21 


" inferior 


8. 


.37 


.97 


60 


1.45 


2.05 


Rubble, without mortar 


3. tc 


) 3.30 



Cost of materials assumed as follows : Cem; 
$4.25 per M ; Sand and Gravel, 80 cents per 
yard ; Labor, $1 per day. 



| 3. to 3.30 

?nt, $1 .25 per barrel ; Lime, $1 ; Bricks, 
ton ; Granite spalls, 55 cents per cubic 



* 300 lbs. net iB the standard barrel, but it usually weighs 308 lba. 



WHEEL GEARING. 509 

WHEEL GEARING. 

The Pitch Line of a wheel, is the circle upon which the pitch is 
measured, and it is the circumference by which the diameter, or the 
velocity of the wheel, is measured. 

The Pitch, is the arc of the circle of the pitch line, and is determ- 
ined by the number of the teeth in the wheel. 

The True Pitch (Chordial), or that by which the dimensions of the 
tooth of a wheel are alone determined, is a straight line drawn from 
the centres of two contiguous teeth upon the pitch line. 

The Line of Centres, is the line between the centres of two wheels. 

The Radius of a wheel, is the semi-diameter running to the periphery 
of a tooth. The Pitch Radius, is the semi-diameter running to the 
pitch line. 

The Length of a Tooth, is the distance from its base to its extremity. 

The Breadth of a Tooth, is the length of the face of wheel. 

A Cog Wheel, is the general term for a wheel having a number of cogs or teeth set 
upon or radiating from its circumference. 

A Mortice Wheel, is a wheel constructed for the reception of teeth or cogs, which 
are fitted into recesses or sockets upon the face of the wheel. 

Plate Wheels, are wheels without arms. 

A Rack, is a series of teeth set in a plane. 

A Sector, is a wheel which reciprocates without forming a full revolution. 

A Spur Wheel, is a wheel having its teeth perpendicular to its axis. 

A Bevel Wheel, is a wheel having its teeth at an angle with its axis. 

A Crown Wheel, is a wheel having its teeth at a right angle with its axis. 

A Mitre Wheel, is a wheel having its teeth at an angle of 45° with its axis. 

A Face Wheel, is a wheel having its teeth set upon one of its sides. 

An Annular or Internal Wheel, is a wheel having its teeth convergent to ita 
centre. 

Spur Gear. — Wheels which act upon each other in the same plane. 

Bevel Gear. — Wheels which act upon each other at an angle. 

When the tooth of a wheel is made of a material different from that of the wheel, 
it is termed a cog : in a pinion it is termed a leaf and in a trundle a stave. 

A wheel which impels another is termed the Spur, Driver, or Leader ; the one im- 
pelled is the Pinion, Driven, or Follower. 

A series of wheels in connection with each other is termed a Train. 

When two wheels act upon one another, the greater is termed the Wheel and the 
lesser the Pinion. 

A Trundle, Lantern, or Wallower is when the teeth of a pinion are constructed of 
round bars or solid cylinders set in to two discs. 

A Trundle with less than eight, staves can not be operated uniformly 
by a wheel with any number of teeth. 

The material of which cogs are made is about one fourth the strength 
of cast iron. The product of their b d* shruld be four times that of iron 
teeth. 

Buchanan : Rules that to increase or diminish velocity in a given pro- 
portion, and with the least quantity of wheel-work, the number of teeth 
in each pinion should be to the number of teeth in its wheel as 1 : 3.59. 
Even to save space and expense, the ratio should never exceed 1:6. 

The least number of teeth that it is practicable to give to a wheel is 
regulated by the necessit}" of having at least one pair always in action, 
in order to provide for the contingenc}- of a tooth breaking. 



510 WHEEL GEARING. 

The teeth of -wheels should be as small and numerous as is consistent 
with strength. 

When a Pinion is driven by a wheel, the number of teeth in the pinion 
should not be less than eight. 

When a Wheel is driven by a pinion, the number of teeth in the pinion 
should not be less than ten. 

The Number of teeth in a -wheel should always be prime to the number 
of the pinion ; that is, the number of teeth in the wheel should not be di- 
visible by the number of teeth in the pinion without a remainder. This 
is in order to prevent the same teeth coming together so often as to cause 
an irregular wear of their faces. An odd tooth introduced into a wheel is 
termed a hunting tooth or cog. 

To Compete tlie Fitch, of* a "Wheel. 
Rule. — Divide circumference at the pitch-line by the number of teeth. 
Examine. — A wheel 40 ins. in diameter requires 75 teeth; what is its pitch? 
3.1416X40 . 

=3 = 1.0*55 ins. 

ro 

To Compute the True or Ch.ord.ial Pitch. 

Rulk. — Divide 180° by the number of teeth, ascertain the sine of the 
quotient, and multiply it by the diameter of the wheel. 

Example. — The number of teeth is 75, and the diameter 40 inches; what is tlie 
true pitch ? 
ISO 

— r=2° 24' and sin. of 2° 24' = .041 S3, which X 40 = 1.6752 ins. 
(5 

To Compute the Diameter of a Wheel. 

Rule. — Multiply the number of teeth by the pitch, and divide the prod- 
uct by 3.1416. 

Example. — The number of teeth in a wheel is 75, and the pitch 1.675 ins. ; what 
is the diameter of it ? 

75x1.675 <A . 

-one- =40ms - 

To Compute the Number of Teeth in a Wheel. 
Rule. — Divide the circumference by the pitch. 

To Compute the Diameter when the True IPitch is given. 

Rulk. — Multiply the number of teeth in the wheel by the true pitch, 
and again by .3184. 

Example. — Take the elements of the preceding case. 
75Xl.6752X-ol84 = 40 ins. 

To Compute the Number of* Teeth in a Trillion or Fol- 
lower to have a given "Velocity-. 

Rulk. — Multiply the velocity of the driver by its number of teeth, and 
divide the product by the velocity of the driven. 

Example. — The velocity of a driver is 16 revolutions, the number of its teeth 54, 
and the velocity of the pinion is 4S; what is the number of its teeth? 

16X54 i„ 

— ~- = 18 teeth. 
4o 
2. A wheel having 75 teeth is making 16 revolutions per minute: whnt is the 
number of teeth required in the pinion to m:ike 24 revolutions in the same time? 

— - — = 50 teeth, 
z4 



WHEEL GEARING. 511 

To Compute the Proportional Radius of a Wheel or 

3Pi.rii.orL. 

j> uljK . — Multiply the length of the line of centres by the number of 
teeth in the wheel for the wheel, and in the pinion for the pinion, and 
divide by the number of teeth in both the wheel and pinion. 

To Compute tire Diameter of a Pinion, when the Di- 
ameter- of the "Wheel and. jSTunfber of Teeth in the Wheel 
and. JPinion are given. 
Rule. —Multiply the diameter of the wheel by the number of teeth in 

the pinion, and divide the product by the number of teeth in the wheel. 
Example. The diameter of a wheel is 25 inches, the number of its teeth 210, and 

the number of teeth in the pinion 30 ; what is the diameter of the pinion ? 
25X30 



210 



:3.57 ins. 



To Compute the Number of Teeth required in a Train 
of Wheels to produce a given Velocity. 

Rule. — Multiply the number of teeth in the driver by its number of 
revolutions, and divide the product by the number of revolutions of each 
pinion, for each driver and pinion. 

Example. —If a driver in a train of three wheels has 90 teeth, and makes 2 revo- 
lutions, and the velocities required are 2, 10, and 18, what are the number of teeth 
in each of the other two? 

10 : 90 : : 2 : 18 = teeth in 2d wheel. 
18 : 90 : : 2 : 10 = teeth in 3d wheel. 

To Compute the Circumference of a Wheel. 

Rule. — Multiply the number of teeth by their pitch. 

To Compute the Revolutions of a "Wheel or IPinion. 

Rule. — Multiply the diameter or circumference of the wheel or the 
number of its teeth, as the case may be, by the number of its revolutions, 
and divide the product by the diameter, circumference, or number of teeth 
in the pinion. 

Example. — A pinion 10 inches in diameter is driven by a wheel 2 feet in diameter, 

making 4G revolutions per minute ; what is the number of revolutions of the pinion ? 

2x12x46 



10 



- = 110.4 revolutions. 



To Compute the "Velocity- of a 3?iirion. 

Rule. — Divide the diameter, circumference, or number of teeth in the 
driver, as the case may be, by the diameter, etc., of the pinion. 

When there are a Series or Train of Wheels and Pinions. 

Rule. — Divide the continued product of the diameter, circumference, 
Or number of teeth in the wheels by the continued product of the diameter, 
etc., of the pinions. 

Example — Tf a wheel of 32 teeth drive a pinion of 10, upon the axis of which there 
is one of 30 teeth, driving a pinion of 8, what are the revolutions of the last? 
32 30 960 cf. ; . 

^Xs=-80= 12reVOlUtl ° nS 
Ex. 2. The diameters of a train of wheels are G, 9, 9, 10, and 12 inches; of the 
pinion?, G, G, G, G, and G inches ; and the number of revolutions of the driving shaft 
or prime mover is 10 ; what are the revolutions of the last pinion ? 
0X0X9X10X12X10 5S3200 M , . 

— 6^6xGxTxT- =-^r = ™ revolutions. 



512 WHEEL GEARING. 



To Compute the Proportion, that tlie "Velocities of* the 
Wlieels in a Train should, bear to one another. 

Rule. — Subtract the less velocity from the greater, and divide the re- 
mainder by one less than the number of wheels in the train ; the quotient 
is the number, rising in arithmetical progression from the less to the great- 
er velocity. 

Example. — What should he the velocities of 3 wheels to produce IS revolutions, 
the driver making 3? 

~~ " — = 7.5 = number to be added to velocity of the driver = 7.5 -f 3 = 10.5, 

and 10.5 + 7.5 = IS revolutions. Hence 3, 10.5, and IS are the velocities of the three 
wheels. 

General Illustrations. 

1. A wheel 96 inches in diameter, having 42 revolutions per minute, is to drive a 
shaft 75 revolutions per minute ; what should he the diameter of the pinion ? 

96 X 42 . 

— — — = 53. » 6 ins. 
75 

2. If a pinion is to make 20 revolutions per minute, required the diameter of an- 
other to make 5S revolutions in the same time. 

53 -i- 20 = 2.9 = the ratio of their diameters. Hence, if one to make 20 revolutions 
is given a diameter of 30 inches, the other will be 30 -j- 2.9 == 10.345 ins. 

3. Required the diameter of a pinion to make 12^ revolutions in the same time as 
one of 32 ins. diameter making 26. 

•£" = 06.66 to. 

12.5 

4. A shaft, having 22 revolutions per minute, is to drive another shaft at the rate 
of 15, the distance between the two shafts upon the line of centres is 45 ins. ; what 
should be the diameter of the wheels? 

Then, 1st. 22 -f- 15 : 22 : :45 : 26.75 == inches in the radius of the pinion. 
2d. 22 -J- 15: ■ 5 : :45 : 18.24 = inches in the radius of the spur. 

5. A driving shaft, having 16 revolutions per minute, is to drive a shaft 81 revolu- 
tions per minute, the motion to be communicated by two geared wheels and two pul- 
leys, with an intermediate shaft ; the driving wheel is to contain 54 teeth, and the 
driving pulley upon the driven shaft is to be 25 inches in diameter ; required the num- 
ber of teeth in the driven wheel, and the diameter of the driven pulley. 

Let the driven wheel have a velocity of "\/l6 X 81 = 36, a mean proportional be- 
tween the extreme velocities 16 and 81. 
Then, 1st. 36 : 16 : : 54 : 24 = teeth in the driven wheel. 
2d. SI : 36 : : 25 : 11.11 bk ins. diameter of the driven pulley. 

6. If, as in the preceding case, the whole number of revolutions of the driving 
shaft, the number of teeth in its wheel, and the diameters of the pulleys are given, 
what are the revolutions of the shafts? 

Then, 1st. IS : 16 : : 54 : 4S =revolutions of the intermediate shaft. 
2d. 15 : 4S : : 25 : SO = revolutions of the driven shaft. 

To Compute the Diameter of a "Wheel for a given 3?itoh 
and. IS'um'ber of* Teeth. 

Rule. — Multiply the diameter in the following table for the number of 
teeth by the pitch, and the product will give the diameter at the pitch circle. 
Example. — What is the diameter of a wheel to contain 4S teeth of 2.5 ins. pitch? 
15. 29 X 2.5 ^3S. 225 ms. 

To Compute the Fitch of a "Wheel for a given Diameter 
and. Number of* Teeth. 

Bulk. — Divide the diameter of the wheel by the diameter in the table 
for the number of teeth, and the quotient will give the pitch. 

Example. — What is the pitch of a wheel when the diameter of it is 50.94 inches, 
and the number of its teeth 80 ? 

50.94 n 

2M = 2mS ' 



WHEEL GEAKING. 



513 



To Compute the TsTumTber of Teeth of* a ^Wlieel for a given 
Diameter and. JPitch. 

Rule. — Divide the diameter by the pitch, and opposite to the quotient 
in the table is given the number of teeth. 

PITCH OF WHEELS. 

A. Ta"ble wherefoy to Compute the Diameter of* a Wheel 
for a given IPitch, or the I*itch. for a given Diameter. 

From 8 to 192 teeth. 



No. of 


Diame- 


No. of 


Diame- 


No. of 


Diame- 


No. of 


Diame- 


No. of 


Diame- 


Teeth. 


ter. 


Teeth. 


ter. 


Teeth. 


ter. 


Teeth. 


ter. 


Teeth. 


ter. 


8 


2.61 


45 


14.33 


82 


26.11 


119 


37.88 


156 


49.66 


9 


2.93 


46 


14.65 


83 


26.43 


120 


38.2 


157 


49.98 


10 


3.24 


47 


14.97 


84 


26.74 


121 


38.52 


158 


50.3 


11 


3.55 


48 


15.29 


85 


27.06 


122 


38.84 


159 


50.61 


12 


3.86 


49 


15.61 


86 


27.38 


123 


39.16 


160 


50.93 


13 


4.18 


50 


15.93 


87 


27.7 


124 


39.47 


161 


51.25 


14 


4.49 


51 


16.24 


88 


28.02 


125 


39.79 


162 


51.57 


15 


4.81 


52 


16.56 


89 


28.33 


126 


40.11 


163 


51.89 


16 


5.12 


53 


16.88 


90 


28.65 


127 


40.43 


164 


52.21 


17 


5.44 


54 


17.2 


91 


28.97 


128 


40.75 


165 


52.52 


18 


5.76 


55 


17.52 


92 


29.29 


129 


41.07 


166 


52.84 


19 


6.07 


56 


17.8 


93 


29.61 


130 


41.38 


167 


53.16 


20 


6.39 


57 


18.15 


94 


29.93 


131 


41.7 


168 


53.48 


21 


6.71 


58 


18.47 


95 


30.24 


132 


42.02 


169 


53.8 


22 


7.03 


59 


18.79 


96 


30.56 


133 


42.34 


170 


54.12 


23 


7.34 


60 


19.11 


97 


30.88 


134 


42.66 


171 


54.43 


24 


7.66 


61 


19.42 


98 


31.2 


135 


42.98 


172 


54.75 


25 


7-98 


62 


19.74 


99 


31.52 


136 


43.29 


173 


55.07 


26 


8.3 


63 


20.06 


100 


31.84 


137 


43.61 


174 


55.39 


27 


8-61 


64 


20.38 


101 


32.15 


138 


43.93 


175 


55.71 


28 


8.93 


65 


20.7 


102 


32.47 


139 


44.25 


176 


56.02 


29 


9.25 


66 


21.02 


103 


32.79 


140 


44.57 


177 


56.34 


30 


9.57 


67 


21.33 


104 


33.11 


141 


44.88 


178 


56.66 


31 


9.88 


68 


21.65 


105 


33.43 


142 


45.2 


179 


56.98 


32 


10.2 


69 


21.97 


106 


33.74 


143 


45.52 


180 


57.23 


33 


10-52 


70 


22.29 


107 


34.06 


144 


45.84 


181 


57.62 


34 


10.84 


71 


22.61 


108 


34.38 


145 


46.16 


182 


57.93 


35 


11.16 


72 


22.92 


109 


34.7 


146 


46.48 


183 


58.25 


36 


11.47 


73 


23.24 


110 


35.02 


147 


46.79 


184 


58.57 


37 


11.79 


74 


23.56 


111 


35.34 


148 


47.11 


185 


58.89 


38 


12.11 


75 


23.88 


112 


35.65 


149 


47.43 


186 


59.21 


39 


12.43 


76 


24.2 


113 


35.97 


150 


47.75 


187 


59.53 


40 


12.74 


77 


24.52 


114 


3629 


151 


48.07 


188 


59.84 


41 


13.06 


78 


24.83 


115 


36.61 


152 


48.39 


189 


60.16 


42 


13.38 


79 


25.15 


116 


36.93 


153 


48.7 


190 


60.48 


43 


13.7 


80 


25.47 


117 


37.25 


154 


49.02 


191 


60.81 


44 


14.02 


81 


25.79 


118 


37.56 


155 


49.34 


192 


61.13 



Note. — The pitch in this table is the true pitch, as before described. 

Change "Wheels in Screw-cutting Lathes. 

— — I = N ; y-rp = S. T representing number of teeth in traverse screw ; 

S number in stud wheel gearing in mandril; t number in wheel upon mandril, 
and t! number in gearing upon stud pinion, gearing in T ; I number of threads 
per inch upon traverse screw ; N number to be cut. 

X x 



514 WHEEL GEARING. 

Teeth of Wheels. 
Epicycloidal. 

In order that the teeth of wheels and pinions should work evenly and 
without unnecessary rubbing friction, the face (from pitch line to top) of 
the outline should be determined by an epicycloidal curve, and the flank 
(from pitch line to base) by an hypocycloidal. 

When the generating circle is equal to half the diameter of the pitch 
circle, the hypocycloid described by it is a straight diametrical line, and, 
consequently, the outline of a flank is a right line and radial to the centre 
of the wheel. 

If a like generating circle is used to describe face of a tooth of other 
wheel or pinion respectively, the wheel and pinion will operate evenly. 

Involute. 

Teeth of two wheels will work truly together when surfaces of their 
face is an involute; and that two such wheels should work truly, the 
circles from which the involute lines for each wheel are generated must 
be concentric with the wheels, with diameters in the same ratio as those 
of the wheels. 

Curves of Teeth. — In the pattern-shop, the curves of epicycloidal or 
involute teeth are denned bj T rolling a template of the generating circle 
on a template corresponding to the pitch line. A scriber on the periphery 
of the template being used to define the curve. 

Least number of teeth that can be employed in pinions having teeth 
of following classes are : involute, 25 ; epicycloidal, 12 ; staves or pins, 6. 

To describe teeth by an epicycloidal curve or an involute, see Shelton's Mechan- 
ics' Guide, London, 1875; Fairbairn's Mechanism and Machinery, Phila., 1872; 
Moseley's Engineering and Architecture, New York, 1875; D. K. Clark's Manual, 
London, 1S77; or Molesworth's Pocket-Book, London, 1876. 

To Construct Circular Teetli. 

/g s\ / N ' Assume A A pitch line, b b line of 

"!<' " / s \/ base of teeth, and tt top. Set off on 

/\_ -/ — V -/^ pitch line divisions both of pitch and 

t-^f^\ I \y / \^ depth of teeth, then, with a radius of 

Q l ,- A-- I f^^-il 1.225 pitch, describe arcs, as osr upon 

A" \ \ )___-( J / "* pitch line forfaces of teeth, then draw 

3-"""~ U ~~~~~"-^-> radial lines ou, r u to centre of wheel 

T > for flanks. Strike arc tt to define 

J ^^^^ •*— ^_ \ l en S tn of tooth and fillet flanks at 

C^-^" ws^J ^^"^w base. 

Proportion of Teeth. 

Pitch =1. Clearances .05 Play — .1 

Length = .75 Depth == .45 Face == .35 

Working lengths .7 Space = .55 Breadth, usual— .25 

Breadth with small pitch=2. ; with large=3. 

Depth, Pitch, and. Breadth. (M. Moein.) 

Cast iron .028/ W=d. .057/ W=P. 

Hard wood .038 /W=d. .079/ W=P. 

W representing weight or stress upon tooth in lbs., d depth of tooth, and P 

pitch in inches. 

Hence, strength of hard wood compared with cast iron is taken as .74 to 1. 

For further illustrations of formation of teeth, bevil gearing, Willis's cdontograph, 
trundles, etc., see Shelton, Moseley, or Fairbairn, as previously referred to. 



WHEEL GEARING. 515 

Face of a tooth is that part of its side which extends from its pitch line 
to its top. 

Flank of a tooth is that part of its side which extends from pitch line to 
line of space between adjacent teeth ; its length, as well as that of face, 
is measured in direction of radius of wheel, and is a little greater than it, 
to admit of sufficient clearance. 

Depth of a tooth is its thickness from face to face, measured at the pitch 
line. 



PROPORTIONS OF TEETH OF WHEELS. 

Tooth. — In computing the dimensions of a tooth, it is to be consid- 
ered as a beam fixed at one end, the weight suspended from the other, 
or face of the beam ; and it is essential to consider the element of ve- 
locity, as its stress in operation, at high velocity with irregular action, 
is increased thereby. 

The dimensions of a tooth should be much greater than is necessary 
to resist the direct stress upon it, as but one tooth is proportioned to 
bear the whole stress upon the wheel, although two or more are actually 
in contact at all times; but this requirement is in consequence of the 
great wear to which a tooth is subjected, the shocks it is liable to from 
lost motion, when so worn as to reduce its depth and uniformity of 
bearing, and the risk of the breaking of a tooth from a defect. 

A tooth running at a low velocity may be materially reduced in its 
dimensions compared with one running at a high velocity and with a 
like stress. 

The result of operations with toothed wheels, for a long period of 
time, has determined that a tooth with a pitch of 3 inches and a 
breadth 7.5 inches will transmit, at a velocity of 6.66 feet per second, 
the power of 59. 16 horses. 

To Compute tlie Dimensions of a Tooth to Resist a 
given. Stress. 
Rule. — Multipl}- the extreme pressure at the pitch-line of the wheel 
b}' the length of the tooth in the decimal of a foot, divide the product by 
the Value of the material of the tooth, and the quotient will give the 
product of the breadth and square of the depth. 

Or — —b d 2 . S representing the stress in pounds, and I the length in feet. 

The Value of cast iron for this or like purposes may be taken at from 50 to TO. 

Note. — It is necessary first to determine the pitch, in order to obtain either the 
length or depth of a tooth. 

Example. — The pressure at the pitch-line of a cast-iron wheel (at a velocity of 
6.66 feet per second) is 488S lbs. ; what should be the dimensions of the teeth, the 
pitch being 3 inches ? 

3X.T=2.1=fm,(/^/i of toothy ivhich-i-l2 = . 175 = length in decimals of a foot; 
3x2.5=7 .b— breadth of tooth. 

The Value of the material in this case is taken at 60. 

*S»X.1T5 and /Ii!5 =1 .3S ins. in deptk. 

60 V 7.5 

When the product b d 2 is obtained, and it is required to ascertain either dimen- 
sion, proceed as follows : 

As d=z.4Q, and as b = <2.5 times the pitch, b is to d as 5.435 is to 1. Assume the 
preceding case where b d 2 ^ 14.25. i < 9 r- 

Then b : d ::5.435: 1 ; .-.6=5.435 d, and 5.435 d3 — 14.25; :.d^=-^'^ =2.6219, 

b d 2 14 05 5.485 

and ^/2.622=1.3S ins., the depth; and —~—b; ;. 1^=7.5 ins., the breadth. 

d 2 1.3s 2 



516 AVHEEL GEARING. 

The following Rule* to ascertain the dimensions of a tooth is the result 
of some consideration of the subject, and is supported by several well- 
defined cases in operation. 

Xo Compute tlie Deptlx of* a Cast-iron. Tootli 
1. When the Stress is given. 
Rule. — Extract the square root of the stress, and multiply it by .02. 
Example. — The stress to be borne by a tooth is 4SS0 lbs. ; what should be its 
depth ? % 4S86X.02=1.4 ins. 

2. When the Horses* Poicer is given. 

RrLK. — Extract the square root of the quotient of the horses' power di- 
vided by the velocity in feet per second, and multiply it by .466. 

Example. — The horses' power to be transmitted by a tooth is 6»», and the velocity 
of it at its pitch-line is 6.166 feet per second: what hhould be the depth of the tooth? 

-XA6Q=:l.d9Sins. 



<& 



6.66 

To Compute tlie Horses' 3?ovver of a Tootli. 

Bulk. — Multiply the pressure at the pitch-line, by its velocity in feet 
per minute, and divide the product by 33 000 

Example. — What is th? horses' power of a tooth of the dimensions and at the ve- 
locity given in the preceding example, page 515 I 

4S5<5x6.66x60" 

=59.16 horses. 

33 000 

To Compute tlie Stress tliat may be "borne "by a Tootli. 
Rule. — Multiply the Value of the material of the tooth to resist a trans- 
verse strain, as estimated for this character of stress, by the breadth and 
square of its depth, and divide the product by the extreme length of it in 
the decimal of a foot. 

* As an exponent of the necessity of an investigation of the stress of a to >th, the 
following deductions by the rules of different authors for like elements are sub- 
mitted: 

Pitch 3 ins. Depth 1.33 ins. Breadth 7.5 ins. Length 2.1 ins. 

FOR CAST IRON. 
Actual power in stress exerted at a velocity of 400 feet per minnte, 4886 lbs. Tooth? 

Ids. 

By above rule /-X. 466 = 1.398t 

" Fairbairn .025^/W = I.W 

/ w 
M Imperial Journal./ yr~ = 1. *6 

» Rankine v /j^; = 



/3P 



14 Buchanan 



3 /H_ 

4\ »• — •■•• 
/.55*> H _ 



1.8 

2.25 

2.24 



II representing horses' poicer (GO), W and P the stress in pounds, and v the ve- 
locity in feet per second, 

t This depth, with a breadth of 7.5 ins., is .1 of the ultimate strength of the aver- 
age strength of American Cast Iron. 



WHEEL GEAKING. 517 

Example The dimensions of a cast-iron tooth in a wheel are 1.3S ins. in depth 

by 7.5 ins. in breadth ; what is the stress it will bear? 

Pitch = 2.174 Xl.3S = 3 ins. Length = .7 of 3 = 2.1 ins. 

Breadth = 2.5X 3 = 7.5 ins. ^^^ = 4S86 WS ' 
PROPORTIONS OF WHEELS 

With six flat arms and Bibs upon one side of them, as m^^ ; or a Web 
in the centre, as es^sa . 

Rim m — Depth, measured from base of the teeth, .45 to .5 of the pitch of 
the teeth, having a web upon its inner surface .4 of the pitch in depth and 
.25 to .3 of it in width. 

Note.— When the face of the wheel is morticed, the depth of the rim should be 
1.5 times the pitch, and the breadth of it 1.5 times the breadth of the tooth or cog. 

Hub. — When the eye is proportionate to the stress upon the wheel, the hub 
should be twice the diameter of the eye. In other cases the depth around 
the eye should be .75 to .8 of the pitch. 

Arm. — Djpth .4 to .45 of the pitch. Breadth at rim 1.5 times the pitch, 
increasing .5 inch per foot of length toward the hub. 

The Ril) upon one edge of the arm, or the Web in its centre, should be 
from .25 to .3 the pitch in width, and .4 to .45 of it in depth. 

When the section of an arm differs from those above given, as with one 

with a plane section, as mzzmzm, or with a double rib, as |^-^| , its dimen- 
sions should be proportioned to the form of the section. 

In a wheel of greater relative diameter, the length of the hub and the 
breadth of the arms, or of the rib or web, according as the plane of the 
arm is in that of the wheel or contrariwise, should be made to exceed the 
breadth of the face of the wheel (at the hub) in order to give it resistance 
to lateral strain. 

The number of arms in wheels should be as follows : 



8.5 to 16 feet in diam 8. 

16 " 24 " " 10. 



1.5 to 3.25 feet in diam 4. 

3.25 " 5 " u 5. 

5 "85 " " 6. 

With light wheels, the number of arms should be increased, in order the 
better to sustain the rigidity of the rim. 

Pitches of Equivalent Strength for Iron and Wood. — Iron 1. Hard 
wood 1.26. 



WINDING ENGINES. 



In Winding Engines, for drawing coals, etc., out of a Pit, where it is re- 
quired to give a certain number of revolutions, it is necessary to know 
the diameter of the Drum and the thickness of the rope, and contrariwise. 

To Compute tlie Diameter of a Drum. 

Where flat Ropes are used, and are wound one part over the other. 
Kui,rc. — Divide the depth of the pit in inches b} T the product of the num- 
ber of revolutions and 3.1416, and from the quotient subtract the product 
of the thickness of the rope and the number of revolutions ; the remainder 
is the diameter in inches. 

Example. — If fin engine makes 20 revolutions, the deptli of the pit being 600 feet, 
and the rope 1 inch, what should be the diameter of the drum ? 

°° X Vl X 20=™ -20 = 94.5^. 



20X3.1410 62.8-J2 

Xx 



518 WINDING ENGINES. DREDGING MACHINE. 

To Compute tlie Diameter of the Itoll. 

Rule. — To the area of the drum add the area or edge surface of tha 
rope ; then ascertain by inspection in the table of areas, or fey calculation, 
the diameter that gives this area, and it is the diameter of the Roll. 

Example What is the diameter of the roll in the preceding example ? 

Area of 94.59 = 7027.2+ area of 7200 X 1 + T200 = 14227.2, and ^14227.2-3- 
.7S54= 134.59 ins. 
Or, the radius of the drum is increased the number of the revolutions multiplied 

by the thickness of the lope; as, *-— [-20x1 = 67.295 ins. 

a 

To Compute tlie IN'nm'ber of Ptevolxitions. 

Rulk. — To the area of the drum add the area of the edge surface of the 
rope ; from the diameter of the circle having that area subtract the diam- 
eter of the drum, and divide the remainder by twice the thickness of the 
rope ; the quotient will give the number of revolutions. 

Example. — The length of a rope is 2600 inches, its thickness 1 inch, and the diam- 
ter of the drum 20 inches ; what is the number of revolutions ? 

Area of 20 + area of rope = 314.16, and 314.16^261)0 = 2914.16, the diameter of 
60.91 — 20 
which is 60.91, and — - — - — = 20.45 revolutions. 
1X2 

Or, subtract the diameter of the drum from the diameter of the roll, and divide 
the remainder by twice the thickness of the rope; as, 134.59 — 94.59 = 40, and 
40 -r- 1x2 = 20 revolutions. 

To Compute tlie IPlace of* jVEeeting of* tlie Ascending and. 
Descending Buckets when two or more are xised. 

Note. —Meetings will always be below half the depth of the pit. 

To Compute this Depth. 

Bulk. — Take the circumference of the drum for the length of the first 
turn ; then, to the diameter of the drum add twice the thickness of the 
rope, multiplied by the number of revolutions, less 1, for a diameter, and 
the circumference of this diameter is the length of the last turn ; add these 
two lengths together, multiply their sum by half the number of revolu- 
tions, and the product will give the depth of the pit. 

Example. — Tlie diameter of a drum is 9 feet, the thickness of the rope 1 inch, and 
the revolutions 20 ; what is the depth of the pit, and at what distance from the top 
will the buckets meet ? 

IvOyOn 1 

9x3.1410 = 28.27 feet length of first turn. 94- * *~ X 3.1416 =3S.23/eef, 

20 *■* 

length of last turn. 2S,27-f S8. 23 X— =60.5 X 10 = 005 feet, or depth of pit. 

2. Divide the sum of the length of the turns of the rope by 2, and to the 
quotient add the length of the last turn; divide the sum l>y 2, multiply 
the quotient by half the number of revolutions, and the product will give 
the distance from the centre of the drum at which the buckets will meet. 

Note. — At half the number of revolutions the buckets will me:t. 

2S.27 4-3S.23 . 00 , " ' 71.4S 20 1429.0 -M> , _ 
J h 3S.23 = 71.4S, and — — X— = — — == 357 .4 feet. 



DREDGING MACHINE. 

In the operation of a Dredging Machine, in 1855, under Lieut. Meade, 
U. S. A., the following elements were obtained : 

Two non-condensing engines, working at a power of 22 horses, exca- 
vated 1075 cubic yards, or 170 tons, of soft and hard clay and mud per 
hour, at a depth of 11 feet from the water line. 



WOOD, TIMBER, ETC. 519 

The coefficients deduced from the friction of the materials raised were 
C =.1 for hard clay with gravel ; =.07 for pure hard clay ; =.05 for com- 
mon clay or sand"; =.04 for soft clay or loose sand; and =.03 for loose 
materials. 

From which Mr. Nystrom furnishes the following formulae : 

( 'W) + C ) W = horses' power, h representing the total height to which the 
material is raised, and W the weight of it in tons. 

700 H -w *+A_ c 

A+-700C"" ' W^700~^ 



WOOD, TIMBER, ETC. 



Selection of Standing Trees. — Wood grown in a moist soil is lighter, 
and decays sooner than that grown in dry, sandy soil. 

The best Timber is that grown in a dark soil intermixed with gravel. 
Poplar, c} T press, willow, and all others which grow best in a wet soil, are 
exceptions. 

The hardest and densest woods, and the least subject to decay, grow in 
warm climates ; but they are more liable to split and warp in seasoning. 

Trees grown upon plains or in the centre of forests are less dense than 
those from the edge of a forest, from the side of a hill, or from open ground. 

Trees (in the U. S.) should be selected in the latter part of July or first 
part of August ; for at this season the leaves of the sound, healthy trees 
are fresh and green, while those of the unsound are beginning to turn ye\- 
low. A sound, healthy tree is recognized by its top branches being well 
leaved, the bark even and of a uniform color. A rounded top, few leaves, 
some of them turned yellow, a rougher bark than common, covered with 
parasitic plants, and with streaks or spots upon it, indicate a tree upon 
the decline. The decay of branches, and the separation of bark from the 
wood, are infallible indications that the wood is impaired. 

Felling Timber. — The most suitable time for felling timber is in mid- 
winter and in midsummer. Recent experiments indicate the latter sea- 
son and in the month of July. 

A tree should be allowed to attain full maturity before being felled. 
Oak matures at 75 to 100 years and upward, according to circumstances. 
The age and rate of growth of a tree are indicated by the number and 
width of the rings of annual increase which are exhibited in a cross-sec- 
tion. 

A tree should be cut as near to the ground as practicable, as the lower 
part furnishes the best timber. 

Dressing Timber. — As soon as a tree is felled, it should be stripped of 
its bark, raised from the ground, the sap-wood taken off, and the timber 
reduced to its required dimensions. 

Inspection of Timber. — The quality of wood is in some degree indicated 
by its color, which should be nearly uniform in the heart, a little deeper 
toward the centre, and free from sudden transitions of color. White spots 
indicate decay. The sap-wood is known b} r its white color ; it is next to 
the bark, anol very soon rots. 

Defects of Timber. — Wind-shakes are circular cracks separating the con- 
centric layers of wood from each other. It is a serious defect. 

Splits, checks, and cracks, extending toward the centre, if deep and 
strongly marked, render the timber unfit for use, unless the purpose for 
which it is intended will admit of its being split through them. 

Brash-wood is generally consequent upon the decline of the tree from 



520 WOOD, TIMBER, ETC. 

age. The wood is porous, of a reddish color, and breaks short, without 
splinters. 

Belted timber is that which has been killed before being felled, or which 
has died from other causes. It is objectionable. 

Knotty timber is that containing many knots, though sound ; usually of 
stunted growth. 

Twisted icood is when the grain of it winds spirally ; it is unfit for long 
pieces. 

Dry-rot. — This is indicated by yellow stains. Elm and beech are soon 
affected, if left with the bark on. 

Large or decayed knots injuriously affect the strength of timber. 

Seasoning and ^Preserving Timloer. 

Timber freshly cut contains about 37 to 48 per cent, of liquids. By ex- 
posure to the air in seasoning one year, it loses from 17 to 25 per cent., and 
when seasoned it yet retains from 10 to 15 per cent. 

Timber of large dimensions is improved and rendered less liable to 
warp and crack in being seasoned by immersion in water for some weeks. 

For the purpose of seasoning, timber should be piled under shelter and 
be kept dry ; it should have a free circulation of air about it, without be- 
ing exposed to strong currents. The bottom pieces should be placed upon 
skids, which should be free from decay, raised not less than 2 feet from 
the ground ; a space of an inch should intervene between the pieces of the 
same horizontal la}-ers, and slats or piling-strips placed between each 
laj r er, one near each end of the pile, and others at short distances, in or- 
der to keep the timber from winding. These strips should be one over 
the other, and in large piles should not be less than 1 inch thick. Light 
timber may be piled in the upper portion of the shelter, heavy timber 
upon the ground floor. Each pile should contain but one description of 
timber. The piles should be at least 2)^ feet apart. 

Timber should be repiled at intervals, and all pieces indicating decay 
should be removed, to prevent their affecting those which are still sound. 

Timber houses are best provided with blinds, which keep out rain and 
snow, but which can be turned to admit air in fine weather, and they 
should be kept entirely free from any pieces of decayed wood. 

The gradual mode of seasoning is the most favorable to the strength 
and durability of timber, but various methods have been proposed for 
hastening the process. For this purpose, steaming timber has been ap- 
plied with success ; and the results of experiments of various processes of 
saturating timber with a solution of corrosive sublimate and antiseptic 
fluids are ver} r satisfactory. This process hardens and seasons wood, at 
the same time that it secures it from dry-rot and from the attacks of 
worms. Kiln-drying is serviceable only for boards and pieces of small 
dimensions, and is apt to cause cracks and to impair the strength of wood, 
unless performed very slowly. Charring or painting is highly injurious 
to any but seasoned timber, as it effectually prevents the drying of the 
inner part of the wood, in consequence of which fermentation and decay 
soon take place. 

Timber piled in badly-ventilated sheds is apt to be attacked with the 
common-rot. The first outward indications are yellow spots upon the ends 
of the pieces, and a 3 T ellowish dust in the checks and cracks, particularly 
where the pieces rest upon the piling-strips. 

Timber requires from 2 to 8 years to be seasoned thoroughly, according 
to its dimensions. It should be worked as soon as it is thoroughly dry, 
for it deteriorates after that time. 



WOOD, TIMBER, ETC. 521 

Oak timber loses one fifth of its weight in seasoning, and about one third 
of its weight in becoming perfectly dry. Seasoning is the extraction or dis- 
sipation of the vegetable juices and moisture, or the solidification of the al- 
bumen. When wood is exposed to currents of air at a high temperature, 
the moisture evaporates too rapidly and the wood cracks ; and when the, 
temperature is high and sap remains, it ferments, and dry-rot ensues. 

Timber is subject to Common-rot or Dry-rot, the former occasioned by 
alternate exposure to moisture and dryness. The progress of this decay 
is from the exterior; hence the covering of the surface with paint, tar, etc., 
is a preservative. 

Painting and charring green timber hastens its decay. 

Dry or Sap-rot is inherent in timber, and it is occasioned by the putre- 
faction of the vegetable albumen. Sap wood contains a large proportion 
of fermentable elements. Insects attack wood for the sugar or gum con- 
tained in it, and Fungi subsist upon the albumen of wood ; hence, to arrest 
dry-rot, the albumen must be either extracted or solidified. 

In the seasoning of timber naturally there is required a period of from 
2 to 4 years. Immersion in water facilitates seasoning by solving the sap. 

The most effective method of preserving timber is that of expelling or 
exhausting its fluids, solidifying its albumen, and introducing an antisep- 
tic liquid. 

The strength of impregnated timber is not reduced, and its resilience is 
improved. 

In desiccating timber by expelling its fluids by heat and air, its strength 
is increased fully 15 per cent. 

In coating unseasoned timber with creosote, tar, etc., the fluids are re- 
tained, and decay facilitated thereby. 

When timber is saturated with creosote, tar, antiseptics, etc., it is also 
preserved from the attack of worms. Jarrow wood, from Australia, is not 
subjected to their attack. 

The condition of timber, as to its soundness or decay, is readily recog- 
nized when struck a quick blow. 

Timber that has been for a long time immersed in water, wdien brought 
into the air and dried, becomes brash}' and useless. 

When trees are barked in the spring, they should not be felled until the 
foliage is dead. 

Timber can not be seasoned by either smoking or charring; but when 
it is to be used in locations where it is exposed to worms or to produce 
fungi, it is proper to smoke or char it. 

Timber may be partially seasoned by being boiled or steamed. 

Impregnation ofWood. 
The several processes are as follows : 
_ Kyan, 1832. Saturated with corrosive sublimate. Solution 1 lb. of chlo- 
ride of mercury to 4 gallons of water. 

Burnett, 1838. Impregnation with chloride of zinc by submitting the 
wood endwise to a pressure of 150 lbs. per square inch. Solution 1 lb. of 
the chloride to 10 gallons of water. 

Boucheri. Impregnation by submitting the wood endwise to a pressure 
of about 15 lbs. per square inch. Solution 1 lb. of sulphate of copper to 
12^ gallons of water. 

^Bethel. Impregnation Ivy submitting the wood endwise to a pressure of 
150 to 200 lbs. per square inch, with oil of creosote mixed with bituminous 
matter. 



522 



WOOD, TIMBER, ETC. 



Louis S. Robbins, 1865. Aqueous vapor dissipated by the wood being 
heated in a chamber, the albumen solidified, then submitted to the vapor 
of coal tar, resin, or bituminous oils, which, being at a temperature not less 
than 325°, readily takes the place of the vapor expelled by a temperature 
of 212°. 

Fluids will pass with the grain of wood with great facility, but will not 
enter it except to a very limited extent when applied externally. 

Absorption of Preserving Solution ~by different Woods 
for a !PeriocL of* T" Days. 

Average Pounds per Cubic Foot. 

Black Oak 3.6 I Hemlock 2.6 f Rock Oak 3.9 

Chestnut 3. [Red Oak 3.9 | White Oak 3.1 

Proportion of Water in various Woods. 



Alder (Betula alnus) 41.6 

Ash (Fraxinus excelsior) 28.7 

Birch {Betula alba) 30.S 

Elm (Ulmus campestris) 44.5 

Horse-che&tmit(JEsculus hippocast.) 38. 2 

Larch (Pinus larix) 4S.6 

Mountain Ash (Sorbus aucuparia) . 28.3 

Oak (Quercus robuf) 34.T 



Pine (Pinus Sylvestris L.) 39.7 

Red Beech (Fagus sylvatica) 39.7 

Red Pine (Pinus picea dur) 45.2 

Sycamore (Acer pseudo-platanus) . 27. 

White Oak (Quercus alba) 36.2 

White Pine (Pinus abies dur) 37. 1 

White Poplar (Populus alba) 50.6 

Willow (Salix caprea) 26. 



Comparative Resilience of Timber. 



Ash 1. 

Beech S6 

Cedar 66 



Chestnut 73 

Elm 54 

Fir 4 



Larch S4 

Oak 63 

Pitch Pine 57 



Spruce 64 

Teak 59 

Yellow Pine ... .64 



"Weiglit and. Strength of Oak and. Yellow IPine. 
Weight of a Cubic Foot. 



Age. 


White Oak,Va. 
Round. | Square. 


Yellow 
Round. 


Pine, Va. 
Square. 


Live Oak. 


Green 

1 Year 

2 Years 


64.7 
53.6 
46. 


67.7 
53.5 
49.9 


47.8 
39.8 
34.3 


39.2 
34.2 
33.5 


78.7 
66.7 



In England, Timber sawed into boards is classed as follows : 

6^ to 7 ins. in width, Battens; 8% to 10 ins., Deals; and 11 to 12 ins., 
Planks. 

In a perfectly dry atmosphere the durability of woods is almost unlim- 
ited. Rafters of roofs are known to have existed 1000 years, and piles 
submerged in fresh water have been found perfectly sound 800 years from 
the period of their being driven. 

Distillation. — From a single cord of pitch pine distilled by chemical 
apparatus, the following substances and in the quantities stated have been 
obtained : 



Charcoal 50 bushel?. 

Illuminating Gas about 10n0 cu. feet. 

Illuminating Oil and Tar . . 50 gallons. 
Pitch or Kesin \% barrels. 



Pyroligneous Acid 100 gallons. 

Spirits of Turpentine 20 " 

Tar 1 barrel. 

Wood Spirit 5 gallons. 



Decrease in Dimensions of Timber by Seasoning. 



Woods. Ins. Ins. 

Cedar, Canada 14 to 13^ 

Elm 11 to V)% 

Oak, English 12 to 11 % 

Pitch Pine, North. .. 10x10 to 9%x9% 



Woods. Ins. Ins 

Pitch Pine, South 1S% to 1S^ 

Spruce 8% to 8% 

White Pine, American 12 to 11% 

Yellow Tine, North IS to 17% 



The weight of a beam of English oak, when wet, was reduced by sea- 
soning from 972.25 to 630.5 pounds. 



HEAT. 523 



HEAT. 



Heat, alike to gravity, is a universal force, and is referred to both 
as cause and effect. 

Caloric, is usually treated of as a material substance, though its 
claims to this distinction are not decided ; the strongest argument in 
favor of this position is that of its power of radiation. Upon touching 
a body having a higher temperature than our own, caloric passes from 
it, and excites the feeling of warmth ; and when we touch a body 
having a lower temperature than our own, caloric passes from our 
body to it, and thus arises the sensation of cold. 

To avoid any ambiguity that may arise from the use of the same 
expression, it is usual and proper to employ the word Caloric to sig- 
nify the principle or cause of the sensation of heat. 

Heat is termed Sensible when it diffuses itself to all surrounding bodies ; 
hence it is free and uncombined, passing from one substance to another, 
affecting the senses in its passage, determining the height of the ther- 
mometer, etc., etc. 

The Temperature of a body, is the quantity of sensible heat in it, present 
at any moment. 

Latent Heat, is that which is insensible to the touch of our bodies, and 
is incapable of being detected by a thermometer. 

When a bod}" passes from a solid to a liquid state, or from a liquid 
to a gaseous, a certain portion of its heat becomes insensible, either by 
feeling or \>y a thermometer, and the portion of heat thus combined with 
the body in its new form is termed latent. Hence, when a gas is converted 
into a liquid, or a liquid into a solid, the same quantity of heat is disen- 
gaged, as was held in a latent state by the body before its change of con- 
dition. 

Specific Heat, is that which is absorbed by different bodies of equal 
weights or volumes when their temperature is equal, based upon the law 
that similar quantities of different bodies require unequal quantities of heat 
at any given temperature. It is also the quantity of heat requisite to change 
the temperature of a body any stated number of degrees compared with 
that which would produce the same effect upon water at 60°. 

The quantity of heat, therefore, is the quantity necessary to change the 
temperature of a body by any given amount (as 1°), divided by the quan- 
tity of heat necessary to change an equal weight or volume of water 60° 
by the same amount. 

Note.— Water has greater specific heat than any known body. 

Mechanical power may be expended in the production of heat either by 
friction or compression, and the quantity of heat produced bears the same 
proportion to the quantity of mechanical power expended, being 1 unit 
for the power necessary to raise 1 lb. 772 feet in height. This number of 
7/2 is termed the mechanical equivalent of heat (Joules). 

Capacity for Heat, is the relative power of a body in receiving and re- 
taining heat, in being raised to any given temperature ; while Specific ap- 
plies to the actual quantity of heat so received and retained. 

Radiation of Heat, is the diffusion of heat by the projection of it in di- 
verging right lines into space, from a body having a higher temperature 
than the space surrounding it, or the body "or bodies enveloping it. 

Reflection of Heat, is the passage of heat from the surface of one sub- 
stance to another or into space, and it is the converse of radiation. 



524 HEAT. 

Heat is reflected from the surface upon which its rays fall in the same 
manner as light, the angle of reflection being opposite and equal to that 
of incidence. The metals are the strongest reflectors. 

Communication of Heat, is the passage of heat through different bodies 
with different degrees of velocity. This has led to the division of bodies 
into Conductors and Non-conductors of caloric ; the former includes such 
as metals, which allow caloric to pass freely through their substance, and 
the latter comprise those that do not give an easy passage to it, snch as 
stones, glass, wood, charcoal, etc. 

The velocity of cooling, other things being equal, increases with the ex- 
tent of surface compared with the volume of substance ; and of two bodies 
of the same material, temperature, and form, but differing in volume. 

Transmission of Heat, is the passage of heat through different bodies 
with different degrees of intensity. Gaseous bodies and a vacuum are the 
highest in the order of transmittents. 

Evaporation or Vaporization, is the conversion of a fluid into vapor. 
Evaporation produces cold, because heat is absorbed to form vapor. 

Distillation, is the depriving of vapor of its latent heat. 

Heat is developed by water when it is violently agitated. 

Heat is developed by the percussion of a metal, and it is greatest at the 
first blow. 

The quantities of heat evolved are nearly the same for the same sub- 
stance, without reference to the temperature of its combustion. 

Latkxt Hkat. — A pound of water, in passing from a liquid at 212° to 
steam at 212°, receives as much heat as would be sufficient to raise it 
through 966.6 thermometric degrees, if that heat, instead of becoming 
latent, had been sensible. 

If b% lbs. of water, at the temperature of 32°, be placed in a vessel, communicating 
with another one (in which water is kept constantly boiling at the temperature of 
212°), until the former reaches the temperature of the latter quantity, then let it be 
weighed, and it will be found to weigh G>£ lbs., showing that 1 lb. of water has been 
received in the form of steam through the communication, and reconverted into wa- 
ter by the lower temperature in the vessel. Now this pound of water, received in 
the form of steam, had, when in that form, a temperature of 212°. It is now con- 
verted into the liquid form, and still retains the same temperature of 212° ; but it 
has caused 5jkf lbs. of water to rise from the temperature of 32° to 212°, and this 
without losing any temperature of itself. Now this heat was combined with the 
steam, but as it is not sensible to a thermometer, it is termed Latent. 

The quantity of heat necessary to enable ice to resume the fluid state is 
equal to that which would raise the temperature of the same weight of 
water 140° ; and an equal quantity of heat is set free from water when it 
assumes the solid form. 



Sensible and Latent Heat of Steam. — (Regnault). 



Temp. 



Deg. 
32 

104 
140 
176 



Latent 
Heat. 



Deg. 

1092.6 
1042.2 

1017 
991.8 



Sum of 

Sensible 

and Latent. 



Deg. 

1124.6 
1146.2 
1157 
1167.8 



Temp. 



Deg. 
212 

230 

248 
266 



Latent 
Heat. 



Deg. 
966.6 

952.2 
936.6 
927. 



Sum of 

Sensible 

and Latent. 



Deg. 
1178.6 

1182.2 
1187.6 
1193. 



Temp. 



Deg. 
302 
338 

374 
410 



Latent 

Heat. 



Deg. 
901.8 

874.8 
849.6 
822.6 



Sum of 

Sensible 

and Latent. 



Deg. 

1203.8 
1212.5 
1223.6 
1232.6 



If to a pound of newly-fallen snow were added a pound of water at 172°, 
the snow would be melted, and 32° will be the resulting temperature. 

When a body h fusing, no rise in its temperature occurs, however great 
the additional "quantity of heat may be imparted to it, as the increased 
heat is absorbed in the operation of fusion. The quantity of heat thus 
made latent varies in different bodies. 



HEAT. 



525 



Latent Heat of various Substances for a Unit of Weight. 

Alcohol 3(54° Ether 163° Phosphorus . 9° I Tin 500° 

Ammonia 860° Ice 142.6° Spermaceti. . 14S° Water 9G6.6 

Beeswax 175° Lead 102° Steam 966.6° Zinc 493° 

Bismuth 22° Mercury. . . 157° Sulphur 17° | 

Specific Hkat. — Every substance has a specific heat peculiar to itself, 
whence a change of composition will be attended by a change of its ca- 
pacity for heat. 

The specific heat of a bod}" varies with its form. A solid has a less capaci- 
ty for heat than the same substance when in the state of a liquid; the specific 
heat of water, for instance, being 9 in the solid state, and 10 in the liquid. 

The specific heat of equal weights of the same gas increases as the 
density decreases ; the exact rate of increase is not known, but the ratio 
is less rapid than the diminution in density. 

Change of capacity for heat always occasions a change of temperature. 
Increase in the former is attended "by diminution of the latter, and con- 
trariwise. 

The specific heat multiplied by the atomic weight of a substance will give the 
constant 37.5 as an average, which shows that the atoms of all substances have equal 
capacity for heat. This is a result for which as yet no reason lias been assigned. 

Thus : The atomic weights of lead and copper are respectively 1294.5 
and 395.7, and their specific heats are .031 and .095. Hence 1294.5 x 
.031 = 40.129, and 395.7 x .095 = 37.591. 

It is important to know the relative Specific Heat of bodies. The most conve- 
nient method of discovering it is by mixing different substances together at different 
temperatures, and noting the temperature of the mixture ; and by experiments it 
appears that the same quantity of heat imparts twice as high a temperature to 
mercury as to an equal quantity of water ; thus, when water at, 100° and mercury at 
40° are mixed together, the mixture will be at 80°, the 20° lost by the water causing 
a rise of 40° in the mercury ; and when weights are substituted for measures, the 
fact is strikingly illustrated ; for instance, on mixing a pound of mercury at 40° 
with a pound of water at 160°, a thermometer placed in it will fall to 155°. Thus 
it appears that the same quantity of heat imparts twice as high a temperature to 
mercury as to an equal volume of water, and that the heat which gives 5° to water 
will raise an equal weight of mercury 115°, being the ratio of 1 to 23. Hence, if 
equal quantities of heat be added to equal weights of water and mercury, their 
temperatures will be expressed in relation to each other by the numbers 1 and 23 ; 
or, in order to increase the temperature of equal weights of those substances to the 
same extent, the water will require 23 times as much heat as the mercury. 

Specific Heat of various Substances. (Air as Unity.) 



Air 

Hydrogen . 



Equal 
Volumes. 



f. 

.903 



Equal 
Weights. 



1. 

12.34 



Oxygen . 
Steam. . . 



Equal 
Volumes. 


Equal 

Weights. 


.976 


.885 



Water as Unity. 



Equal 

Weights. 

Water 1. 

Air 267 

Alcohol ... .7 
Bismuth . . .023 

Brick 2 

Charcoal . . .241 
Coke and) 202 

Clay . . S 
Copper 095 



Equal 

Weights. 

Cast iron.. .13 
Carbonic ) 21 g 
acid ... ) 

Ether 517 

Gold 032 

Glass 198 

Hydrogen. 3.405 

Ice 504 

Iron 115 



Equal 

Weights. 

Lead 031 

Lime 217 

Linseed oil. .53 
Mercury . . . .033 
Oxvgeri . . . .218 
Olive-oil ... .31 
Petroleum . .468 
Phosphorus .189 
Platinum . . .034 



Equal 

Weights. 

Steam* 475 

Steel 116 

Sulph. acid . .335 

Silver 057 

Spts. Turp'e .467 

Sulphur 203 

Tin 056 

Woods 54 

Zinc 095 



* Steam tinder a constant volume, that is confined, is estimated at 866. 

Yy 



526 



HEAT. 



Illusteation. — If 1 lb. of coal will heat 1 lb. of water to 100°, —~ = i — r of a 
lb. will heat 1 lb. of mercury to 100°. 

To Compute tlie Temperature of a Mixture of lilce 

Substances. 



WT + wt 



:T 



W (t' — t) 



sw\**&&-vr>. 



: T. W representing the 



W + w ~~ v ' T— t ^ " V W 

weight or volume of a substance of the temperature T, w the weight or volume 
of a like substance of the temperature t, and t' the temperature of the mixture 
W + w. 

iLLrsTBATiON. — When 5 cubic feet of water at a temperature of 150° is mixetf 
with 7.5 cubic feet at 5J° (0, what is the resultant temperature of the mixture? 
5 X 150° + 7.5 X 50° __1125_ 
5 + 7.5 7" 12.5 7 

2. How much water at (T) 100° should be mixed with 30 gallons (w) at 00°, the 
temperature required being S0° ? 

30(30°— 60°) 600 0rt . 

— — -=30 gallons. 



= 90°. 



-80° 



' 20 



To Compute tlie Temperature of a Mixture of XJulilre 

Substances. 



WST + WS* 



:£. 



*(t>- 



-f) w t'(W8 + ws)-wst 



WS + wi 
iv representing the weights, and S and s the specific heat of the substances. 

Illustration To what temperature should 20 lbs. iron (W) be heated to raise 

150 lbs. (w) of water at a temperature (t) of 50° to 60° ? 
.s = l, and S = .114. 
60° (20 X .114 + 150 X 1) —150 X 1 X 50° _ 1636.8 

20 X. H4 ~ 2.28 ' 

Capacity fob Heat. — W T hen a body has its density increased, its capacity for 
heat is diminished. The rapid reduction of air to one fifth of its volume evolves 
heat sufficient to inflame tinder, which requires 550°. 



. Equal 
Volumes. 





Relative 

Equal 
Weights. 


Capacity^ 

Equal 
Volumes. 


for Heat c 

Gold... 

Ice 

Iron . . . 
Lead . . 


/ various 

Equal 
Weights. 


Bodies. 

Equal 
Volumes. 


(Water i 

Mercury 
Silver . 
Tin . . . 
Zinc. . 


is Unity.) 

Equal 
Weights. 


Water. . 
Brass . . 

Copper . 
Glass . . 


1. 
.116 
.114 
.1ST 


1. 

.971 
1.027 

.44S 


.05 
.9 

.126 
.043 


.903 

.993 

.487 


.036 
.082 
.06 
.102 



.833 



The rule for ascertaining by calculation, combined with experiment, 
the relative capacities of different bodies, is as follows : 

Multiply the weight of each body by the number of degrees of temper- 
ature lost or gained b}- the mixture, and the capacities of the bodies will 
be inversely as the products. 

Or, if the bodies be mingled in unequal quantities, the capacities of the 
bodies will be reciprocally as the quantities of matter, multiplied into their 
respective changes of temperature. 

If 1 lb. of water at 150° is mixed with 1 lb. of mercury at 40°, the re- 
sultant temperature is 152°. 

Thus, lxl5G°-152 = 4°, and Ix40°cvl52°= 112°. Hence the capacity 
of water for heat is to the capacity of mercuiy as 112° to 4°, or as 28 to 1. 

Radiation of Caloric. — Radiation is affected by the nature of the 
surface of the body; thus, black and rough surfaces radiate and absorb 
more heat than light and polished surfaces. Bodies which radiate heat 
best absorb it best. 



HEAT. 



521 



Blackened tin. 1. 
Bright lead ... .19 

Clean tin 12 

Copper 12 



Radiating Power of various Bodies. 

Glass .9 

Gold 12 

Ice 85 

India ink SS 



Lamp-black. . 1. 

Lead 45 

Mercury 2 

Polished iron. .15 



Silver 12 

Tin 12 

Water 1 . 

| Writing-paper 1. 



1.T 



Loss by Radiation. 
To Compute tlie Loss of* Heat per Sq.-a.are Foot. 
(T-t) 



dv 



-=: R. T representing temperature of pipe, which is assumed to be ^ 

less than that of the steam ; t temperature of the air ; I length of the pipe in feet; 
d diameter in inches ; v velocity of the heat in feet per second; and R radiation in 
degrees per second. 

Rkflection of Hicat is the converse of Radiation; the one increases 
as the other diminishes. 

Reflecting Power of various Substances. 



Brass 1. 1 Glass 


. .1 
. .0 
. .0 


Silver 
Steel 


9 

7 


Tin 8 


Glass, waxed) „ r Lamp-black. . . 
or oiled j * | Lead 


Tinfoil 85 

" with Mer'y .14 



Conduction or Convection of Hicat. — Air and gases are very im- 
perfect conductors. Heat appears to be transmitted through them almost 
entirely by conveyance, the heated portions of air becoming lighter, and 
diffusing the heat through the mass in their ascent. Hence, in heating a 
room with air, the hot air should be introduced at the lowest part. The 
advantage of double windows for the retention of heat depends, in a great 
measure, upon the sheet of air confined between them, through which heat 
is very slowly transmitted. 

The Convection of heat refers to the transfer and diffusion of heat in a 
fluid mass, by means of the motion of the particles of the mass. 



Relative Conducting Power of various Bodies. 



Bismuth 061 

Cast iron 359 

Copper, rolled. .S45 
Fire-brick Oil 



Fire-clay Oil 

Gold 981 

Iron, wrought . . 436 
Lead 18 



Marble 024 

Mercury 677 

Platinum G8 

Porcelain 012 



Silver 1. 

Steel 397 

Tin 303 

Zinc 641 



Woods (with Water as Unity). 

Water 1. I Apple 28 I Elm 32 j 

Ash 31 | Ebony 22 | Lime 39 j 



Oak . . 
Pine . 



.33 
.39 



Alcohol 0232 



Of Fluids. 
Mercury 1. | 



Water 0357 



Various Substances compared with each other. 



Air 576 

Ashes (wood) . .927 
Beaver's fur. . 1.226 



Charcoal 937 

Cotton 1.046 

Eider-down.. 1.305 



Hare's fur... 1.315 
Lamp-black. 1.117 
Lint 1.032 



Silk, raw.... 1.284 
Silk, sewing.. .917 
Wool 1.11S 



Various Substances compared with Slate.— (J. Hutchinson.) 



Substance. 


Conducting 
Power. 


Cooling 
Power. 


Substance. 


Conducting 
Power. 


Cooling 
Power. 


Slate 

Lath and plaster . 

Asphaltum 

Brick 


1. 

.26 
.45 
.6 


1. 
.75 

1.06 
.97 


Hair and lime. . . 

Pine wood 

Plaster 

Stone 


1.09 

.28 

1 


.38 
.69 
.63 
.95 



Relative Power of various Substances to Transmit Heat. 
All bodies capable of transmitting heat are more or less transparent, 
though their powers of transmitting heat and light are not in the same 
relative proportions* 



528 



HEAT. 



Air 1. 

Alcohol 15 

Crown-glass.. .49 



Flint-glass 67 

Gypsum 2 

Nitric acid. . . *. .15 



Rock-crystal .. 
Rape-seed oil . 
Sulphuric acid 



Sulphuric ether .21 

Turpentine 31 

Water 11 



Weight of Steam Condensed by various Substances per Square Foot per Hour. 



In Air. 


Tempera- 
ture. 


Steam 
in Lbs. 


In Air. 


Tempera- 
ture. 


Steam 
in Lbs. 


Cast iron 


59° 
59° 
59° 
59° 


.36 
.2S 
.35 
.36 


Tin-plate 

In Water. 
Copper 


59° 

72° 


.21 


Copper 

Glass 


Plate iron 


21.5 



Practical Deductions from above and preceding Results. 

Aspkaltum is the best composition for resisting moisture, and, being a 
slow conductor of heat, it is best adapted where economy of heat and dry- 
ness are required. 

Slate is a very dry material, but, from its quick conducting power, it is 
not adapted for the retention of heat. 

Cements. Plaster of Paris and Woods are well adapted for the lining 
of rooms, having low conductive powers, while Hair and lime, being a 
quick conductor, is one of the coldest compositions. 

Fire-brick absorbs much heat, and is therefore well adapted for the lining 
of fire-places, furnaces, etc. ; while contrariwise, Iron, being a high con- 
ductor of heat, is one of the worst of substances for this purpose. 

Common brick is not a very slow conductor of heat ; it is 1.8 times higher 
in the scale than oak wood, and about .027 lower than fire-brick. 



Relative Transmitting Powers of Various Substances. 
Air 1. Glass .67. Alcohol .15. Water .11. Ice 



06. 



The heat which passes through one plate of glass is less subject to ab- 
sorption in passing through a second and a third plate. Of 1000 rays, 451 
were intercepted by 4 plates as follows : 

1st. 381. 2d. 43. 3d. 18. 4th. 9. 

Evaporation proceeds only from the surface of fluids, and therefore, 
other things equal, must depend upon the extent of surface exposed. 

When a liquid is covered by a stratum of dry air, evaporation is rapid, 
even when the temperature is low. 

As a large quantity of heat passes from a sensible to a latent state during 
the formation of vapor, it follows that cold is generated b}- evaporation. 

Fluids evaporate in a vacuum at from 120° to 125° below their boiling 
point. 

Distillation is the depriving vapor of its latent heat, and, though 
it ma} T be effected in a vacuum with very little heat, no advantage in re- 
gard to a saving of fuel is gained, as the latent heat of vapor is increased 
proportionately to the diminution of sensible heat. 

A temperature of 70° is sufficient for the distillation of water in a vessel 
exhausted of air. 

Congelation and Liqukfactiox. — Freezing water gives out 140° of 
heat. All solids absorb heat when becoming fluid. 

The particular quantity of heat which renders a substance fluid is termed 
its caloric of fluidity, or latent heat. 

Fluids boil in a vacuum with less of heat than when under the pressure of 
the atmosphere. On Mont Blanc water boils at 187° ; and in a vacuum 
water boils at 98° to 100°. according as it is more or less perfect. 



HEAT. 



529 



Water may be reduced to 5° if confined in tubes of from .003 to .005 inch 
in diameter : this is in consequence of the adhesion of the water to the sur- 
face of the tube, interfering with a change in its state. It may also be re- 
duced in its temperature below 32° if it is kept perfectly quiescent. 

Effect ixpoii Various Bodies "by Heat. 
Wedgewood's zero is 1077° of Fahrenheit, and each degree = 130°. 
In the designation of degrees of temperature, the symbol -f is omitted when the 
temperature is above ; but when it is below it, the symbol — must be prefixed. 

Degrees. 

Acetification ends 88 

Acetous fermentation begins 78 

Air Furnace 3300 

Ambergris melts 145 

Ammonia boils . . . 140 

Ammonia (liquid) freezes —46 

Antimony melts 951 

Arsenic melts 305 

Beeswax melts. 151 

Bismuth melts 476 

Blood (human), heat of 98 

u freezes 25 

Brandy freezes —7 

Brass melts 1900 

Cadmium melts 6t'-0 

Charcoal burns 800 

Coal Tar boils 325 

Cold, greatest artificial —166 

" greatest natural . . . —56 

Common fire 790 

Copper melts 2548 

Glass melts 2377 

Gold, fine, melts 2590 

Gutta-percha softens 145 

Heat, cherry red 15 10 

" " (Daniell) 1141 

" bright red 1800 

" red, visible by day 1077 

" white 2900 

Highest natural temperature, Egypt 117 

Ice melts 32 

India-rubber and Gutta-percha vul- 
canize 293 

Iron (cast) melts 3479 

u (wrought) melts 2980 

" bright red in the dark 752 

" red hot in twilight 884 



Degrees. 

Lard melts 95 

Lead melts 594 

Mercury boils 662 

" volatilizes 680 

" melts -39 

Milk freezes 30 

Naphtha boils 186 

Nitric Acid (sp. grav. 1.424) freezes —45 

Nitrous Oxide freezes —150 

Olive-oil freezes 36 

Petroleum boils 306 

Phosphorus melts 108 

" boils 560 

Pitch melts 91 

Platinum melts 3080 

Potassium melts 135 

Proof Spirit freezes -7 

Saltpetre melts 606 

Sea-water freezes 28 

Silver, fine, melts 1250 

Snow and Salt, equal parts 

Spermaceti melts 112 

Spirits Turpentine freezes 14 

Steel melts 2500 

" polished, blue 580 

" " straw color 460 

Strong Wines freeze 20 

Sulphur melts 226 

Sulph. Acid (pp. grav. 1.641) freezes —45 

Sulphuric Ether freezes —46 

" u boils 98 

Tallow melts 97 

Tin melts 421 

Vinegar freezes . . . 28 

Vinous fermentation 60 to 77 

Water in vacuo boils 98 

Zinc melts 740 



Boiling Points of 'Vario^^s 3n.-u.ids. 



Degrees. 

Ether 96 to 104 

Alcohol, sp. grav. 813 173.5 

Nitric Acid, " 1.5 210 

" " 1.42 248 

Sea Salt 224.3 

Common Salt 226 

Sulphuric Acid, sp. grav. 1.848. . . 600 
" " 1.3 ... 240 



Degrees. 

Rectified Petroleum 31 6 

Oil of Turpentine £-04 

Phosphorus 554 

Sulphur 570 

Linseed Oil 640 

Sweet Oil 412 

Sea-water 213.2 

Water, distilled 212 



Volume of several Liquids at tlieir Boiling 3?oint. 

Steam. I Steam 

1 Water 1700 1 Ether 298 

1 Alcohol 528 I 1 Turpentine 193 

Yy* 



530 



HEAT. 



Water may be heat. 



For boiling point of salt water, see Water, p. 541. 
ed in a Digester to 400° without boiling. 

Boiling Points corresponding to Altitudes of the Barometer between 26 and 

31 Inches. 



Barom. 


Boiling Point. 


Barom. 


Boiling Point 


| Barom. 


Boiling Point. 


Barom. 


Boiling Point 


26. 

20.5 

or 


204.91° 
205.79° 
200.07° 


27.5 

28. 
28. 5 


207.55° 

208.43° 
209.31° 


29. 

29.5 

30. 


210.19° 
211.07° 
212.° 


30.5 
31. 


212.83° 
213.76° 



IVLelting Point of Alloys. 



Lead 2, Tin 3, Bismuth 5 212° 



1, 
1, 



Lead 2, 



3, 
4, 
1, 
2, 
3.. 



21o° 
240° 
280° 
330° 
334° 



Tin 8, Bismuth 1 392° 

Lead 2, Tin 1 (solder) 475° 

" 1, " 2 (soft solder) 360° 

Zinc 1, " 1 399° 

Leadl, " 1 368° 

" 1, " 1, Bism'h 4, Cadm'm 1 155° 



For Alloys and Fusible Compounds, see pp. 627 62S. 

iFrigorifLc nVEixtnres. 



^duld. 01 ' Temperature falls 



Nitrate of Ammonia 1 part \ ^ From 5QO ^ 

Water 1 " j 



71° From 50° to -21° 



Phosphate of Soda 9 parts 

Nitrate of Ammonia 6 

Dilute Nitric Acid 4 

Sulphate of Soda J P*?* 8 }. ... 50° .. .. From 50" to 0° 

Muriatic Acid 5 ) 

Snow... 2parts) >53 o From -15° to -68° 

Muriate of Lime 3 " j 

Snow. ....... . ........ 8 parts) ^ Q From _ 6 g to _ 90 o 

Dilute Sulphuric Acid 10 " J 

LINEAL EXPANSION OR DILATATION OF SOLIDS. — {Faraday.') 

At 212°, the length of the bar at'32°= 1. 



Antimony 1 001 OS 

HSs'niuth 1.00139 

Urns* 1 00191 

Brick, common 1 00055 

Cast iron 1.00111 

Cement 1.00144, 

Copper 1.00175 



Fire-brick 1.0175 

Glass 1.00085 

Gold 1.0015 

Granite 1.00-79 

Lead 1.00284 

Marble 1.0011 

Platinum 1.00095 



Sandstone 1.00174 

Silver 1.00201 

Slate 1.00114 

Steel 1.00119 

Tin 1.002 

Wrought iron 1.00126 

Zinc 1.00294 



Rankine gives rates somewhat higher than these. 
1.00183 : Brass, 1.00216 ; Bronze, 1.0018. 



Thus, for Copper, 



Expansion, and Dilatation of Gases and 3TTu.ids. 
At 212°, the volume at 32° = 1. 



Air 1.305 

Hydrogen 1.300 

Steam 1.305 



Water 1.04775 

do. sea 1.020 

Alcohol 1.1112 



Mercury 1.01S15 

Oil 1.08 

Oxygen 1.367 



To Compute the Expansion of a Substance in Parts of its Length. 

Divide 1. by the decimal given in the above Table (omitting the unit of 1), and the 
quotient will give the proportion. 

Llluxtration.—A. rod of copper will expand 1-^.00175 = 571.4 = 1 part or foot in 
571.4 parts or feet for ISO . 



HEAT. 



531 



To Compute the Expansion for any Number of Degrees under 180°. 

Divide the decimal given in the preceding Table by the number of degrees, and the 
quotient will give t e expansion for the decrees required. 

Illustration.— A rod of copper will expand for each degree .00175 -f- 180 = .00000972 
= 972 parts in 100 000 000, or 1 part in 10JB SS0. 



To Compute the Volume of a Grasr, at any- Temperature, 
its Volume at 32° being Isno-wn, and the Pressure toeing 
constant. 

Rule. — Divide the difference between the number of degrees in temper- 
ature and 32° by 490. Add the quotient to 1 if it is above 32°, and sub- 
tract 1 if it is below 32°. Multiply the volume of the gas at 32 c by the 
resulting number, and the product will give the number required. 

Example. — What volume will 1000 cubic feet of air at 32° acquire by being heated 
to 1000° ? 
1000°— 32°= 968°, which -4- 490 — 1.9755, to which add 1 = 2.9755. 
Then, 1000x2.9755 = 2975.5 cubic feet. 



Expansion of .A.ir.— (Dai/ton.) 



Temp. 


Expan- 
sion. 


Temp 


32° .. 


1. 


40° . 


33° . 


1.002 


45° 


34° . 


1.004 


50° 


35° . 


1.007 


55° 



Expan- 
sion. 
, 1.021 
. 1.032 
. 1.043 
. 1.055 



60° .. 1.0G6 

65° .. 1.077 

70° .. 1.089 

75° .. 1.099 



Temp. 

80° . 
85° . 
90° . 
95° . 



Expan- 
sion. 
. 1.110 
. 1.121 
. 1.132 
. 1.142 



Temp. 

100°, 
200°. 
212°, 

302°. 



Expan- 
sion. 
. 1.152 
. 1.854 
. 1.376 
. 1.558 



Temp. 

392° 

482° 
6S0° 
772° 



Expan- 
sion. 
. 1.739 
. 1.912 
. 2.028 
. 2.312 





Expansion of* Water.- (Dalton.) 




Temperature. 


Expansion. 


Temperature. 


Expansion. 


Temperature. 


Expansion. 


12° 


1.00236 


82° 


1.00312 


152° 


1.01934 


22 


1.00090 


92 


1.00477 


162 


1.02245 


32 


1.00022 


102 


1.00672 


172 


1.02575 


*40 


1. 


112 


f 1.00880 


182 


1.02916 


52 


l.ono2l 


122 


1.01116 


192 


1.03265 


62 


1.00083 


132 


1.01367 


202 


1.03634 


72 


1.00180 


142 


1.01638 


212 


1.04012 



Hence, at 72°, water expands ' = 555. 55th part of its original bulk. 
.0018 

The ratio of Expansion for Solids and Liquids increases with the temperature ; 
that of the Gases is uniform for all temperatures. 

To Compute the Temperature to which a Substance of 
a given. Length or Dimension, must "be Submitted! or 
Reduced, to give it a Greater or Less Length or "Vol- 
ume by Expansion or Contraction. 

Linkal. — When the Length is to be increased. 

-Yrj--\-t = T. L representing the whole length of the substance when in- 
creased, and I the primitive length of it in like denominations. T and t the 
temperatures of L and /, and C the expansion of the substance for each de- 
gree of heat. 

Illustration.— A -copper rod at 32° is 100 feet in length; to what temperature 
must it be subjected to increase its length 1.1633 ins. ? 

The expansion for a unit of length of copper for 180° is .001745. Hence . 001745 -r- 
180 = .000009694 for each degree. 



100 X 12 -f- 1.1633 — 100 X 12 



.000009694 X 100 X 12 



-f 32: 



1.1633 
".011633 



-f- 32 = 132°. 



* Water is held to be at its greatest density when at 39.83.° 



532 



HEAT. 



When the Length is to be reduced. 



L-l 

'Vi 



T=t 



/ ( l + C(T- )=L;and 1 + (C(T _ or ^ 

Illustration — Take the elements of the preceding ca?e. Then, to ascertain L, 

1-200 X (l + .000009094 X (132— 32) ) — 1200 X 1 + .0009694 = 1200 X 1.00D9694 = 
1201.1633 ins 

To Compute the Expansion of Fluids in Yolnme. 

Rulic— Proceed bj r the preceding formulae for computing the length of 
a substance. Substitute V and v for the volume, instead of L and /, the 

length. 

Illustration. — A closed vessel contains 6 cubic feet of water at a temperature of 
40° ; to what height will a column of it rise in a pipe 1.152 ins. in area, when it is ex- 
posed to a temperature of 130° ? 

1.152 ins. - .00S square feet. C for water — .00023325. 

6 (l + ."00023325(130 — 40)) = 6.12595, and C ' 12 ^ 9 Q g~ 6 — 15.744 lineal feet. 



TEMPERATURE BY AGITATION. 

Results of Experiments with Water inclosed in a Vessel and violently Agitated. 
Temperature of Air, 60.5° ; of Water, 59. 5°, 



Duration 
of Agitation. 



Increase 
of Temperature 



Hours. 

.5 
1. 



10. ° 
14.5° 



Duration 

of Agitation. 



Hours. 

2 
3 



Increase 
of Temperature. 



19.5° 

29.5° 



Duration 
of Agitation 



Hours. 

5 
6 



Increase 
of Temperature. 



39.5° 
42.5° 



IVXean Temperatures dfvarious I^ocalities. 



London 51° 

Edinburgh... 41° 



I Home 60° 

I Equator 82° 



Poles 

Torrid Zone. . 



-13° 

75° 



I Polar Regions. . 36° 
Globe 50° 



Line of Perpetual Congelation, or Snow Line. 



Latitude. 


Height. || Latitude. 


Height. 


10° 
20° 


Feet. 

14764 30° 
13478 || • 40° 


Feet. 
11484 
9000 



Latitude. 


50° 
6j° 



Height. 



Feet. 

63,4 
i.818 



Latitude. | Height. 



70° 
80° 



Feet. 
1278 
451 



At the Equator it is 15260 feet; at the Alps, 8120 feet; and in Iceland, 
3084 feet. At the Polar Regions ice is constant at the surface of the earth. 

To Reduce the Degrees of a Fahrenheit Thermometer to those of Reaumur 
and the Centigrade, and contrariwise. 

Fahrenheit to Reaumur. — If above zero. • Subtract 32 from 

the number of degrees; multiply the remainder by 4, and divide the 
product by 9. 

Thus, 212° - 32° = 180°, and 180° x4-f9 = 80°. 

If below zero. — Add 32 to the number of degrees ; multiply 

the remainder by 4, and divide the product by 9. 

Tims, - 40° +32° = 72°, and 72 c x 4 -4- 9 = - 32°. 

Reaumur to Fahrenheit. — Multiply the number of degrees by 9, and di- 
vide the product Iry 4. Then, when thej' are above the freezing point, add 
82 to the quotient, and when the}' are below, subtract 32. 

Thus, 80° x 9 -=- 4 -- 180, and 180 + 32 = 212°. 
44 -32° x 9-^4 = 72, and 72-32 = 40°. 



HEAT. 533 

Fahrenheit to Centigrade.— If above zero. Subtract 32 from 

the number of degrees ; multiply the remainder by 5, and divide the 
product by 9. 

Thus, 212° - 32° x 5^- 9 = 180 x 5 + 9 = 100°. 

If below zero. — Add 32 to the number of degrees ; multiply 

the remainder by 5, and divide the product by 9. 

Thus, - 40° + 32° x 5 -f- 9 = 72 x 5 '4- 9= - 40°. 

Centigrade to Fahrenheit. — Multiplj T the number of degrees by 9, and di- 
vide the product by 5. Then, when they are above the freezing point, add 
32 to the quotient, and when they are below, subtract 32. 

Thus, 100° x9-r5 = 180, and 180 + 32 = 212°. 
4t - 10° X 9 4- 5 = 18, and 18-32 = 14°. 

Reaumur to Centigrade. — Multiply by .25, and add the product ; or di- 
vide bj T 4, and add that product. 
Thus, 80° X .25 = 20, and 20 + 80 = 100°. 

Or, 80° -f- 4 = 20, and 20 + 80 = 100°. 

Centigrade to Reaumur. — Divide by 5, and subtract the product. 
Thus, 100° -h 5 = 20, and 20-100 = 80°. 

Corresponding Degrees upon the Three Scales. 



Fali. 


Cent. 


f Reaum. i 


Fah. 


| Cent. '. 


1 Reaum. 


Fah 


Cent 


Reaum. 


212 


100 


1 80 1 


32 


\ o 


1 o 


-40 


-40 


1 -32 



Temperature of the Earth. — The ratio of increase in its temperature is 
directly as the depth from the surface, being about 1° for every 65 feet. 

WARMING BUILDINGS AND APARTMENTS. 

By Low Pressure Steam (1% to 2 lbs.) or Hot Water. 

One square foot of plate or pipe surface will heat from 40 to 100 cubic 
feet of inclosed space to 75° in a latitude where the temperature ranges 
from —10°, or 10° below zero. 

The range from 40 to 100 is to meet the conditions of exposed or corner 
buildings, of buildings less exposed, as the intermediate ones of a block, 
and of rooms intermediate between the front and rear. 

As a general rule, 1 square foot will heat 75 cubic feet of air in outer or 
front rooms and 100 in inner rooms. 

By High Pressure Steam. 

When steam at a pressure exceeding 2 lbs. per square inch is used, the 
space heated by it will be in proportion to its increase of temperature above 
that pressure, less the increased radiation of heat in its course to the place 
of application. 

One cubic foot of water evaporated is required for ever}- 2000 cubic feet 
of inclosed space. 

By Hot Water of Low or High Temperatures. 

(P — t) (T — t) 

7 p _ rp X .005 V— square feet of surf ace ofplcte or pipe. P rep- 
resenting temperature of plate or pipe, T and t the requv, ed temperature and 
that of the external air, and V the volume or cubic feet of inclosed space. 

Ventilation. 
Each person requires from 3 to 4 cubic feet of air per minute. Win- 
dows, as ordinarily constructed, will admit about 8 cubic feet per minute. 



534 



LIGHT. 



LIGHT. 






Light is similar to Heat in many of its qualities, being emitted in the 
form of ra} r s, and subject to the same laws of reflection. 

It is of two kinds, Natural and Artificial; the one proceeding from the 
Sun and Stars, the other from heated bodies. 

Solids shine in the dark only at a temperature from 600° to 700°, and 
in daylight at 1000°. 

The Intensity of Light is inversely as the square of the distance from the 
luminous body. 

The Velocity of the Light of the Sun is 192 500 miles per second. 

The Standard of Intensity or of comparison of light between different 
methods of Illumination is a Sperm Candle " short 6," burning 120 grains 
per hour. 

Loss of Light by Use of Shades.— (F. H. Storer.) 



American enamelel 

Crown 

Crystal plate 

English " 

Porcelain transparency . . 



Thick- 
ness. 



Ins. 
% 



PerCent. 

51.23 

13. OS 

8.61 

6.15 

97.68 



Window, double, Eng. . . 
" " German. 
u single, u 
" u ground.. 
u green 



Thick- 
ness. 



X 



Loss. 



PerCent 
9. 3 J 

13. 
4.27 
65 75 

81.95 



ILLUMINATION.— GAS, LAMPS, AND CANDLES. 

Comparison of several Varieties of Lamps, Fluids, and Candles with Coal* 
Gas, deduced from Reports of Com. of Franklin Institute, and of A. Fry e, 
3t.D.,etc.,etc. 



Lamp and Fluid. 



Intensity 


Ratio 


Light 


Time of [ 

Burning 

1 Pint of 

Oil. 


of 
Light. 


of Cost 
per Hour. 


at Equal 
Cost. 








Hours. 


1.75 


.57 


3.08 


9.31 


2.15 


1.22 


1.8 


6.32 


1.22 


.86 


1.35 


9.87 


.69 


.67 


1.2 


14.6 


.v / 


.76 


.97 


11.3 


1. 


1. 


1. 





1.15 


1.25 


.93 


6.75 


1.76 


1.09 


1.55 


8.42 



Relative 

Costs for 
Equal 
Lights. 



Camphene 

Carcel, Sperm oil, maximum 
" " mean 
" " minimum. 
" Lard oil 

Gas 

Semi-solar, Sperm oil 

Solar kt 



$ Cts. 
.32 
.56 
.74 
.83 

1.03 

1. 

1.07 
64 



Diaphane 

Palm oil 

Spermaceti, short 6's 

Tallow, short 6's, single wick 

double " .... 

Wax, short 6's 

" long 4's 





Intensity 


Light 


Cost with 


Burns. 


of 


at Equal 


Equal 




Light.} 


Costs. 


Light. 


Hours. 








6.6 


.7 


.5 


2.08 


6.6 


.7 


.77 


1.32 


8. 


.8 


.54 


2.16 


6. 


.58 


.85 


1. 


5.5 


1. 


1. 


1.46 


9. 


.8 


.61 


1.96 


13. 


— 


— 


— 



Cost com- 
pared with 
Gas for 
•Equal Light. 

15.1 
10.5 
16.2 
7.5 
7.1 
14.4 



* City of Philadelphia. 

f Compared with a fish-tail jet of Edinburgh gas, containing 12 per cent, of condensable matter 
and consuming 1 cubic foot per hour. 



LIGHT. 



535 



Dimensions, Consumption, and Comparative Intensify of Light of Candles. 



No. in a 
Pound. 


Diame- 
ter. 


Length. 




Inch. 


Inches. 


3 


1. 


12 


3 


% 


15 


6 


.8 


9 


3 


.9 


15 


4 


.8 


WK 


6 


.84 


§X 


3 


1. 


m 


3 


.9 


15 


4 


.8 


13X 



Consumption 
per Hour. 



Light compared 
with Carcel. 



Wax 

u 

Spermaceti . 

{< 
Tallow 



Grains. 
135 

156 

204 



.09 

.09 

.06 to .08 



The illuminating power of coal gas varies from 4.4 to 1.6 times that of 
a tallow candle 6 to a pound ; the consumption being from 2.3 to 1.5 cubic 
feet per hour, and the specific gravitj" from .58 to .42. 

The higher the flame from a burner the greater the intensity of the light, 
the most effective height being 5 inches. 

English Cannel coal produces the greatest quantity and the best quality 
of coal gas. Scotch Parrot coal is next in order. 

Water absorbs its own volume of carbonic acid gas. 

The greater the proportion of hydrogen, and the less oxygen and sul- 
phur, the better the coal is adapted for generating gas. 

Pine-wood gas will give when burning 4.6 cubic feet per hour — a light equal to 
1S.3 sperm candles per hour; and Oak-wood gas, under like- conditions, will give a 
light equal to 19. IT candles. 

Philadelphia City gas is equal to IT. 5 sperm candles. 24T2 lb3. pine wood pro- 
duced 123S0 cubic feet of gas, 46.8 bushels charcoal, and 4.5 gallons coal tar. 

A mean of Coal and Mineral Oils gave an expenditure of 1.6 galls, oil for 1000 cu- 
bic feet of gas at .6 the intensity of the light, and 2.6 galls, gave an equal light of 1000 
cubic feet of gas per hour. l.S galls. Burning Fluid gave .15 the intensity of a gas- 
light for 1000 cubic feet of gas, and for equal light there was required 11. T galls, for 
1U00 cubic feet of gas. 

In the combustion of oil in an ordinary lamp, a straight or horizontally cut wick 
gives great economy over an irregular cut wick. 

Relative Intensity of Hjiglit from different Candles. 

Burned during Flow Giving equal Light to Amount burned 
of 1000 cub. feet of Gas. 1000 cub. feet of Gas. for equal Light. 



Paraffine 

Sperm 

Adamantine . . 
Tallow 



.098 
.005 
.10S 
.0T4 



Lbs. 
3.5 
3.9 
5.1 
5.1 



35 5 
41.1 

4T;2 
53. S 



103 

120 
137 
155 



Intensity- of Light -witn Equal Yolumes of Gras from 
different Burners. 





Expenditure in 


Cubic Feet per 


At most 






Hour. 




Effective 


Burners. 








Height of 
Flame. 




1. 


2. 


3. 


4. 


Single jet, 1 foot = candles* 


2.6 





— 


— 


100. 


Fish-tail No. 3, 1 foot = candles. 


3.5 


4. 


4.2 





138. 


Bats'-wing 1 u = " 


3. 


4.1 


4.3 


4.5 


135. 


Argand, 16 holes, 1 " = " 


.32 


1.9 


3.3 


3.8 


— 


Argand, 24 holes, 1 " = M 


— 


— 


— 


— 


183.5 


Argand, 28 holes,l " =r " 


.34 


2.3 


3.5 


5.8 


188'. 


Argand, 42 holes,! " = " 


— 


— 


— 


— 


182.8 



* Spermaceti candle burning 120 grains per hour. 



536 



LIGHT. 



Volume of G-as in. Ctfbic Feet reqnired to 3Prod.ia.ee the 

Liglxt ofOne Spermaceti Candle. 





Expenditure 


n Cubic Feet per Hour. 






.4. | .6. 


.9. | 1.5. 


2.6. 


Argand 26 holes 


.37 
.37 


.28 
.28 


.2 
.2 


.8 
.17 


.3 


Fish-tail No. 1* 


" No.2f 


.16 



Relative Intensity, Consumption, and. Cost of various 
Modes of Illiamination. 

Oil at 11 cents per lb. Tallow at 14 cents per lb. Wax at 52 cents 
per lb. Stearine at 32 cents per lb. 100 cubic feet coal gas at 14 cents, 
100 cubic feet of oil gas at 52 cents. 



Illuminator. 


Intensity. 


Consumption 
of Material 
per Hour. 


Illumination. 

Carcel Lamp 

= 100. 


Actual 

Cost per 

Hour. 


Cost per Hour 
for equal 
Intensity. 


Carcel Lamp 

Lamp with inverted 
reservoir 


100. 

90. 
31. 
6.65 
14.6 
14.4 
10.7 
16. 

127. 
127. 


42. 

43. 

26.7 
8. 
9.6 
9.3 
8.5 
8.8 

Cubic feet. 

8.7 
2.4 


100. 

57.8 

48.7 

33.6 

61.6 

66.6 

54. 

67.5 


Cents. 
.87 

.89 
.56 
.16 
.92 
.59 
.25 
.89 

1.16 
1.26 


.87 

.99 
1.78 
2.49 
6.31 
4.13 
2.34 
5.7 

.91 


Astral Lamp 

Petticoat Lamp 

Wax Candle 6 to lb. 
Stearine " 5 " 
Tallow " 6 " 
Sperm " 6 " 

Coal Gas 


Oil Gas 


.73 



1000 cubic feet of 13-candle coal gas is equal to 7.5 gallons sperm oil, 52.9 lbs. mold 
candles, and 44.6 lbs. sperm candles. 



A retort produces about 600 cubic feet of gas in 5 hours with a charge 
of about X% cwt. of coal, or 2800 cubic feet in 24 hours. 

In estimating the number of retorts required, )^th should be added for 
being under repairs, &c. 

Purifiers. — Wet purifiers require 1 bushel of lime mixed with 48 bushels 
of water for 10000 cubic feet of gas. 

Dry purifiers require 1 bushel of lime to 10000 cubic feet of gas, and 1 
superficial foot for every 400 cubic feet of gas. 

A cubic foot of good gas, from a jet ■£% of an inch in diameter and height 
of flame of 4 inches, will burn for 65 minutes. 

Internal lights require 4 cubic feet, and external lights about 5 cubic 
feet per hour. When large or Argand burners are used, from 6 to 10 cubic 
feet will be required. 

The pressure with which gas is forced through pipes should seldom exceed 2*£ 
inches of water at the Works, or the leakage will exceed the advantages to be ob- 
tained from increased pressure. 

When pipes are laid at an inclination either, above or below the horizon, a correc- 
tion will have to be made in estimating the supply, by adding or deducting -^^ of 
an inch from the initial pressure for every foot of rise or fall in the length of the pipe. 

* Fullv spread at 85 inch pressure at 1.4 cubic feet per hour. 
+ • « i9 «< 2.4 " " 



LIGHT. 



537 



In Winter the average of duration of internal lights per day is 5.08 
hours ; in Summer it is 2.83 ; in Spring it is 3.41 ; and in the Fall, 4.16. 

Street-lamps in the city of New York consume 3 cubic feet of gas per 
hour. In some cities 4 and 5 cubic feet are consumed. Fish-tail burners 
for ordinary coal gas consume from 4 to 5 cubic feet of gas per hour. 

The standard of gas burning is a 15-hole Argandlamp, internal diameter 
.44 inch, chimney 7 inches in height, and consumption 5 cubic feet per 
hour, giving a light from ordinary coal gas of from 10 to 12 candles, with 
Cannel coal from 20 to 24 candles', and with the rich coals of Virginia and 
Pennsylvania of from 14 to 16 candles. 

In Philadelphia, with a fish-tail burner, consuming 4.26 cubic feet per 
hour, the illuminating power was equal to 17.9 candles, and with an Ar- 
gand burner, consuming 5.28 cubic feet per hour, the illuminating power 
was 20.4 candles. 

Gas, which at the level of the sea would have aValue of 100, would have 
but 60 in the city of Mexico. 

Loss of Light by Glass Globes. 
Clear glass, 12 per cent. \ Half ground, 35 per cent. | Full ground, 40 per cent. 

Resin Gas. — Jet ^, flame 5 inches, 1}£ cubic feet per hour. 

1 Chaldron Newcastle coal, 3136 lbs., will furnish 8600 cubic feet of gas 
at a specific gravity of .4, L454 lbs. coke, 14.1 gallons tar, and 15 gallons 
ammoniacal liquor. 

Volumes of G-as obtained, from a Ton of* Coal, liesin, etc. 



Boghead Cannel. ,.\\ 
Wigan Cannel. \ 

Cannel -j 

Cape Breton, " Cow\ 

Bay," etc j" 

Cumberland 

English, mean 

Newcastle i 



Cubic 
Feet. 


Specific 
Gravity. 


13 334 

15426 

8960 

15 000 


.42 
.73 
.42 

.58 


9500 


- 


11 000 

9 500 

10 000 


.24 

,4 

.5 



Oil and Grease 

Pictou and Sidney. . . 

Pine wood 

Pittsburg . . 

Resin ...... i 

Scotch ■! 

Virginia . 



Western. 
Walls-end ... 



Cubic 


Specific 


Feet. 


Gravity. 


23 000 


.67 


8 000 


_ 


11800 


.66 


9 520 


_ 


15600 


.66 


10 300 


.55 


15000 


.64 


8960 


_• 


9 500 


_ 


12 000 


.42 



Australian Coal is superior to Welsh in the furnishing of gas. 
1 lb. Peat will supply gas for 1 hour's lia;ht. 1 ton Wigan Cannel has 
produced coke, 1326 lbs. ; gas, 338 lbs. ; tar, 250 lbs. ; loss, 326 lbs. 

To Compute the "Volumes of Gas discharged through 
I?ipes.— (Clegg.) 



1350 d? 



Ihd 
Vi7 



- V. d representing diameter of the pipe and h the height 



of the water in inches, denoting the pressure upon the gas, I length of pipe in 
yards, g specific gravity of the gas, and V the volume in cubic feet per hour. 
g may be assumed for ordinary computation at .42. 

Volumes of* G-as discharged per Hour under a Pressure 
ofHalfan Inch. ofWater, Specific Gravity of Gas .48. 



Diameter 
of Opening. 


Volume. 


Diameter 
of Opening. 


Volume. ILPA^T 6 ? 6 : 
01 Opening. 


Volume. 


Ins. 


Cubic Feet. 

-so 

321 


Ins. 


Cubic Feet. Ins. 

723 \.y z 
12ST U l.X 


Cubic Feet. 
1625 
2010 



Diameter 
of Opening 



Ins. 
5. 



Volume. 



Cubic Feet 

28S5 

46150 



Zz 



538 



LIGHT. 



GAS PIPES. 

Flow of G-as ixx Pipes. 

The flow of Gas is determined by the same rules as govern that of the 
flow of Water. The pressure applied is indicated and estimated in inches 
of water. 

Diameter am.d. Length of Gas-pipes to transmit given 

Volumes of Gas to Branch. IPipes.— (Dr. Uke.) 



Volume 
per Hour. 


Diam- 
eter. 


Length- 


Volume 
per Hour. 


Diam- 
eter. 


Length. 


Volume 
per Hour. 


Diam- 
eter. 


Length. 


Cub. Feet. 

50 

250 

500 

700 


Ius. 
.4 
1. 

1.9T 
2.65 


Feet. 
100 
200 
600 

1000 


Cub. Feet. 
1000 
1500 

2000 
2000 


Ins. 
3.16 
3.87 
5.32 
6.33 


Feet 
1000 
1000 
2000 
4000 


Cub. Feet. 
2000 
6000 
6000 
8000 


Ins. 

7. 

7.75 

9.21 

8.95 


Feet. 
6000 
1000 
2000 
1000 



The volumes of gases of like specific gravities discharged in equal times 
by a horizontal pipe, under the same pressure and for different lengths, 
are inversely as the square roots of the lengths. 

The velocity of gases of different specific gravities, under like pressure, 
are inversely as the square roots of their gravities. 

By experiment, 30000 cubic feet of gas, specific gravity of .42, were dis- 
charged in an hour through a main 6 ins. in diameter and 22.5 feet in length ; 
and 852 cub. feet, specific gravity .398, were discharged, under a head of 
3 ins. of water, through a main 4 ins. in diameter and 6 miles in length. 

The loss of volume of discharge by friction, in a pipe 6 ins. in diameter 
and 1 mile in length, is estimated at 95 per cent. 

In distilling 56 lbs. of coal, the volume of gas produced in cubic feet 
when the distillation was effected in 3 hours was 41.3, in 7 hours 37.5, in 
20 hours 33.5, and in 25 hours 31.7. 

For Rules and Results of Velocities, etc., see Appleton's Dictionary of Mechanics 
and Engineering, and Hughes's Treatise on Gas Works. London. 

GAS ENGINES. 

In the Lenoir engine, the best proportions of air and gas are, for common 
gas, 8 volumes of air to 1 of gas, and for cannel gas, 11 of air to 1 of gas. 

The time of explosion is about the 27th part of a second, and the result- 
ant temperature 2474°. 

An engine, having a C3 T linder 4% ins. in diameter and 8% ins. stroke of 
piston, making 185 revolutions per minute, develops a power of half a 
horse. 







Services for Lamps 








Lamps. 


Length 


Diameter 


Lamps. 


Length 


Diameter 


Lamps. 


Length 


Diameter 


from Main. 


of Pipe. 


from Main. 


of Pipe. 


from Main. 


of Pipe. 


No. 


Feet. 


Ins. 


No. 


Feet. 


Ins. 


No. 


Feet. 


Ins. 


2 


40 


X 


10 


100 


% 


25 


180 


\x 


4 


40 


X 


15 


130 


1 


30 


200 


VX 


6 


50 


% 


20 


150 


IK 









Average Composition of London Gas by Voltime. 



Aqueous vapor 
Carbonic acid . 
Carbonic oxide 
Hydrogen 



Common 


Cannel 


Gas. 


Gas. 


2. 


2. 


.7 


.1 


7.5 


6.8 


46. 


27.7 



Light carb'd hyd. . . 

Nitrogen 

Olefiant, etc #< 



Common 
Gas. 



39.5 
.5 

3.8 



Cannel 
Gas. 



50. 

.4 
13. 



LIGHT. — WATER. 



539 



Combustion, Temperature, and. Power of Gases. 



Alcohol 

Camphene 

Cannel gas 

Carbon 

Carbonic oxide 

Common coal gas. 

Ether 

Hydrogen 

Marsh gas 

defiant gas 

Paraffine 

Rape oil 

Sperm oil 

Spermaceti 

Stearine 

Sulph. hydrogen. . 

Wax 

Wood spirit.. .... 



Per Pound 
of Gas. 


Water Heated 1 Degree. 


Temp, of 

Combustion. 


Air 












Oxygen 


Per lb. of 


Per Cub. Ft. 


Open 


1 Degree. 


used. 


Material. 


of Gas. 


Flame. 




Cub. Feet. 


Lbs. 


Lbs. 


Deg. 


Cub. Feet. 


24.6 


12929 


1597 


4831 


— 


38.9 


18573 


7134 


5026 





31. 


20140 


760 


5121 


365S5 


31. 


14544 


— 


3026 





6.7 


4825 


320 


5358 


15403 


8T.5 


21060 


650 


. 5228 


31299 


30.9 


13219 


3217 


5150 





93.4 


62080 


329 


5744 


15837 


47.2 


23543 


996 


4762 


47946 


40.5 


21344 


1585 


5217 


76290 


40.5 


21327 


— 


5239 


— 


38.7 


17752 


— 


50S7 


— 


38.7 


17230 





4937 


— 


37. 


17589 





4413 


— 


34.4 


1S001 





5095 


Lt 


16.7 


7414 


671 


4388 


— 


37.7 


15809 


— 


4122 


— 


25.3 


9547 


819 


4641 


— 



Ifcegu.lati.ori. of the Diameter and. Extreme Length, of Tub- 
ing and. Number of Burners permitted. 



Diameter 




Number 


Capacity 


Number 


Diameter 




Number 


Capacity 


Number 


of 


Length. 


of 


of 


of 


of 


Length. 


of 


of 


of 


Tubing. 




Burners. 


Meters. 


Burners. 


Tubing. 




Burners. 


Meters. 


Burners. 


Ins. 


Feet, 




Light. 




Ins. 


Feet. 




Light. 




5r 


6 


1 


3 


6 


1 


70 


35 


45 


90 


% 


20 


3 


5 


10 


IX 


100 


60 


60 


120 


% 


30 


6 


10 


20 


m 


150 


100 


100 


200 


% 


40 


12 


20 


40 


2 


200 


200 






% 


50 


20 


30 


60 













Temperature of Gases. — The combustion of a cubic foot of common gas 
will heat 65 gallons of water 1°. 



WATER. 
Fresh Water. The constitution of it by weight and measure is 

By Weight. By Measure. I By Weight. By Measure. 

Oxygen 88.9 1 j Hydrogen... 11.1 2 

One cubic inch of distilled water at its maximum density of 39°. 83, 
the barometer at 30 inches, weighs 252.6937 grains, and it is 828.5 
times heavier than atmospheric air. 

A cubic foot weighs 998.068 ounces, or 62.37925 lbs. avoirdupois. 

Note — For facility of computation, the weight of a cubic foot of water is taken at 
1000 ounces and 62.5 lbs. 

2. By the British Imperial Standard, the weight of a cubic foot of water at 62°, 
the barometer at 30 ins. =998. 224 ounces. 

At a temperature of 212° its weight is 59.675 lbs. Below 39°. 83 
its density decreases, at first very slow, but progressing rapidly to the 
point of congelation, the weight of a cubiofootof ice being but 57. 25 lbs. 



540 WATER. 

It expands. 089 = j|i| of its bulk in freezing. From 40° to 12° it ex* 
pands .00236 of its bulk; and from 40° to 212° it expands .04012, = 
.00023325 for every degree, giving an increase in volume (from 40° to 
212°) of -^ = 1 cubic feet in 24.92 feet. 

JF^Anf u ( \° 1Um * °L W t at , er ^M 1 lb. per square inch, is 2.306 feet 
60° (62.4491 lbs.), equivalent to the 1 the at P m osphere is .... . 33.949 » 
pressure of ( 

35.84 cubic feet of water weigh a ton. 

39.03 " ice * " 

When water is pure it will not become turbid, or produce a precipitate with any 
of the following Re-agents. 

Baryta Water, If a precipitate or opaqueness appear, Carbonic Acid is present 

Chloride of Barium, Indicates Sulphates. 

Nitrate of Silver, Indicates Chlorides. 

Oxalate of Ammonia, Indicates Lime salts. 

Sulphide of Hydrogen, slightly acid, Indicates Antimony, Arsenic, Tin, Copper, 
Gold, Platinum, Mercury, Silver, Lead, Bismuth, and Cadmium. 

Sulphide of Ammonium, solution alkaloid by ammonia, Indicates Nickel, Cobalt, 
Manganese, Iron, Zinc, Alumina, and Chromium. 

Chloride of Mercury or Gold and Sulphate of Zinc, Indicate organic matter. 

Mineral Waters are divided into 5 groups, viz. : 

1. Carbonated, containing pure carbonic acid — as, Seltzer, Germany; Spa, Bel- 
gium; Pyrmont, Westphalia ; Seidlitz, Bohemia ; and Sweet Springs, Virginia. 

2. Sulphurous, containing sulphuretted hydrogen — as, Harrowgate and Chelten- 
ham, England ; Aix-la-Chapelle, Prussia ; Blue Lick, Ky. ; Sulphur Springs, Va., etc 

3. Chalybeate, containing carbonate of iron — as, Hampstead, Tunbridge, Chel- 
tenham, and Brighton, England ; Spa, Belgium ; Ballston and Saratoga N. Y. ; and 
Bedford, Penn. 

4. Alkaline, containing carbonate of soda — these are rare, as, Vichy, Ems. 

5. Saline, containing salts — as, Epsom, Cheltenham, and Bath, England; Baden- 
Baden and Seltzer, Germany; Kissingen, Plombieres, France; Seidlitz, Bohemia; 
Lucca, Italy; Yellow Springs, Ohio; Warm Springs, N. C. ; Congress Springs, N. Y. ; 
and Grenville, Ky. 

Brief Rules for the Qualitative Analysis of Mineral Waters. 

The first point to be determined, in the examination of a mineral water, is to 
which of the above classes does the water in question belong. 

1. If the water reddens blue litmus paper before boiling, but not afterward, and 
the blue color of the reddened paper is restored upon warming, it is carbonated. 

2. If it possesses a nauseous odor, and gives a black precipitate, with acetate of 
lead, it is sulphurous. 

3. If, after the addition of a few drops of hydrochloric acid, it gives a blue pre. 
cipitate, with yellow or red prussiate of potash, the water is a chalybeate. 

4. If it restores the blue color to litmus paper after boiling, it is alkaline. 

5. If it possesses neither of the above properties in a marked degree, and lesvtt 
a large residue upon evaporation, it is a saline water. 

River or canal water contains & \ o{ Ug Tolume rf ng mM ^ 

Spring or well water " ^ j 

Sea-Water. A cubic foot of it weighs 64.3125 lbs. 

Height of a column of water^) , 1U . , . n non ^ • 
at 60° fspecific gravity, 1029.), 1 }J b -f er T™ - lnCh ' ^i'^ 
equivalent to the pressure of....) the atm ° s P h ere is 32.966 « 

34. 83 cubic feet weigh a ton. 



WATER. 



541 



Sea-water contains from 4 to 5 % ounces of salt in a gallon of water 

Sahne Contents of Sea-Water from several Localities. 



Baltic 6.6 

Black S^a 21.6 

Arctic 28.3 



British Channel 35.5 

Mediterranean 39.4 

Equator 39.42 



South Atlantic 41.2 

North Atlantic 42.6 

Dead Sea 385. 



There are 62 volumes Of carbonic acid in 1000 of sea-water. 



Destructive Effect of Sea-Water upon Metals and Alloys per Square Foot. 



Steel. 
Iron . 



Grains. I Grains. 

. 40 Copper 9 

. 38 | Zinc 8 



Grains. 
Galvanized Iron. . . 1.5 
Tin.... 2. 

Sea^water, according to the analysis of Dr. Murra3 T , at the specific grav- 
ity of 1.029, contains 

Muriate of soda ,... 220 . 01 I Muriate of magnesia 42 . 08 

Sulphate of soda 33.16 J Muriate of lime 7 .-84 

303.09 
Or, 1 part sea-water contains .030309 parts of salt= ^ part of its weight. 

Boiling Points at different Degrees of Saturation. 



Salt, by Weight, 
in 100 Parts of Sea- 
water. 



Boiling 
Point. 



Salt, by Weight, 
in 100 Parts of Sea- 
water. 



Boiling 
Point. 



Salt, by Weight, 
in 100 Parts of Sea- 
water. 



Boiling 
Point. 



6.06 =A 

9.09 =& 

12.12 =t* 



213.2° 
214.4° 
215.5° 
216.7° 



15. 15= 3% 

18.18=3% 



24.25= 



217.9° 
219. ° 
220.2° 
221.4° 



27.28=3 9 3 
30.31=M 



*36.37= 



222.5° 
223.7° 
224.9° 
226. ° 



Deposits at different Degrees of* Saturation and. Tem- 
perature. 

When 1000 Parts are reduced hy Evaporation, 

Volume of Sea-water. I Boiling Point. Salt in 100 Parts. Nature of Deposit. 



1000 
299 
102 



214° 
217° 

228° 



3. 
10. 
29.5 



None. 

Sulphate of lime. 

Common salt. 



WAVES OF THE SEA. 

Arnott estimated the extreme height of the waves of an ocean, at a di&- 
tance from land sufficiently great to be freed from any influence of it upon 
their culmination, to be 20 feet. 

The French Exploring Expedition computed waves of the Pacific to be 
22 feet in height. 

The average force of the waves of the Atlantic Ocean during the sum- 
mer months, as determined by Thomas Stevenson, was 611 lbs. per square 
foot ; and for the winter months 2086 lbs. During a heavy gale a force 
of 6983 lbs. was observed. 

By the observations of Mr. Douglass in 1853, he deduced that when 
waves had heights of 

8 feet, there were 35 in number in one mile, and 8 per minute. 
15 " " 5 and 6 " " 5 

20 " " 3 " " 4 '* 

Tidal Waves. — Professor Airej T declares that when the length of a wave 
is not greater than the depth of the water, the velocity depends only upon 
its length, and is proportionate to the square root of its length. 

* Saturated. 

Zz* 



542 



GUNNERY. 



When the length of a wave is not less than 1000 times the depth of the water, the 
velocity of it depends only upon the depth, and is proportionate to the square root 
of it ; the velocity being the same that a body falling free would acquire by falling 
through a height equal to half the depth of the water. The diurnal and other tidal 
waves, so far as they are free, may be all considered as running with the same veloci- 
ty, but the column of the length of the wave must be doubled for the diurnal wave. 









Length oi 


Wave in Feet 






Depth of Water 


1 


10 


100 


1000 


10000 


100 000 




Velocity per Second in Feet. 


1 

10 

100 

1000 

10000 


2.26 
2.26 


5.34 

7.15 
7.15 


5.67 
16.88 
22.62 
22.62 


17.92 
53.19 
71.54 
71.54 


17.93 

56.67 
168.83 
226.24 


56.71 
179.21 
533.9 



The wave produced by the action of the sun and moon is termed the 
Free Tide Wave. The semi-diurnal tide wave is this, and has a period 
of 12 hours 24-J- minutes. 





Semi-ID inrrLal ITree-T 


ide Wave. 




Depth 
of Water. 


Velocity 
per Second. 


Length. 


Space described 
per Hour. 


Depth 
of Water. 


Velocity 
per Second. 


Length. 


Space described 
per Hour. 


Feet. 


Feet. 


Miles. 


Miles. 


Feet. 


Feet. 


Miles. 


Miles. 


1 


5.7 


47.9 


3.9 


100 


56.7 


429.5 


38.7 


4 


11.3 


95.9 


7.7 


400 


113.4 


959. 


77.3 


10 


17.9 


151.6 


12.3 


800 


160.4 


1356. 


109.4 


20 


25.4 


214.4 


17.3 


1000 


179.3 


1516. 


122.3 


40 


35.9 


303.2 


24.5 


2000 


253.6 


2144. 


172.9 


60 


43.9 


371.4 


29.9 


4000 


358.7 


3032. 


244.5 



GUNNERY. 

A heavy bodj 7 impelled by a force of projection describes a parabola, the 
parameter of which is four times the height due to the velocity of the pro> 
jection. 

It has been ascertained by experiment that the velocity of a shot pro- 
jected from a gun varies as the square root of the charge direct!} 7 , and a?, 
the square root of the weight of the shot reciprocally. 

To Compute tlie "Velocity of* a Sliot or Shell. 

Rule. — Multiply the square root of treble the weight of the powder in 
pounds by 1G00 ; divide the product by the square root of the weight of 
the shot; and the quotient will give the velocity in feet per second. 

Example. — What is the velocity of a shot of 196 lbs., projected with a charge of 
9 lbs. of powder? 

V9X3 = 5.2, and ^196 = 14. Then, 5.2xl600-M4^594/^. 



e, or the Charge for 



= 14. 

To Compute the Irtange for a Charj 
a Range. 

When the Range for a Charge is given. — The ranges have the same pro- 
portion as the charges of powder ; that is. as one range is to its charge, so 
is any other range. to its charge, the elevation of the gun being the same 
in both cases. Consequently, 

Rule 1. To Compute the Range. — Multiply the range determined by 
the charge in pounds for the range required, and divide the product by 
the given charge ; the quotient will give the range required. 



GUNNERY. 



543 



Rulic 2. To Compute the Charge. — Multiply the given range by the 
charge in pounds for the range determined, divide the product by the range 
determined, and the quotient will give the charge required. 

Example. — If, with a charge of 9 lbs. of powder, a shot ranges 4000 feet, how 
far will a charge of 0.75 lbs. project the same shot at the same elevation ? 
4000 X 6.75 ^9 = 3000 feet. 

Ex. 2. — If the required range of a shot is 3000 feet, and the charge for a range of 
400 J feet has been determined to be 9 lbs. of powder, what is the charge required to 
project the same shot at the same elevation ? 

8000X9 -7- 4000 = 6.75 lbs. 

To Compute tlie Range at on.e Elevation, "When the 
Ptange for another is given. 

Role. — As the sine of double the first elevation in degrees is to its 
range, so is the sine of double another elevation to its range. 

Example.— If a shot range 1000 yards when projected at an elevation of 45°, how 

far will ifc range when the elevation is 30° 16', the charge of powder being the same? 

Sine of 45° X 2 = 100 000 ; sine of 30° 16' X 2 = 87 064. 

Then, as 100 000 : 1000 : : 87 064 : 870.64/eetf. 

To Compute the Elevation at one Range, When the Ele- 
vation, for another is given.. 

Rule. — As the range for th& first elevation is to the sine of double its 
eleva.tion, so is the range for the elevation required to the sine for double 
its elevation. 

Example. — If the range of a shell at 45° elevation is 3750 feet, at what elevation 
must a gun be set for a shell to range 2810 feet with a like charge of powder ? 
Sine of 45° X 2 = 100 000. 

Then, as 3750 : 100 000 : : 2810 : 74 933 — sine for double the elevation — 24° 16'. 

INITIAL VELOCITY AND RANGES OF SHOT AND SHELLS. 

The Range of a shot or shell is the distance of its first graze upon a horizontal 
plane, the piece mounted upon its proper carriage. 

Projectile. 



Arms and Ordnance. 



Rifle Musket 

Musket, 1841 

6-Pounder 

6 " 

12 " 

24 " 

32 " 

42 " 

42 " 

8-inch Columbiad. . 

10 " " 

10 " " 

10 " Mortar 

13 " " 

15 " Columbiad.. 

15 " " 

RIFLED. 

10-pounder Parrott, 

20 

30 " " 

60 " " 
100 " * 
100 " M 
200 " M 
12-inch Rodman 
Hall's Rockets 



Description. 



Elongated. 
Round. 



Shell. 



Elongated. 
Shell. 

M 

o inch. 



Weight. 



Grains. 
510 
412 
Lbs. 
6.15 
6.15 
12.3 
24.25 
32.3 
42.5 
42.5 
65. 
127.5 
127.5 
98. 
200. 
302. 
315. 

9.75 

19. 

29. 

60. 
100. 
101. 
150. 

16. 



Pow- 
der 




Time of 
Flight. 



1.75 



14.19 
14.32 



36. 



23.29 

21. 
17.25 

27. 

29. 
28. 

5X 



Eleva- 
tion. 



2 

5 
1 

2 
1 
1 
5 

15 
15 

39 15 
45 
45 
7 
25 

20 
15 
25 



25 
4 

40 

47 



Range. 



800 
1523 

575 
1147 

713 

775 
1955 
3224 
3281 
5654 
4250 
4325 
1948 
46S0 

5000 
44')0 
6700 

6910 
6820 
2200 

1720 



544 



GUNNERY. 



Approximate Rule for Time of Flight. 
Under 4000 j^ards, velocity of projectile 900 feet in one second ; under 
6000 yards, velocity 800 feet ; and over 6000 yards, velocity 700 feet. 



PENETRATION OF SHOT AND SHELL. 

Experiments at Fort Monroe, 1S39, and at West Point, 1853. 





o 


"3 


8 

B 
m 

.9 

p 






Mean Penetration. 






Ordnance. 




it fa 
|4| *$ 




| 


.2 

« 


6 
"5 

1 


§1 


32-Pounder 

32 " 

42 " 

42 « 

42 " 

42 " 

8-inch Howitzer . . . 

8 u Columbiad. . 
10 " "' .. 
10 " " ... 
15 " " 

(( u 
RIFLED. 

10-pounder Parrott. 

20 " " 

30 " " 

60 M " 

100 " " 

200 "- " 

12-inch Rodman 


Lbs. 

8. 

11. 

10.5 

7. 

6. 

12. 

18. 
IS. 

55. 


Shot. 
<( 
t( 

Shell. 

Shell. 
Shot. 

Shell. 
Shot. 


Yds. 
S80 
100 
100 
100 

8S0 
200 
114 
100 

1 
400 


Feet. 

17K 


Ins. 

8 
33 


Ins. 

60. 

54.75 

40.75 

63.5 
56.75 


Ins. 

15.25 

18. 

S.5 
44. 


Ins. 

12. 
4.5 


Ins, 

3.5 
4. 

if 
7.75 


Ins. 

24 


Ins. 

4.* 






The solid shot broke against the granite, but not against the freestone or brick, 
and the general effect is less upon brick than upon granite. 
The shells broke into small fragments against each of the three materials. 

The penetrations in other kinds of earth are found by multiplying the 
above by .63 for sand mixed with gravel ; by .87 for earth mixed with 
sand and gravel, weighing 125 lbs. per cubic foot; by 1.09 for compact 
mold and fresh earth mixed with sand, or half clay ; by 1.44 for wet pot- 
ter's c\slj ; by 1.5 for light earth, settled ; and by 1.9 for light earth, fresh. 

The penetration in other kinds of earth and stone may be obtained bv 
using the coefficients given for the other tables. For woods, use for beecn 
and ash 1, for elm 1.3, for white pine and birch 1.8, and for poplar 2. 

Penetration in Ball Cartridge Paper, No. 1. 

Musket, with 134 grains, at 13.3 yards 653 sheets. 

Common rifle, 92 grains, at 13.3 yards 500 sheets. 

Experiments — England. — (Holley.) 



Ordnance. 


Charge. 


Projecti 


le. Weight 


Velocity. 


Ranjre 


Target and Effects. 




Lbs. 




Lbs. 


Feet. 


Yards 




11-inch U. S. Navy. . 


30 


Shot 


169 


1400 


50 


Iron plates, 14 ins. — 
loosen f d. 


15-ineh Rodman. . . . 


60 


" 


400 


14S0 


53 


Iron plates 6 ins.— 
destroyed. 


Rifled. 














7-inch Whitworth . . 


25 


Shot 


150 


1241 


200 


Inglis'st — destroyed 


10.5-inch Armstrong 


45 


" 


307 


1228 


200 


U 11 


13-inch " 


90 


X " 


344.5 


1760 


200 


Solid plates, 11 ins. 
thick — destroyed. 



* Passed through. t 8 " in - vertical and 5-in. horizontal slabs, and 7-in vertical and 5-in. 

horizontal slabs, 9X5 in. ribs and 3-in. ribs. J Steel. 



GUNNERY. 



545 



Penetration of Lead Balls in Small Arms. 
Experiments at Washington Arsenal in 1S39, and at West Point in 183T. 

I Penetration. 



Diam. 
of Ball. 



Charge. 
Powder. 



White Oak. White Pine. 



Musket 

Common rifle. 
Hall' s rifle . . . 



Hall's carbine, musket cal- 
ibre 



.5775 



Pistol 

Rifle musket 

Altered musket 

Rifle, Harper's Ferry. 

Pistol carbine 

Sharpe's carbine 

Burnside's " 



Grains. 
134 
1-14 
100 

92 
100 

70 

70 

SO 

90* 
100* 

51 

60 
70 
40 
60 
55 



Yards. 

9 

5 

5 

9 

5 

9 

5 

5 

5 

5 

5 

200 

200 

200 

200 

30 

30 



Grains. 
397.5 
397.5 
219. 

219. 

219. 

219. 



219. 
500. 
730. 
500. 
450. 
463. 
350. 



Inches. 

1.6 

3. 

2.05 

1.8 

2. 

.6 
1.7 

.8 
1.1 
1.2 

.725 



11. 

10.5 
9.33 
5 75 

7.17 
6.15 



The musket discharged at 9 yards distance, with a charge of 134 grains, 1 ball 
and 3 buckshot, gave for the ball a penetration of 1.15 in., buckshot, .41 in. 

"Weight and. Dimensions of Leaden Balls. 
Number of Balls in a Pound, from \%fhs to . 237 of an Inch Diam. 

No. 



Diam. 


No. 


Diam. 


No. 


Diam. 


No. 


Diam. 


No. 


Diam. 


No 


Diam. 


Inch 




Inch. 




Inch. 




Inch. 




Inch. 




Inch. 


1.67 


1 


.75 


11 


.57 


25 


.388 


80 


.301 


170 


.259 


1.326 


2 


.73 


12 


.537 


30 


.375 


88 


.295 


180 


.256 


1.157 


3 


.71 


13 


.51 


35 


.372 


90 


.29 


190 


.252 


1.051 


4 


.6 f >3 


14 


.505 


36 


.359 


100 


.285 


200 


.249 


.977 


5 


.677 


15 


.4S8 


40 


.348 


110 


.281 


210 


.247 


.919 


6 


.662 


16 


.469 


45 


.338 


120 


.276 


220 


.244 


.873 


7 


.65 


17 


.453 


50 


.329 


130 


.272 


23" 


.242 


.835 


8 


.637 


18 


.426 


60 


.321 


140 


.268 


240 


.239 


.802 


9 


.625 


19 


.405 


70 


.314 


150 


.265 


250 


.237 


.775 


10 


.615 


20 


.395 


75 


.307 


160 


.262 


260 





270 
2S0 
290 
300 
310 
320 
330 
340 
350 



The decimals for the dimensions in the division of an inch are 

1^.... 1.3125 1% S75 1% 6875 \%, 4375 I fc 25 

% 9375 |% 8125 |^ 5 |& 3125 | % 187 

Heated shot do not return to their original dimensions upon cooling, but retain a 
permanent enlargement of about .02 per cent, in volume. 



Powder. 


Loss of Fore 

Ball 


e by Windage (24- Pounder Gun). 

Initial Velocity of Ball in Feet per Second. 


Without 
Windage. 


Windage, 
.135 Inch. 


Windage, 
.245 Inch. 


Windage, 
.355 Inch. 


Lbs. 

4 
6 


Lbs. 

24 25 
24.25 


Feet. 

1631 
1963 


Feet. 

1450 
1702 


Feet. 

1332 
1596 


Feet. 

1197 
1465 



A comparison of these results shows that 4 lbs. of powder give to a ball without 
windage nearly as g'eat a velocity as is given by 6 lbs. to a ball having .14 inch 
windage, which is the true windage of a 24-pound ball; or, in other Avords, this 
windage causes a loss of nearly one third of the force of the charge. 



* Charges too great for service. 



546 



GUNNERY. 



Vents. — Experiments show that the loss of force by the escape of gas 
from the vent of a gun is altogether inconsiderable when compared with 
the whole force of the charge. 

The diameter of the Vent in U. S. Ordnance is in all cases .2 inch. 

Guns and Howitzers take their denomination from the weights of the ; r 
solid shot in round numbers, up to the 42-pounder; larger pieces, rifled 
guns, and mortars, from the diameter of their bore. 

Effect of different Descriptions of Wadding with a Charge of 11 Grains 
of Powder. 

Velocity of Ball 
per Second. 



Feet 

1308 
1377 
1346 
1482 
1132 
1200 
1100 



Ball wrapped in cartridge paper, crumpled into a wad . . -j 

1 felt wad upon powder and 1 upon ball 

2 felt wads upon powder and 1 upon ball 

1 elastic wad upon powder and 1 upon ball 

2 pasteboard wads upon powder 

2 elastic wads upon powder 

The felt wads were cut from the body of a hat, weight 3 grains. 

The pasteboard wads were .1 of an inch thick, weight 8 grains. 

The cartridge paper was 3x4.5 inches, weight 12.82 grains. 

The elastic wads were u Baldwin's indented, 1 * a little more than .1 of an inch 
thick, weight 5.127 grains. 

The most advantageous wads are those made of thick pasteboard, or of 
the ordinary cartridge paper. 

Number of Pellets in an Ounce of Lead Shot of the different Sizes. 



A A 


.... 40 


No. 2 


.... 112 


No 


7 


... 341 


A 


.... 50 


3 


.... 135 




8 


... 600 


B B 


.... 58 


4 


.... 177 




9 


... 984 


B 


75 


5 


.... 218 




10.... 


... 1726 


No. 1 


.... 82 


6 

So. 14 


.... 280 
31f 


)0 


12 ... 


... 2140 



Proportion of Powder to Shot for the following Numbers of Shot, as de- 
termined by Experiment. 



No. 


Shot. 


Powder 


No. 


Shot. 


Powder. 


| No. 


Shot. 


Powder. 


2 
3 


Oz. 

2. 
1.75 


Drama. 

1.5 
1.625 


4 
5 


Oz. 

1.5 
1.375 


Drams. 
1% 

2>£ 


6 

7 


Oz. 

1.25 
1.125 


Drams. 

2% 



Note. —2 oz. of No. 2 shot, with 1.5 drams of powder, produced the greatest effect. 
The increase of powder for the greater number of pellets is in consequence of the 
increased friction of their projection. 

Numbers of Percussion Caps corresponding with the Birmingham Numbers 
Eley's 5 6 | 7 8 9 | 24 10 | 11 | 18 12 13 14 



Birmingham 43 



44 I 46 



4S 



49 | 50 51and52|53and54|55and5G 



57 



53 



Where there are two numbers of the Birmingham sizes corresponding with only 
one of Eley's, it is in consequence of two numbers being of the same size, varying 
only in the length of the caps. 



GUNPOWDER. 



Gunpowder is distinguished as Musket, Mortar, Cannon, Mammoth, and Sporting 
powder; it is all made in the same manner, of the same proportions of materials, 
and differs only in the size of its grain. 



GUNNERY. 



547 



Bursting or Explosive Energy —By the experiments of Captain Rodman, U. S. 
Ordnance Corps, a pressure of 45000 lbs. per square inch was obtained with 10 lbs. 
of powder, and a ball of 43 lbs. 

Also, a pressure of 185 000 lbs. per square inch was obtained when the powder was 
burned in its own volume, in a cast-iron shell having diameters of 3.85 and 12 ins. 

Properties and Results of Gunpowder, determined by Experiments of Captain 
A. Mobdeoai, U. S. A. 



24-Pounder Gun. 

Weight of ball and wad 24.25 lbs. 

" " poAvder 6. " 

Windage of ball 135 inch. 



Composition. 



. Musket Pendulum. 

Weight of ball 397.5 grains. 

" " powder 120. " 

Windage of ball 09 inch. 



Salt- Char- Sul- 
petre. coal. phur. 





o 

2 03 bJD 

cr 


77 


275 


569 


314 


1134 


214 


61T4 


142 


5344 


282 


1642 


__ 


13152 


_ 


166 


183 


103 


182 


72808 


100 


295 


212 


2378 


204 


11600 


- 



Cannon, large . . 

" small.. 

Musket 

Rifle 

Rifle 

Musket 

Rifle 

Cannon, uneven. 

" large . . 

Sporting 

Blasting, uneven 

Rifle 

Sporting 

Rifle 



Per cent. 

2.77 .677 
3.35 .72 



V76 



14. 



12.5 

13. 
15. 

15. 

15. 



12. 



12.5 

10. < 
15.1 



10, 



-{ 



* Dupont's Mills, 
Del. 



t Dupont's Mills, 

Del. 
* Dupont's Mills, 

Del. 
Loomis, Hazard, 

& Co., Conn.* 
Waltham Abbey, 

England.* 



3.55 



2.09 
1.91 
4.42 



.907 

.728 
.834 
.943 
.78S 
.756 
1. 

.82 



.865 



Comparison of tlie Force of a Charge in various .A.rm.s. 
Arm. Lock. Powder, Windage. ^gfn* Velocity. 



Ordinary rifle . . . 

t Hall's rifle 

j Hall's carbine . 
t Jenks's carbine 
Cadet's musket. . 
Pistol 



Percussion. 



Flint. 
Percussion. 



Flint. 
Percussion. 



Grains. 
100 
70 
70 
70 
70 
70 
35 



Inch. 

.015 

.015 

.0 

.0 

.0 

.045 

.015 



Grains. 
219 
219 
219 
219 
219 
219 
218.5 



Feet. 
2018 
1755 
1490 
1240 
1687 
1690 
947 



Deductions of Captain Mordecai. 

Proof of Powder. — The common eprouvettes are of no value as instruments for 
determining the relative force of different kinds of gunpowder. 

In the proof of gunpowder a cannon pendulum should be used. 

In a 24-pounder gun, new cannon powder should give, with a charge of 6 lbs., an 
Initial velocity of not less than 1600 feet to a ball of medium weight and windage. 

For the proof of powder for small arms, a small ballistic pendulum is be?t adapted. 

The initial velocity of a musket ball (18 to the pound), of .05 inch windage, with 
a charge of 120 grains, should be, 

With new musket powder, not less than 1500 feet. 
With new rifle powder, not less than 1600 feet. 
Witli fine sporting powder, not less than 1800 feet. 

Manufacture of Powder— The powder of greatest force, whether for cannon or 
small arms, is produced by incorporation in the u cylinder milK" 



f Rough. 



J Loaded at the breach. 



548 



GUNNERY. 



in all re- 



Effect of Wads.— In the service of cannon, heavy wads over the ball are in i 
spects injurious. 

For the purpose of retaining the hall in its place, light grommets should be used. 

On the other hand, it is of great importance, and especially so in the use of small 
arms, that there should be a good wad over the powder for developing the full force 
of the charge, unless , as in the rifle, the ball has but very little windage. 

Effect of the Size of the Gram. — Within the limits of the difference in the size of 
grain, which occurs in ordinary cannon powder, the granulation appears to exercise 
but little influence upon the force of it, unless the grain be exceedingly dense and hard. 

Effect of Glazing.— Glazing is favorable to the production of the greatest force, 
and' to the quick combustion of the grains, by affording a rapid transmission of the 
flame through the mass of the powder. 

Effect of using Percussion Primers. — The increase of force by the use of primers, 
which nearly closes the vent, is constant and appreciable in amount, yet not of suf 
ficient value to authorize a reduction of the charge. 



Bore, Weight 

Arm. | Bore. 


of Charges, ar 

Windage of Ball. 


,d Ranges 

Powder. 


for U. S. Small Arms. 

Ball. 


Musket. . . . 

Eifle 

Pistol 


Inch 
.09 
.69 
.54 
.54 
.54 


Inch. 

* .05 
t .04 
.015 
.015 
.015 


Grains. 

120 

110 
75 
35 
30 


397.5 grains. Wad, 10.2 grains. 
470.2 " " 10.2 " 
218.5 " " 8.4 " 
218.5 " " 5.5 " 
218.5 " " 5.5 " 



Ranges for Small Arms.— Musket. With a ball of 17 to the pound, and a charge 
of 110 grains of powder, etc., an elevation of 36' is required for a range of 200 yards ; 
and for a range of 500 yards, an elevation of 3° 30' is necessary, and at this distance 
a ball will pass through a pine board 1 inch in thickness. 

Rifle. With a charge of 70 grains, an effective range of from 300 to 350 yards is 
obtained ; but as 75 grains can be used without stripping the ball, it is deemed better 
to use it, to allow for accideutal lo3s, deterioration of powder, etc. 

Pistol. With a charge of 30 grains, the ball is projected through a pine board 1 
inch in thickness at a distance of 80 yards. 

Proof of* Powder. 

Ordinary Proof of Powder. — One oz. with a 24-lb. ball. The mean range of new, 
proved at any one time, must not be less than 250 yards ; but none ranging below 
225 yards is received. 

Powder in magazines that does not range over ISO yards is held to be unserviceable. 

Good powder averages from 2S0 to 300 yards ; small grain, from 300 to 320 yards. 

Restoring Unserviceable Powder .—When powder has been damaged by being 
stored in damp places, it loses its strength, and requires to be worked over. If the 
quantity of moisture absorbed does not exceed 7 per cent., it is sufficient to dry it to 
restore it for service. This is done by exposing it to the sun. 

When powder has absorbed more than 7 per cent, of water it should be sent to a 
powder mill to be worked over. 

Dimensions of Powder Barrels for 100 lbs. of Powder 



Whole length 20.5 ins. 

Length, interior in the clear. . . 18. " 
Interior diameter at the head. .14. u 



Interior diameter at the bilge . . 16. ins. 
Thickness of staves and heads. . .5 u 
Weight of barrels about 25 lbs. 



Diameter of Holes in Sieve to determine the Class of Powder. 

Musket powder, No. 1, .03 in.; No. 2, .06in.|Cannon powder, No. 4, .25 in.; No. 5, .35 in. 
Mortar " No. 2, .06 in.; No. 3, .10 in., Mammoth " No. 6,. 6 in.; No. 7,.9in. 

Musket Powder. — None should pass through sieve No. 1 ; all through No. 2. 
Mortar Powder. — None should pass through sieve No. 2 ; all through No. 3. 
Cannon Powder. — None should pass through sieve No. 4 ; all through No. 5. 



.05679 lbs., or 18 to the pound. 



f .05861 lbs., or 17 to the pound. 



RAILWAYS AND ROADS. 



549 



RAILWAYS AND ROADS. 

Table of Gradients, Rise per Mile, and Resistance to Gravity. 



Gradient of 1. in.. J 20 



Rise in ft. per mile 264 
Resistance in lbs. 
per ton of train . .J112 



25 



211 

89.6 



30 35 

151 



176 
'4. 



64 



40 | 45 


50 


60 


70 


80 


90 


100 


132 117 


106 


88 


75 


66 


59 


53 


56 1 50 


14.8 


37.3 


32 


28 


24.8 


22.4 



2240 

Resistance due to gravity upon any inclination H± — - — l os . per 

rate of grad. * " 

ton of tram. 

Resistance of Trains upon a Level at different Speeds. 

jyj + 8 = R. V representing velocity in miles per hour, and R resistance 
in lbs. per ton of train. 

The resistance of curves may be taken at 1 per cent, for each degree of 
the curve covered by a train. 

Irregularities of roads vary from 5 to 40 per cent. Strong side winds 
resist 20 per cent. & ' 

Miles. 

Velocity of train per hour 



Resistance upon straight 
line per ton 

Ditto, with sharp curves 
and strong wind* 



10 



Lbs. 

m 

13 



15 



Lbs. 

14 



20 



Lbs. 

15K 



30 



Lbs. 

13}^ 
20 



40 



Lbs. 

m 

26 



50 1 CO 



Lbs. 

34 



Lbs. 

29 



70 



36^ 
55 



To Compute th.e "Weight of Rails. 

h «J 2 = y* , L r ?Pf e senting greatest load upon one driving wheel in tons, 
ana W weight of rail in lbs. per yard. 

Sectional area of rail in inches x 10.08 = weight of rail in lbs. per vard. 

Weight of rail m lbs. per yard x 1.571 =-- weight of rails per mile of sin- 
gle line in tons. ^ 

Points and Crossings, Ordinary Crossing, Narrow Gauge. 

Length of outer switch = 15 feet. 

Throw of outer switch at point = 4 ins. 
Clearance " " =3>£ins. 

Length of guard rail — S feet. 

Clearance of ditto — 1 % ins. 



Length from point to crossings 75 feet. 
Total length from point to points 165 " 

Radius — 600 " 

Angle of crossing = 1 in 10. 

Length of inner switch =z 10 feet. 



Railway Sidings, etc. 

-jr d - r ~7 ^ d) 2 =L. d representing distance between centres of lines of 
siding in feet, r radius of curves, and L length over the points. 

Coefficients of Adhesion of Locomotives per Ton upon the Driving Wheels. 

When the rails are very dry... eVo Tn mistv weather . . . 350 

I Wnen the rails are very wet. . . 600 In frost"or snow 200 

In coupled engines the adhesion is due to the load upon all the wheels 
coupled to the drivers. 

The adhesion must exceed the traction of an engine upon the rails 
otherwise the wheels will slip. ' 

* Equal to 50 per cent, added to resistance upon a straight line. 

3 A 



550 



RAILWAYS AND ROADS. 



To Compvite the Load, ^vvhicli a Locomotive will draw 
Tip an Inclination.. 

T 

— W = L. T representing tractive power of locomotive in lbs., r re- 
sistance due to gravity, and r' resistance due to assumed velocity of train in 
lbs. per ton, W weight of locomotive and tender, and L load the locomotive 
can draio in tons, exclusive of its own weight and tender. 

Coefficient of Traction of Locomotives. 

Railroads in good order, etc 

Railroads in ordinary condition 

Tredgold estimates the resistance to a train from concussions, at a ve- 
locity of 10 miles per hour and above this, at }{ the velocity. 

To Compute tlie Traction, Retraction, and. .A^dliesive 
Power of a Locomotive or Train. 



J"4 lbs. 
16 " 



„ TT _ _ a s P 

When upon a Level. ^ = T. 



a representing area of one cylinder in 



sq. ins., s stroke of piston in feet, P mean pressure of steam in lbs. per sq. in., 
D diameter of driving wheels, and T traction in lbs. 

C ic = A. C representing coefficient in lbs. per ton, w weight of locomotive 
upon driving wheels in tons, and A adhesion in lbs. 



When upon an Inclination. 



asF 
D 



■rwh—T. r representing resistance 



per ton of locomotive, and h height of rise in feet per 100 feet of road, 
r zv h — R, representing retraction in lbs. 
Cw b 
10ft = A. b representing base of inclination in feet per lWfeet of road. 

When the Velocity of a Train is considered. 

When upon a Level, W (c + -/V} = R; When upon an Inclination, 
W (r A + c +V V) = R. V representing velocity of train in miles per hour. 

Illustration — A train weighing 300 tons is to be driven up a grade of 52.8 feet 
per mile, with a velocity of 16 miles per hour; required the retractive power? 

52. S per mile 3= 1 in 100 feet = r = 22. 4 lbs. C = 5. 

200 (22.4X1 + S -f VIC) = 200 X 22. 4 -f- 9 = 0280 lbs. 

The resistance to traction upon a level is doubled by a radius of curve 
of 400 feet, and 13 lbs. per ton is the additional friction upon a curve of 
300 feet. 

By the experiments of Mr. Gooch with a Dynamometer, the resistances of a train 
rere determined to be as follows : 



ity per 
Hour. 


Engine 
Cars. 


jnd Tender 50 tons. Train 100 tons. 

Resistance in lbs. per Ton. 




Engine and 
Cars. 


Engine and 
Tender. 


Atmosphere 
for Cars. 


Oscillation of 
Cars. 


Miles. 
13.1 

46.8 

01. 3 


Lbs. 

8.19 
14.30 
21.8 
19.8 


Lbs. 
9.04 

12.28 
21 . S4 
31.10 
32. S6 


Lbs. 

11.97 
20.43 
37.36 
46.11 
50.68 


Lbs. 
.02 
1.47 
7.37 
11.53 
13.52 


Lbs. 
.87 
1.35 
3.02 
3.77 
4.09 



The atmospheric resistance per bulk of cars alone is estimated as equal to the 
•o.liu't of .001 
our squared. 



Eroduct of .00002 the bulk of the cars in cubic feet, and their velocity in miles per 
< 



RAILWAYS AND ROADS. 551 

The oscillating resistance to cars alone is estimated at the quotient of the product 
of the weight of the cars and their velocity in miles per hour, divided by 1.5. 

The resistance of the engine and tender alone, is estimated by the sum of . 5 the 
velocity in miles per hour, added to 5 for the friction of the axles and parts. To this 
sum add the product of . 00004 times the square of the velocity and the weight of the 
cars, and multiply their sum by the weight of the engine and tender in tons. 

Illustration. — Assuming a train of 150 tons (engine and tender 50 tone, cars 
100), at a velocity of 55 miles per hour and a bulk of cars of 18000 cubic feet. 

Then, .00002x18000x552 =1089. lbs. atmospheric resistance, 

100x55 

= 366.6 u oscillating resistance. 



15 



55x . 5 -f- 5 + 5^X. 00004X100X50 = 2230. " engine and tender, 
100X6 = 600. " friction of cars. 

42S5.6 " 
4285 6 
Hence = 28 . 5T lbs. per ton of the train. 

Experiment gave a resistance of 29. lbs. per ton. 

Grades of 200, and even 250 feet, can be advantageously overcome. 

To Complete tlie Maximum Load, tnat can "be drawrn "by 
an Engine, up the IVXaxixnnxn Grade tliat it can attain, 
tlie ^Weight and. GJ-rade Tbeing given.— (Majok McClkllan, U. S. a.) 

.2 A _ , .2A-8L n . _ _ . 

4949 r -l « = a — 4949 T — P representing the adhesive 

weight of the engine in lbs., Q the grade in feet per mile, and L the Load in tons. 

Note. — When the rails are out of order, and slippery, etc., for .2 A, put .143 A. 

2. With an engine of 4 drivers, put . 6 as the weight resting upon the drivers ; 
with 6 drivers the entire weight rests upon them. 

Illustration. — An engine weighing 30 tons has 6 drivers; what are the maxi- 
mum loads it can draw upon a level, and upon a grade of 250 feet, and what is its 
maximum grade for that load ? 

.2X2240X30 13440 

- = rs ,-,..,» — 1595.4 tons upon a level. 



.4242 {-8 ~ 8.4242 " 
.2x2240x30 13440 



.4252x250 + 8 114.05 
.2X2240X30 — 8X111.8 1249T 



117.8 tons up a grade of 250 feet. 
= 250.1 tons. 



.4242xli7.8 49.97 

The adhesion of a 4-wheeled locomotive, compared with one of 6 wheels, is as 5 to 8. 

Regulations for Railways— {English Board of Trade). 

Cast-iron girders to have a breaking weight = 3 times the permanent load, added 
to 6 times the moving load. 

Wrought-iron bridges not to be strained to more than 5 tons per square inch. 

Minimum distance of standing work from the outer edge of rail at level of carriage 
steps, 3.5 feet in England and 4 feet in Ireland. 

Minimum distance between lines of railway, 6 feet. 

Stations. — Minimum width of platform, 6 feet. Minimum distance of columns 
from edge of platform, 6 feet. Steepest gradient for stations, 1 in 300. Ends of plat- 
forms to be ramped (not stepped). Signals and distant signals in both directions. 

Carriages. — Minimum space per passenger 20 cubic feet. Minimum area of glass 
per passenger, 60 superficial ins. Minimum width of seats, 15 ins. Minimum 
breadth of seat per passenger, 16 ins. Minimum number of lamps per carriage, 2. 

Requirements. — Joints of rails to be fished. Chairs to be secured by iron spikes. 
Fang bolts to be used at the joints of flat-bottomed rails. 

Friction of Railway Carriages. 
The least resistance in parts of the weight, as determined by experiments, is the 
^■g part, and the greatest resistance the j^g part, equal to a resistance of 7.48 lbs. 
and of 20.2 lbs. per ton. 



552 



RAILWAYS AND ROADS. 



The average resistance of a great number of experiments being the ^ J^ part, equal 
toS.S3 lbs. per ton. 

By the experiments of Mr. Geo. Rennie, he determined that the resistance of an 
axle was directly as its diameter, and but one half in the terms of its length ; alsG 
that the length of the bearing should be twice its diameter, and that the area of the 
bearing surface should not be subject to a greater insistent weight than 90 lbs. per 
square inch. 

The resistance to the leading car of a train is about 12 lbs. per ton, and of the in- 
termediate cars, S lbs. 
The friction of locomotive engines is about 9 per cent., or 2 lbs. per ton of weight, 
Case-hardening of wheel-tyres reduces their friction from .14 to .OS part of the load, 
The resistance of the atmosphere to a train is as the square of its velocity, being 
^ lb. p?r square foot for a velocity of 10 miles per hour, 1 lb. for 20 miles, etc. 

Roads. 
Relative Capacities of different Roads. 



Load upon horse's back 1. 

Inferior gravel or earth roads ... 3. 
Macadamized road 9. 



Plank road 25. 

Stone track 33. 

Railway 54. 



Coefficients of Friction in proportion to Load upon Road Surfaces. 



Gravel road, new 

Sand road 

Broken stone, rutted 

u fair order. 

" perfect order 

Macadamized road 

Earth, good order 



Per 100. 


Per Ton 


.0S3 


1S6 


.003 


141 


.052 


117 


.023 


63 


.015 


34 


.033 


T4 


.025 


50 



Pavement, street 

" very smooth. 

Plank 

Stone track 

Railway 

Common road, bad order 



Per 100. 


Per Ton. 


.015 


34 


.01 


22 


.006 


13 


.01 


22 


.05 


112 


.0036 


8 


.07 


157 



For other elements, see Friction, pp. 347-349. 

Resistance of Gravity at different Inclinations. 



Rise. Load. 

1 in a 100 9 

1 in 50 81 

1 in 44 75 



Rise. Load. 

1 in 40 72 

1 in 30 G4 

1 in 26 54 



an 
ise 


at ions. 


Load. 


1 


in 24 . . . 


... .0 


1 


in 20 . . . 


... .4 


1 


in 10 .... 


... .25 



Inclination of Roads. — The limit of practicable inclination varies with 
the character of the road and the friction of the vehicle. For the best car- 
riages on the best roads, the limit is 1 in 35. 

To secure effective drainage of a road, it should incline 1 in 125 in the 
direction of its length. The transverse section of a Macadamized road 
should have an inclination of 1 in 50. 

In the construction of Roads the advantage of a level road over that of 
an inclined one, in the reduction of labor, is superior to the cost of an in- 
creased length of road in the avoiding of a hill. 

In the construction of a Macadamized road none but cubes of stone 
should be used, and none, the longest diameter of which exceeds 2>£ ins., 
and when the stone is very hard this may be reduced to X% and X% ins. 

The dimensions of a hammer for breaking the stone should be, head 
6 ins. in length, weighing 1 lb., handle 18 ins. in length; and an average 
laborer can break from 1).< to 2 cubic yards per day. 

The thickness or depth of the stones, i. e. the metaling, should be 6 
ins., in 2 layers of 3 inches, laid at an interval, enabling the first layer to 
be fully consolidated before the second is laid on. 

A horse ran draw upon a plank road three times the load that he can 
upon an ordinary broken stone or Macadamized road. 



CANALS. SEWERS. 



553 



To Compute tlie Tractive Power of a Horse Team. 

When upon a Level. L (c -j- -y/V) = T ; L representing Load in tons, and c co> 

efficient as before. — — j-— T ; d representing duration of travel in hours. 
V yd 

375 w' h 
When upon an Inclination. L (r h + c + V V) = T, and — — - — T ; r rep* 

resenting resistance in lbs. per ton, h vertical rise in 100 feet, v velocity in miles pe7 
hour, and w' weight of horses in lbs. 

Horses upon Turnpike Roads. 

At a speed of 10 miles per hour, a horse will perform 13 miles per day 
for 3 years. In ordinary staging, a horse will perform 15 miles per day. 

Comparative Effect of Horses upon Roads and Canals. 



Duty. 


Rate per 
Hour. 


Force. 


Distance per 
Day. 


Duration per 
Day 


Effect. 


Railroad 

Turnpike 

Canal 


Miles. 

10 
9 


Lbs. 

125 

42 

133 


Miles. 

20 
13 
10 


Hours. 

8. 

1.3 

1.11 


2500 

546 

1330 



CANALS. 

Resistance of Boats at Low and High Velocities. 
Low Velocities. 



Speed per 
Hour. 


Weight which 
1 lb. will draw. 


Resistance 
per Ton. 


Speed per 
Hour. 


Weight which 
1 lb. will draw. 


Resistance 
per Ton. 


Miles. 

4 

3X 

3^ 


Lbs. 
200 
243 

299 


Lbs. 
11.2 
9.22 
7.5 


Miles. 
3 

2 


Lbs. 

474 

819 

1600 


Lbs. 
4.73 
2.73 
1.4 



Speed per Hour. 


High Velocities. 

Resistance per Ton. 




Maximum Load. 


Minimum Load. 


Average Load. 


Miles. 
4 

I0X 


Lbs. 
7.1 

49.8 
56.8 


Lbs. 
13.1 
74.9 

92.8. 


Lbs. 

9.2 

58.53 

72.45 



SEWERS. 

Sewers are classed as Drains, Sewers, and Culverts. 
Drains are the small courses, as from one or more locations leading 
to a sewer. 

Sewers are the courses from a series of locations. «. 

Culverts are the courses that receive the discharge of sewers. 

The greatest fall of rain is 2 inches per hour = 54308.6 galls, per acre. 

Drainage of Lands "by IPipes. 



Coarse gravel sand . . . 
Light sand with gravel 

Light loam 

Loam with clay 



Depth 
of Pipes 



Ft Ins. 
4 6 



Distance 
apart. 



Feet. 
60 
50 
33 
21 



3 A* 



Loam with gravel . . 

Sandy loam 

Soft clay • 

Stiff clay 



Depth 


Distance 


of Pipes 


apart. 


Ft. Ins. 


Feet. 


3 3 


27 


3 9 


40 


2 9 


21 


2 6 


15 



554 



SEWERS. 



Circular. 55 Vxx 2/= v, , and vx«=V; # representing area of sewet 
-4- ^e wetted perimeter, f inclination of do. per mile, and v velocity of flow, in 
feet per minute : a area of flow in square feet, and V volume of discharge ifi 
cubic feet per minute. 

D 2D 

Egg. — = w, —^- — w', and D = r. D representing 

r height of sewer, w and w' width at bottom and top, and 
--- > r ra di us of sides. 

In culverts less than 6 feet in depth,* the brick 
work should be 9 ins. thick. When they are above 
6 feet and less than 9 feet, it should be 14ins. thick. 

If the diameter of top arch = 1, the diameter of in- 
verted arch — .5, and the total depth s= the sum of the 
two diameters, or 1.5 ; then the radius of the arcs which are tangential to 
the top, and inverted, will be 1.5. 

From this any two of the elements can be deduced, one being known. 
Oval. Top and bottom* should be of equal diameters. The diameter 
.76 depth of culvert; the intersections of the top and bottom circles, as n, 
Fig. 14, p. 168, form the centres for striking the courses connecting the 
top and bottom circles. 

The inclination of sewers should not be less than 1 foot in 240. 




Dimensions, Areas, and Volume of Work per Lineal Foot of Egg-shaped Sewers 
of different Dimensions. 



Internal Dimensions. 



Volume of Brick-work. 



Depth. 



Feet. 
3. 

6. 



Diameter of 
Top Arch. 



Feet. 

1.5 

2. 

2.5 

3. 

3.5 

4. 

4.5 

5. 

5.5 



Diameter 
of Invert. 



Feet. 

75 
1. 

i.2b 

1.5 

1.T5 

2. 

2.25 

2.5 

2.75 

3. 



A 1 /-, Inch 
Thick. 



Sq. Feet. 
2.5-3 
4.5 
7.03 
10.12 
13.78 
IS. 
22.78 
28.12 
34.03 
40.5 



Cub. Feet. 

2.81 

3.5(5 
4.31 
5.06 
5. SI 
6.56 
7.31 



9 Inch 
Thick. 



0.56 
10.87 
12.75 
14.25 
15.75 
17.06 
18. 
19.69 



13k, Inch 
Thick. 



24.75 
27. 
28.41 
30.94 



In laying large sewers through quicksands, cast-iron inverts are some- 
times employed, and with success, to connect the foundation of the whole 
work together. 

Area of Surface from which Circular Servers will discharge Water equal in Volume 
to One Inch in Depth upon surface per Hour, including ordinary City Drainage. 







Diameter of Sewers in Feet. 












in Feet. 


2 


2^ 


3 


4 


5 


6 




Acres. 


Acres. 


Acres. 


Acres. 


Acres. 


Acres. 


None 


3S% 


67^ 


120 


277 


5.0 


1020 


1 in 480 


4S 


75 


135 


30S 


630 


1117 


1 in 240 


50 


87 


155 


355 


735 


1318 


1 in 160 


63 


113 


203 


460 


950 


1692 


1 in 120 


7S 


143 


257 


590 


1200 


2180 


1 in 80 


DO 
125 


165 
1S2 


205 

31S 


570 
730 


inss 

15 


24S6 


1 in 60 


2675 



* Internal dimensions. 



AEOHES AND ABUTMENTS. 



555 



ARCHES AND ABUTMENTS. 

Approximate Rules and Tables for tlie Deptli of Arches 
and. Thickness of .A/bntments. 

C v*r= D. C representing coefficient, r radius of arch at crown, t thick* 
ness of abutment, h height of abutment to spring, and D depth of crown in 
feet. 

In single arches, Stone C = .3, Brick .4, and Rubble .45. 



Depths required for the Crowns of Arches. 



Radius 

of 
Curve. 



Feet. 

2 

3 

4 

4^ 

5 

5^ 

C 

7 



Feet. 
.42 
.47 
.52 
.56 
.6 
.64 
.67 
.71 
.74 
.8 
.85 



Feet. 
.56 
.63 
.69 
.75 
.8 
.85 
.9 
.94 
.98 

1.06 

1.13 

1.2 



Radius 






Radius 






Radius 






of 


Stone. 


Brick. 


of 


Stone. 


Brick. 


of 


Stone. 


Brick. 


Curve. 






Curve. 






Curve. 






Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


10 


.95 


1.26 


24 


1.47 


1.96 


80 


2.68 


3.5S 


11 


1. 


1.33 


25 


1.5 


2. 


85 


2.77 


3.69 


12 


1.04 


1.33 


30 


1.64 


2.19 


90 


2.85 


3.S 


13 


1.08 


1.44 


35 


1.78 


2.37 


95 


2.92 


3.9 


14 


1.12 


1.5 


40 


1.9 


2.53 


100 


3. 


4. 


15 


1.16 


1.55 


45 


2.01 


2.6S 


110 


3.15 


4.2 


16 


1.2 


1.6 


50 


2.12 


2.S3 


120 


3.29 


4.38 


17 


1.23 


1.65 


55 


2.22 


2.97 


130 


3.42 


4.56 


18 


1.27 


1.7 


60 


2.33 


3.1 


140 


3 55 


4.73 


19 


1.31 


1.74 


65 


2.42 


3.22 


150 


3.67 


4.9 


20 


1.34 


1.79 


70 


2.51 


3.35 


160 


3.8 


5.06 


22 


1.41 


1.S8 


75 


2.6 


3.46 


170 


4.13 


5.22 



Minimum Thickness of Abutments for Arches of 120°, where their Depth 
does not exceed 3 Feet. (Computed from the Formula — 



%/"■+©- 



o r 
2h~~ 



Radius 


Heigl 


t of Abutment to 


Spring in Feet. 


Radius 

of 
Arch. 


Height of Abutment to Spring in Feet. 


Arch. 


5 


7.5 


10 


20 


30 


5 


7.5 


10 


20 


30 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


4. 


3.7 


4.2 


4.3 


4.6 


4.7 


12. 


5.6 


6.4 


6.9 


7.6 


7.9 


4.5 


3.9 


4.4 


4.6 


4.9 


5. 


15. 


6. 


7. 


7.5 


8.4 


8.8 


5. 


4.2 


4.9 


4.S 


5.1 


5.2 


20. 


6.5 


7.7 


8.4 


9.6 


10. 


6. 


4.5 


4.7 


5.2 


5.6 


5.7 


25. 


6.9 


8.2 


9.1 


10.5 


11.1 


7. 


4.7 


5.2 


5.5 


6. 


6.1 


30. 


7.2 


9.7 


0.7 


11.1 


12. 


8. 


4.9 


5.5 


5.8 


6.4 


6.5 


35. 


7.4 


9.1 


10.2 


11.8 


12.9 


9. 


5.1 


5.8 


6.1 


6.7 


6.9 


40. 


7.6 


9.4 


10.6 


12 8 


13.6 


10. 


5.3 


6. 


6.4 


7.1 


7.3 


45. 


7.8 


9.7 


11. 


13.4 


14.3 


11. 


5.5 


6.2 


6.6 


7.3 


7.6 


50. 


7.9 


10. 


11.4 


14. 


15. 



Note — The abutments firs assumed to be without counterforts or wing walls. 



KEYSTONES. 



To 



Compute tlie Deptli of Keystones for Segmental 
.A.rches of Stone.— (Tkautwine.) 

First Class of Arch. .36 V of the radius at the crown. 
Second Class of Arch. A V of the radius at the crown. 
Brick or Rubble. .45 V of the radius at the crown. 
In Viaducts of several Arches. Increase the above units to .42, .46, 
and .51. 

RAILWAY BRIDGES. 

For Spans between 25 and 70 feet, 

Rise, J of the span. Depth of Arch, .02 of the span. 

Thickness of Abutments, from % to J of the Span. Batter, 1 in. per foot. 



556 



IRON WORKS, FLOUR AND SAW MILLS, ETC. 



COST OF TUNNELS PRIOR TO 1855. — (Major McClellan, U. S. A.) 



Location. 


Per Cubic 
Yard. 


Location. 


Per Cubic 
Yard. 


Black Rock, U. S., greywacke ) 


$ Cts. 

6.60 

3.18 
1.55 
4. 


England, freestone, marble, ) 
clay, etc., lined \ 


% Cta. 
3.46 


Blaisley, France, lined 

Blis worth, Eng. , blue clay, lined 
Blue Kidge, U. S 


Lehigh, U. S., hard granite 

Schuylkill, U. S., slate 

Union, U. S., slate 


4.36 
2. 

2. OS 



Railway Tunnels, 

In soft sandstone, U. S., without lining, per lineal yard $ 88. 

In loose ground, thick lining, per lineal yard 710. 

Ordinary brick lining, including centering, per cubic yard 8.50 

Shafts. 

Blaisley Tunnel, clay, chalk, and loose earth, per yard in depth $139.11. Deep- 
est 646 feet. 

Black Rock, 7 feet in diameter and 139 in depth, hard slate, per yard in depth 
$79.50, or per cubic yard $1S.72. 

The time required to drive the heading of the Black Rock Tunnel for 1782.5 feet 
was 23S7 turns of 12 hours each. 



iron works (england). 

Temperature of hot blast 600° 

Density of blast and of refining furnace . . . 2% to 3 lbs. per sq. inch. 
Revolutions of puddling rolls per minute, 60 ; rail rolls, 100 ; rail saw 
800. 

Horse-power (Indicated) required for different Processes. 



Blast furnace 

Refining " 

Puddling rolls with squeezers i 
and shears < 



Rail rolling train 250 

Small bar train 60 

Double rail saw 12 

Straightening 7 



10 tons bar iron per day 



ROLLING-MILLS. 

80 | Plates, for each sq. foot rolled ... 5 



FLOUR MILLS, SAW MILLS, WOOD-WORKING MACHINERY. 
Flonr IVlills. 

For each pair of 4-feet stones, with all the necessary dressing machinery, etc., there 
is required 15 horse's power. 

One pair of 4-feet stones will grind about 5 bushels of wheat per hour. Each 
bushel of wheat so ground per hour requires .87 actual or 1.11 indicated horses 1 
power, exclusive of dressing and other machinery. 

Stones, 4 feet diam., 120 to 140 revolutions per minute. 

Dressing Machines, 21 ins. diam., 450 to 500 revolutions per minute. 

Creepers, ?>% ins. pitch, 75 revolutions per minute. 

Elevator, 18 ins. diam., 40 revolutions per minute. 

Screen, 16 ins. diam., 300 to 350 revolutions per minute. 

7SS cubic feet of water, discharged at a velocity of 1 foot per second, are necessary 
to grind and di\ ss 1 bushel of wheat per hour = 1.40 horses' power per bushel. 

2000 feet per minute, for the velocity of a stone 4 feet in diameter, may be con« 
aidered a maximum speed. 



WOOD-WORKING MACHINERY, MINING, AND BLASTING. 557 

Sa-v^-mill. 

Gang saiv, 30 sq. feet of dry oak, or 45 sq. feet of dry pine, per hour. . 1 horse-power. 

Circular saw, 2.5 feet in diani., 270 revolutions per minute, 40 sq. 
feet of oak, or 70 of dry spruce 1 " 

300 revolutions per minute. 1.33 square feet of dry pine per minute, kerf ^ 
inch and 6 ins. deep, requires the power of 1 horse for the saw alone ; and 1 square 
foot, kerf 3^ inch and 1 foot in depth, requires a like power. 

4.5 feet in diameter, kerf 3^, and 1 foot in depth, requires 1 horse's power for 1.33 
feet per minute. 

Oak requires nearly one half more power than pine. 

With a kerf of % inch, 1 horse's power will saw 2.66 square feet per minute. 

The speed of the periphery should he ahout 50 feet per minute. 

"Velocities of Wood-working IVIacliiiiery in Feet or 
Ttevoliatioias per ^Vlintite. 

Circular saws, at periphery, 6000 to 7000 feet. 

Band saw, 2500 feet. 

Gang saws, 20 in. stroke, 120 strokes per minute. 

Scroll saws, 300 strokes per minute. 

Planing-machine cutters at periphery, 4000 to 6000 feet. 

Work under planing machine, ^yth of an inch for each cut. 

Molding-machine cutters, 3500 to 4000 feet. 

Squaring-up-machine cutters, 7000 to 8000 feet. 

Wood-carving drills, 5000 revolutions. 

Machine augers, \% diam., 900 revolutions. 

Machine augers, % diam., 1200 revolutions. 

Gang saws require for 45 super, feet of pine per hour, 1 horse power. 

Circular saws require for 75 super, feet of pine per hour, 1 horse power. 

In oak or hard wood, %ths f ^ ne ahove quantity require 1 horse power. 

Sharpening Angles of Machine Cutters. 
Adzing soft wood across the grain. . . 30° 1 Gouges and ploughing machines . . . 40° 
Planing machines, ordinary soft wood 35° J Hard-wood tool cutters 50° to 55° 



MINING AND BLASTING. 
MINING. 



Z 3 
In ordinary Soil, — = charge of powder in pounds, I representing half the depth 

of the line of least resistance. 

In Masonry, I 3 X C = charge in pounds; C representing a coefficient depending 
upon the structure. 

In a plane Wall, C=.15, in one witli counterforts =2, and under a foundation 
when it is supported upon two sides = .4 to .6. 

BLASTING. 

In small Masts 1 lh. of powder will loosen ahout 4>£ tons. 

In large hlasts 1 lb of powder will loosen about 2% tons. 

50 or 60 lbs. of powder, inclosed in a resisting bag, hung or propped up against a 
gate or barrier, will demolish any ordinary construction. 

One man can bore, Avith a bit 1 inch in diameter, from 50 to 100 ins. per day of 10 
hours in granite, or 30,0 to 400 ins. pfi' day in limestone. 

Two strikers and a holder can bore with a bit 2 inches in diameter 10 feet in a 
day in rock of medium hardness. 



558 WATER POWER, HYDRAULIC RAM, PUMPS, ETC. 

PROJECTION OF WATER. 

Heights to ^v\Kh.icli Water may- "be Projected, through. 
Engine IPipes -u.11.der Pressure. 



Pressure 

per Square 

Inch. 


Equivalent 

Head 
of Water. 


Height 
of Jet. 


Ratio of 
Compression 

of Air in 
Air-chamber. 


Pressure 

per Square 

Inch. 


Equivalent 

Head 
of Water. 


Height 
of Jet. 


Ratio of 

Compressioa 

of Air in 
Air-chamber. 


Lbs. 
30 
45 
60 
75 


Feet. 
68 
102 
136 
1T0 


Feet. 
33 
66 
99 

132 


.5 
.33 
.25 
.2 


Lbs. 
90 
105 
120 
150 


Feet. 
204 
238 
272 
340 


Feet, 
165 
198 
231 
29T 


.IT 

.14 

.125 

.1 



Power recjriired to raise Water from Wells by a Douhle' 
acting .Lifting-pump. 



Diameter 

of 

Pump. 



Ins. 

2 

2^ 

3 

3^ 

4 



Gallons. 
265 
420 
620 
830 
1060 



Depth from which this Volume can be raised by each Unit 
of Power. 



Man turning Donkey Horse One Horse- 

a Crank. working a Gin. working a Gin. power Engine. 



Feet. 
80 
50 
35 
25 
20 



Feet. 
160 
100 

70 
50 
40 



Feet. 
560 
350 
245 
175 
140 



Feet. 
880 
550 
385 
275 
220 



WATER POWER. 



To Compute Water-power, 
5^S HP 
.00189 Yh — horse's power, and -^ — = V; V representing volume of water. 

in cubic feet, per minute, and h head of water from race in feet. 

Effective Horse-power for different Motors. 

Theoretical power t j 

Undershot wheels — .4 I Reaction wheel = ' 2 

Poncelet's undershot wheel ... = .6 Impact wheel "5 

Breast wheel (high) = .55 ^ ,. t V, 

(low). = .6 Turbines = {.% 

Overshot wheel — / • 84 Tremont turbine = ^79 

i -64 I 



882 HP 



Turbines . 

Tremont t 
Hydraulic ram. 

HYDRAULIC RAM. 



= .6 



— _ V, .00113 V fc = HP; V representing volume of water in cubic feet per 
minute, h head of water in feet, and HP actual horsepower. 

JET PUMP. 

The greatest effect of a Jet Pump is when the depth from which the water & 
drawn through the supply or suction pipe is .9 of the height from which the water 
fell to give the jet. 

Theflow up the suction-pipe being .2 of that of the volume of the jet; hence, tin, 
effect — ." x & ^^ • 18. 

Imperial Gralloiis. 
6.2355 Gallons in a Cubic Foot. 



WAVES. 

The undulations of waves are performed in the same time as the oscillations of a 
pendulum, the length of which is equal to the breadth of a wave, or to the distance 
between two neighboring cavities or eminences. 



DAMS, TUNNELS, WIND-MILLS, AND ICE. 



559 



DAMS AND TUNNELS. 

DAMS (Earthwork). 

Width at top in high dams from 7 to 20 ft. | Breast slopes r= 3 to 1 

"Width at top in low dams . . = height. | Back slopes =z 2 to 1 

Height above surface of water not less than 3.5 feet. 

Proportion of Laborers in Bank, Fillers, and. Wheelers, 
in different Soils, Wheelers being Estimated for a Dis- 
tance of £50 Yards. 



Get- 


Fill- 


Wheel- 


ters. 


ers. 


ers. 


1 


1 


1 


1 


2 


2 


1 


2 


2 



In hard clay 

In compact gravel . . 
In rock 



Get- 
ters. 


Fill- 
ers. 


Wheel- 

ers. 


1 
1 

3 


1 
1 


1 
1 



In loo~e earth, sand, etc. 

In compact earth 

In marl 

Masonry. 

Width at bottom = .7 height ; at middle = .5 height ; and at top ±= .3 height. 

TUNNELS. — (From actual practice in Brick-work). 



Purpose. 



Canal 

Canal 

Thames Tunnel. 
Railway 



Canal . 



Formation of 
Strata, 



Various 

Clay 

Clay 

Chalk 

Various 

Shale 

Green sand 

Freestone 

Chalk and earth. 



Extreme 
Height. 



Feet. Ins. 

16 2 

21 6 

22 3 
26 6 
2T 6 
30 

30 G 

36 

39 



Extreme 
Width. 



Feet. Ins. 

17 

20 

37 6 

27 

27 

30 

30 

36 

35 6 



Depth 
at Crown. 



Feet Ins. 

1 3 

1 6 

2 6 
1 6 

110^ 

1 lGtf 

2 3 
2 3 



23° — 



18rf2 



WIND-MILLS. — (Molesworth.) 

To Compute the Angles of the Sails. 

.angle of the sail with the plane of motion at any part of the sail ; 

r representing radius of sail in feet, and d distance of any part of the sail from the 
axis. 

Axis of Shaft of Wind-mill with Horizon. 

8° upon level ground. 

Breadth of whip at axis, ^ length of whip. 

Depth " 



40 
JL 
60 
_1_ 
80 



Depth " 

Width of sail 

Divided by the whip in the proportion of 5 to 3, the narrow portion being nearest to 
the wind. 

Width of sail at axis, i length of whip ; distance of sail from axis, ith length 
of whip. 

Cross-bars from 16 to 18 inches apart. 



STRENGTH OF ICE. 

Thickness, 2 in3. will bear infantry. 

u 4 JJ cavalry or light guns. 

u 6 u heavy field-guns. 

" 8 " upon sledges, a weight not exceeding 1000 lbs per sq. ft. 



560 BEAMS, REVOLVING DISC, CASTINGS, AND SCALE. 

STIFFNESS OF BEAMS. 

Stiffness of Beams.-(TBKDGOLD.) 

V- == d ; ; — = b ; b representing breadth, and d depth in inches, I lei 
b d'^ 
in feet, and W load in lbs. upon the middle. 
C = Pine .01, Ash .01, Beech .013, Elm .015, Oak .13, Teak .008. 
When the beam is uniformly loaded, put .625 W instead of W. 

Resistance to Detrusion. 

"When one beam is let in, at an inclination to the depth of anotber, so as to bear 
in the direction of the fibres of the beam tbat is cut, the depth of the cut at right 
angles to the fibres should not be more than J- of the length of the piece, the fibres 
of which, by their cohesion, resist the pressure. 

To Oompnte the T^engtn necessary to resist a given Hor- 
izontal Thrush, as in the Case of a Xrtafter let into a Tie- 
Beam. 

4 T 

— - = 1; b representing the breadth of the beam in inches, T the horizontal thrust 

be 
in lbs., c the cohesive resistance of the material in lbs. per sq. inch, and I the length 
in inches. 



REVOLVING DISC. 
To Compute the Power. 

Rule. — Multiply one half the weight of the disc by the height due to 
the velocity of its circumference in feet per second. 

Example. — A grind-stone Z% feet in diameter, weighing 2000 lbs., is required 
to make 362^ revolutions per minute ; what power must be communicated to it? 

Circum. of 3% = 10.6/eef, which X 362.25 and ^- 60 = 04 feet per second. Then 
2000-^2 X 64=64000 lbs. raised lfoot. 

Note. — If the revolving disc is not an entire or solid wheel, being a ring or annu- 
lus, it must first be computed as if an entire disc, and then the portion wanting 
must be computed and deducted. 

Power Concentrated, in IVtoving Bodies. 

Simple power is force multiplied by its velocity. Power concentrated in a moving 
body is tbe weight of the body multiplied by the square of its velocity; and the prod- 
uct divided by the accelleratrix, or the power concentrated in a moving body, is 
equal to the power expended in generating the motion. 



SHRINKAGE OF CASTINGS. 



Iron, small cylinders = ^ in. per ft. 

" Pipes =y s " 

u Girders, beams, etc. = % in 15 ins. 
" Large cylinders, } 

the contraction > — ^ per foot. 

of diam. at top. ) 
u Ditto at bottom — 3^2 per ^ oot ' 



Ditto, in length = % in 1C ins. 

Brass, tbin = % in 9 ins. 

Brass, thick = > 4 r in 10 ins. 

Zinc = ^ in a foot. 

Lead = %. in a foot. 

Copper = %> in a foot. 

Bismuth = ^ * n a f°°t- 



VERNIER SCALE. 

The Vernier Scale is ^*, divided into 10 equal parts ?o tbat it divides a scale 
of lOths into lOCths when the lined meet in the two scales. 



TONNAGE OF VESSELS. 561 

Measurement and. Computation of tlie Tonnage of* Ves- 
sels under tlie A.ct of Congress of* 6th. May, 1864r. 

Measurements are expressed in feet and decimals of a foot, and tonnage 
in tons and hundredths of a ton. 

The " Tonnage Length" is the length along the middle line of the ves- 
sel upon the under side of the tonnage-deck plank, but for convenience is 
measured upon the top of the deck, and is the length between these ex- 
tremities, which is divided into a number of parts, according to the classi- 
fication under the law. 

The depths are perpendicular and the breadths horizontal ; the upper 
breadth, which in eveiy case passes through the top of the tonnage depth, 
being at a distance below the deck, at its middle line, equal to one third 
of the spring of the beam at that point, and thus passing through the deck 
upon each side ; and the lower breadth, which is at the bottom of the ton- 
nage depth, being at a distance above the upper side of the floor timber 
at the inside of the limber-strake, equal to the average thickness of the 
ceiling, and thus passing through the keelson. 

The " spring of the beam" is the perpendicular distance from the crown 
of the tonnage deck at the centre to a line stretched from end to end of the 
beam, and must be ascertained at each point where it is to be used in the 
measurement. 

The Register of every vessel expresses her length and breadth, together 
with her depth, and the height under the third or spar deck is ascertained 
in the following manner : The tonnage deck, in vessels having three or 
more decks to the hull, is the second deck from below ; in all other cases 
the upper deck of the hull is the tonnage deck. The length from the fore 
part of the outer planking, upon the side of the stem, to the after part of 
the main stern-post of screw steamers, and to the after part of the rudder- 
post of all other vessels, measured upon the top of the tonnage deck, is ac- 
counted the vessel's length. The breadth of the broadest part upon the 
outside of the vessel is accounted the vessel's breadth of beam. A meas- 
ure from the under side of tonnage-deck plank, amidships, to the ceiling 
of the hold (average thickness), is accounted the depth of hold. If the 
vessel has a third deck, then the height from the top of the tonnage-deck 
plank to the under side of the upper-deck plank is accounted as the height 
under the spar deck. 

The register tonnage of a vessel is her internal cubical capacity in tons 
of 100 cubic feet each, to be ascertained as follows : From the inside of 
the inner plank (average thickness) at the side of the stem to the inside 
of the plank upon the stern timbers (average thickness), deducting from 
this length what is due to the rake of the bow in the thickness of the deck, 
and what is due to the rake of the stern timber in the thickness of the 
deck, and also what is due to the rake of the stern timber in one third 
of the spring of the beam. 

Classes* 

Class 1. Vessels of which the tonnage length is 50 feet or under. 

2. Over 50 feet, and not exceeding 100 feet in length. 

3. Over 100 feet, and not exceeding 150 feet in length. 

4. Over 150 feet, and not exceeding 200 feet in length. 

5. Over 200 feet, and not exceeding 250 feet in length. 
Go Over 250 feet in length. 

If there is a break, a poop, or any other permanent closed-in spacs upon the upper 
decks, or upon the spar deck, available for cargo, or stores, or for the berthing or ac- 
commodation of passengers or crew, the tonnage of such space is computed. 

If a vessel has a third deck, or spar deck, the tonnage of the space between it and 
the tonnage deck is computed. 

In computing the tonnage of opan vessels, the upper edge of the upper strake is 
to form the boundary-line of measurement, and the depth shall be taken from an 

3B 



562 TONNAGE OF VESSELS. 

athwart-ship line, extending from the upper edge of said strake at each division of 
the length. 

The register of a vessel expresses the numher of decks, the tonnage under the 
tonnage deck, that of the between decks, above the tonnage deck ; also that of the 
poop or other inclose dspaces above the deck, each separately. In every registered 
IT. 6. vessel the number denoting the total registered tonnage must be deeply carved 
or otherwise permanently marked upon her main beam, and shall be so continued ; 
and if it at any time cease to be so continued, such vessel shall no longer be recog- 
nized as a registered U. S. vessel. 

Recapitulation of Measurements. 

Register Length. — Length at the middle of the 2d deck from below, in vessels of 
two or more decks, and in all other vessels of the upper deck, measured from the 
fore part of the outer planking upon the side of the stem, to the after part of the main 
stern-post of single screw propeller steamers, and to the after part of the rudder-post 
of other vessels, measured upon the top of the tonnage deck. 

Tonnage Length. — Length at upper side of tonnage-deck beams, from the inside 
of the inboard plank, at its average thickness at the side of the stem to the inside of 
the plank upon the stern timbers at its average thickness, deducting from this length 
that which is due to the rake of the boAv in the thickness of the deck, and of the 
stern timber in the thickness of the deck, and one third the spring of the beam. 

Breadth of Beam — At the broadest part of the outside of the vessel. 

Depth of Hold.— Height measured from the under side of tonnage-deck plank 
amidships from a point at a distance of one third the spring of the beam to the ceil- 
ing of the hold at its average thickness. 

Height under Spar Deck. — The mean height from top of tonnage-deck plank to 
the under side of the upper-deck plank. 

Open Vessels. — The upper edge of the upper strake is to be the boundary -line of 
measurement of length, and the depth is to be measured from a line running athwart..- 
ships from the upper edge of the upper strake at each division of the length. 

By an Act of Congress of 2Sth February, 1865, the preceding rule of admeasure- 
ment was amended as follows : No part of any ship or vessel shall be admeasured 
or registered for tonnage that is used for cabins or state-rooms, and constructed en- 
tirely above the first deck, which is not a deck to the hull. 

carpenters' measurement. 

For a Single-deck Vessel. 

RtTLE.— Multiply the length of keel, the breadth of beam, and the depth of the hold 
togather, and divide by 95. 

For a Double-deck Vessel. 

Rule.— Multiply as above, taking half the breadth of beam for the depth of the 
hold, and divide by C5. 

BRITISH MEASUREMENT. 

Divfde the length of the upper deck between the after part of the stem and the fore part of the 
stern-post into 6 equal parts, and note the foremost, middle, and aftermost points of division. Meas- 
ure the depths at these three points in feet and tenths of a foot, also the depths from the under side of 
the upper deck to the ceiling at the limber-strake ; or, in case of a break in the upper deck, from a 
line stretched in continuation of the deck. For the breadths, divide each depth into 5 equal parts, 
and measure the inside breadths at the following points, viz. : at .2 and 8 from the upper deck of the 
foremost and aftermost depths, and at .4 and .8 from the upper deck of the nmidship depth. Take the 
length, at half the amidship depth, from the after part of the stem to the fore part of the stern-post. 

Then, to twice the amidship depth, add the foremost and aftermost depths for the sum of the dep>ths ; 
and add together the foremost upper and lower breadths, 3 tim*.s the upper breadth with the lower 
breadth at the midship, and the upper and twice the lower breadth at the after division for the sum 
of the breadths. 

Multiply together the 6nm of the depths, the sum of the breadths, and the length, and divide the 
product by 3500, which will give the number of tons, or register. 

If the vessel has a poop or half deck, or a break in the upper deck, measure the inside mean length, • 
breadth, and height of such part thereof as may be included within the bulkhead ; multiply these 
three measurements together, and divide the product by 92.4. The quotient will be the number of 
tons to be added to th» result, as above ascertained. 

For Open Vessels. — The depths are to be taken from the upper edgre of the upper strake. 

For Steam Vessels. — The tonnage due to the engine-room is deducted from the total tonnage com- 
puted by the above rule. 

To determine this, measure the inside length of the engine-room from the foremost to the aftermost 
bulkhead ; then multiply this length by the amidship depth of the vessel, and the product by the inside 
•midship breadth at .4 of the depth from the deck, and divide the finaf product by 92.4. 



WEOUGHT-IRON BEAMS, ETC. 



563 



General Ifcnle to 



Compute tlie 
!M!acliine. 



"Work done "by- any 



Ascertain the distance through which the power, P, applied to the machine has 
operated in one minute, and represent it by a. 

Ascertain the distance through which the weight, W, producing useful work, has 
operated in one minute, and represent it by b. 

Then, a P — b W i± work done by friction per minute. 
a P = work applied per minute. 

bW = useful work done per minute. 

[Mechanical Laws of Elastic Flu-ids. 

Boyle's or Mariotte's Law. — The elastic force of a gas or air at a given tempera- 
ture is inversely proportional to the space which it occupies. 

Letp and P represent elastic forces of a gas when they occupy the spaces s and S. 



The elastic force of any gas at a given temperature is proportional to its density. 



"Wronglxt Iron Beams. 

{Trenton Iron Works, Cooper, Hewitt, & Co., N. Y.) 











Load 










Load 




Thick- 


Width 


Weight 


borne with 
Safety. 

c 




Thick 


Width 


Weight 


borne with 

Safety. 

7= w - 


Depth. 


ness of 


of 


per Lineal 


Depth. 


ness of 


of 


per Lineal 




Web. 

/ 


Flanges. 


Foot. 


t= w . 




Web. 


Flanges 


Foot 


Ins. 


Ins. 


Ins. 


Lbs. 


C in Lbs. 


Ins. 


Ins. 


Ins. 


Lbs. 


C in Lbs. 


6 


X 


3 


13.3 


76 000 


9 


% 


4 


30 


246 000 


C 


«" 


3^ 


1G.G 


9;2 000 


9 


% 


5% 


50 


44S0H0 


7 


% 


3X 


20 


124000 


nx 


X, 


*# 


40 


390 000 


9 


% 


oH 


23.3 


192 oeo 


15 


±Xe 


51.6 


640 000 


9 


Xs 


4 


28 


240 000 


15 


% 


5% 


6G.6 


90S 000 



Load uniformly distributed, Beam resting upon two supports, I representing length 
in feet, and W weight in pounds = .3 of breahing or ultimate strain. 

Illustration. — What is the weight, uniformly distributed, that may be borne 
with safety by floor beams of the above description resting upon two supports, 20 
feet in length,' 9 ins. in depth, % in. width of web, and 4 ins. width of flange ? 

C = 240 000. 24 ^°° = 12 000 lbs. 

See page 4T0 for other Formulae and Illustrations. 



564 FUEL. 

FUEL. 

With equal weights, that which contains most hydrogen ought, in its 
combustion, to produce the greatest volume of flame where each kind is 
exposed under like advantageous circumstances. Thus, pine wood is 
preferable to hard wood, and bituminous to anthracite coal. 

When wood is emploj^ed as a fuel, it should be as dry as practicable. 
To produce the greatest quantity of heat, it should be dried b}- the direct 
application of heat ; as usually employed, it has about 25 per cent, of 
water mechanically combined with it, the heat necessary for the evapora- 
tion of which is lost. 

Different fuels require different volumes of oxygen ; for the different 
kinds of coal it varies from 1.87 to 3 lbs. for each lb. of coal. 60 cubic 
feet of air is necessary to furnish 1 lb. of oxygen ; and, making a due al- 
lowance for loss, nearly 90 cubic feet of air are required in the furnace of 
a boiler for each lb. of oxygen applied to the combustion. 

Bitvimiixcms Coal. 

Lignite. Brown Coal or Bituminous Wood. — Presents a distinct woody 
structure ; is devoid of taste, brittle, and burns readilv, leaving a white 
ash. This coal contains and absorbs moisture in some cases fully 40 per 
cent. 

Caking Coal. — Fractures uneven ; color varying from a resinous to a 
gray-black, and when heated breaks into small pieces, which afterward 
agglomerate and form a compact body. When the proportion of bitumen 
is great, it fuses into a past} r mass. This coal is unsuited where great 
heat is required, as the draught of a furnace is impeded by its caking. 
It is applicable for the production of gas and coke. 

Splint or Hard Coal. — Color black or brown-black, lustre resinous and 
glistening. When broken, the principal fracture appears irregular and 
slaty, the transverse being fine grained, uneven, and splintery. It kin- 
dles less readHy than caking coal, but when ignited produces a clear and 
hot fire. 

Cherry or Soft Coal. — Alike to splint coal in its fracture and appear- 
ance, but its lustre is more splendent. It does not fuse when heated, is 
very brittle, ignites readilv, and produces a bright fire with a clear } T ellow 
flame, but consumes rapidly. 

Cannel or Parrot Coal. — Color jet, or gray or brown black, compact 
and even texture, a shining, resinous lustre. Fractures smooth or flat, 
conchoidal in ever}- direction, and polishes readily. From its decrepita- 
tion when exposed to heat it is termed parrot coal. 

Experiments upon the practical burning of this description of coal in 
the furnace of a steam-boiler give an evaporation of from 6 to 10 lbs of 
fresh water, under a pressure of 30 lbs. per square inch for 1 lb. of coal ; 
Cumberland (Md., U. S.) coal being the most effective, and Scotch the 
least. 

Couls that contain sulphur, and are in progress of decay, are liable to spontaneous 
combustion. 

The limit of evaporation from 212° for 1 lb. of the best, assuming all 
of the heat evolved from it to be absorbed, would be 14.9 lbs. 

.Ajntliracite Coal. 

Anthracite or Glance Coal, or Culm. — Is hard, compact, lustrous, and 
sometimes iridescent, the most perfect being entirely free from bitumen; 
it ignites with difficulty, and breaks into fragments when heated. 

The evaporative power of this coal, in the furnace of a steam-boiler and 
under pressure, is from 7W to 9)^ lbs. of fresh water per lb. of coal. 

Coals from one pit will vary 6 per cent, in evaporative value. 



FUEL. 



565 



Coke. 

Coke. — Coking in a close oven will give an increase of yield of 40 per 
cent, over coking in heaps, the gain in bulk being 22 per cent. Coals 
when coked in heaps will lose in bulk. 

Cannel and Welsh (Cardiff) coals when coked in retorts will gain 30 
per cent, in bulk and lose 36.5 per cent, in weight. 

The relative costs of coal and coke for like results, as developed by an 
experiment in a locomotive boiler, are as 1 to 2.4. 

Its evaporative power, in the furnace of a steam-boiler and under press" 
aire, is from 1% to 8/4 l° s - °f fresh water per lb. of coke. 

Charcoal. 

Charcoal. — The best quality is made from Oak, Maple, Beech, and 
Chestnut. 

Wood will furnish, when properly burned, about 23 per cent, of coal. 

Charcoal absorbs, upon an average of the various kinds, about 5.5 per 
cent, of water, Oak absorbing about 4.28, and Pine 8.9. 

Its evaporative power, in the furnace of a boiler and under pressure, is 
5J£ lbSo of fresh water per lb. of coal. 

The volume of air chemically required for the combustion of 1 lb. of 
charcoal is 129.5 cubic feet. 

138 bushels charcoal and 432 lbs. limestone, with -2612 lbs. of ore, will 
produce 1 ton of pig iron. 

Produce of Charcoal from various Woods. 



Apple 23.8 

Ash 26.7 

Beech 21.1 



Birch 24.1 I Oak 22.85 

Elm. 25.1 W young . . 33.3 

Maple 22.9 | Poplar 20.5 



Red Pine 23. 

White Pine . . 23.5 
Willow 18.6 



The produce of charcoal by a slow process of charring is very nearly 
50 per cent, greater than by a quick process. 

Wood. 
Weights and Comparative Values of different Woods. 

Woods. Cord. Valu 



Shell-bark Hickory . . 
Red- heart Hickory . . 

White Oak 

Red Oak 

Virginia Pine 

Southern Pine 

Hard Maple 



Lbs. 
4469 
8705 
8821 

3251 
2689 
3375 

2S78 



1. 

.SI 
.SI 
.69 



Woods. 


Cord. 


Value. 


New Jersey Pine 

Yellow Pine 

White Pine 


Lbs. 

2137 
1904 

1S68 


.54 
.43 
.42 


Beech 


7 


Spruce 


52 


Hemlock 

Cottonwood 


.44 



The evaporative power of 1 cubic foot of pine wood is equal to that of 
1 cubic foot of fresh water; or, in the furnace of a steam-boiler and under 
pressure, it is 4% lbs. fresh water for 1 lb. of wood. 

Northern Wood. — One cord of hard wood and one cord of soft wood, 
such as is used upon Lakes Ontario and Erie, is equal in evaporative 
effects to 2000 lbs. of anthracite coal. 

Western Wood. — One cord of the description used by the river steam- 
boats is equal in evaporative qualities to 12 bushels (9G0 lbs.) of Pittsburg 
coal. 

9 cords cotton, ash, and cypress wood are equal to 7 cords of yellow 
pine. 

The solid portion (ligniri) of all woods, wherever and under whatever 
circumstances of growth, are nearly similar, the specific gravity beinir as 
1.46 to 1.53. 

3B* 



566 



FUEL. 



The densest woods give the greatest heat, as charcoal produces greatei 
heat than flame. 

for every 14 parts of an ordinary pile of wood there are 11 parts of 
space ; or a cord of wood in pile has 71.68 feet of solid wood and 56.32 feet 
of space. 

Trees in the early part of April contain 20 per cent, more water than 
the}- do in the end of January. 

Ash. 

Proportion of Ash in 100 lbs. of several Woods. 

Wood. 



Ash... 
Beech . 
Birch . 



Per Cent. 
.5 
.35 
.34 



Leaves. 


Woods. 


Wood. 


Leaves. 


Per Cent. 

5T4 
5. 


Elm 

Oak 

Pitch Pine 


Per Cent. 

1.8S 

.21 

.25 


Per Cent 
11.8 
4. 
3.15 



Feat. 

Peat. — The proportion of ash in peat varies very much, ranging from 
1.25 per cent, in grass peat to 18.47 per cent, in other varieties, the mean 
of Irish peat being about 3.5 per cent. 

The distillation of peat produces, upon an average, Water 31 parts, 
Tar 3, Charcoal 29, and Gas 37. 

In the distillation "of peat, the following products have been obtained : 

Charcoal, 41.1 per cent.; Watery Liquor, 19.3; Tar, .6; and Gaseous 
matter, 39, 

Its evaporative power, in the furnace of a steam-boiler and under press- 
ure, is from 0% to 5 lbs. of fresh water per lb. of fuel. 

Average Composition of* Fuels. 



Bituminous Coals. 

Welsh 

Duffryn 

Newcastle 

Scotch 

Derbyshire 

Lancashire 

Sydney, S. W 

Borneo 

Formo-a Island 

Vancouver s Island 

Chili, Conception Bay . . 

" Chiriqni 

Patngonia 

V. Diemen's) S. Cape . . 

Land / Adv'teB.. 

Cannel, Wigau 

Cumberland 

Anthracite 

Oak 

White Pine 

Birch 

Charcoal. Oak 

" Pine 

44 Maple 

Peat, den?e 

Patent, Warlich'a 

11 Wylara'3 



Specific 
Grav- 




Hydro- 


Nitro- 


Sul- 


Oxy- 


Ash. 


] Percent- 
age of 


ity. 




gen. 


gen. 




gen. 




1 Coke. 


1.32 


S3. 78 


4.79 


.98 


1.43 


4.15 


4.91 


72.6 


1.33 


S8 26 


4.66 


145 


1.77 


.6 


3.26 


84.3 


1.26 


82.24 


5.42 


1.61 


1.35 


6.44 


2.94 


60.67 


1.26 


78.53 


5.61 


1. 


1.11 


9.69 


4.03 


54 22 


1.29 


79.85 


4 84 


1.23 


.72 


10.S6 


2.4 


59 32 


1.28 


78. 


5.23 


1.32 


1. 


8 75 


5.69 


60.22 





82.39 


5.32 


1.27 


.07 


8.32 


2.04 


58. 


1.2S 


64.52 


4.74 


.8 


14S 


20.75 


7.74 


— 


1.24 


7S.26 


5.7 


.64 


.49 


10.95 


3.96 


— 





66.93 


5.32 


1.02 


2.2 


S.7 


15.83 


— 


1.29 


70.55 


5.76 


.95 


1.98 


13.24 


7.52 


— 





3S.9S 


4.01 


.58 


6.14 


13. 3S 


36.91 


— 





62.25 


5.05 


.63 


1.13 


17.54 


13.4 


— 





63.4 


2.89 


1.27 


.98 


U»l 


30.45 


— 





80.22 


3.<5 


1.30 


1.9 


4.S 


8.67 


— 


1.23 


79.23 


6.08 


1.1s 


1.43 


7.24 


4 84 


60.33 


— 


93.81 


1.S2 





— 


2.77 


1.6 


— 


1.5 


88.54 








.52 


— 


8.67 


— 





48.13 


5.25 


— 





44.5 


1.3 


— 





49.95 


6.41 








43.05 


.31 


— 





48.12 


6.37 


• — 





45. 


.48 


— 





S7.68 


2.83 


— 


— 


6.43 


3.06 


— 


— 


71.36 


5.95 


— 





22.19* 


.3 


— 





70.07 


4.61 


— 


— 


24. S9* 


.43 


— 





61.02 


5.77 


.SI 





32.4 


— 


— 


1.15 


90.02 


5 56 





1.62 


— 


2.91t 


85.1 


1.1 


79.91 


5 69 


1.6S 


1.25 


6.63 


4.84 


65.8 



* Including Nitrogen. 



t Including Oxygen. 



FUEL. 



567 



Weights, Evaporative Powers per "Weight and. Bulk, 
etc., of different Fuels. — (W.R.Johnson and others,) 



Spec. 
Grav. 



Weight 

per 

Cubic Foot. 



Steam from 

Water at 

212° by 1 lb, 

of Fuel. 



Clinker 
from 100 lbs. 



Cubic Feet 

required 

to Stow a 

Ton. 



Bituminous. 
Cumberland, maximum 
" minimum 

Duffryn 

Cannel, Wigan 

Blossburgh 

Midlothian, screened. . . 
u average . . . 

Newcastle, Hartley 

Pictou. 

Pittsburg 

Sydney 

Carr's Hartley 

Clover Hill, Va 

Cannelton, Ind 

Scotch, Dalkeith 

Chili 

Japanese, Takasmia . . . 

Anthracite. 

Peach Mountain , 

Forest Improvement. . . 
Beaver Meadow, No. 5 . 

Lackawanna 

Welsh, Jones & Co 

Beaver Meadow, No. 3 . ', 

Lehigh , 

Patent, Warlich's 

Coke. 

Natural Virginia , 

Midlothian 

Cumberland 

Charcoal 

Peat 

Wood. 

Pine wood, dry 



1.313 

1.33T 

1.326 

1.23 

1.324 

1.283 

1.294 

1.257 

1.318 

1.252 

1.338 

1.262 

1.285 

1.273 

1.519 

1.231 



1.464 

1.477 

1.554 

1.421 

1.375 

1.61 

1.59 

1.15 



1.323 



Lbs. 

52.92 

54.29 

53.22 

48.3 

53.05 

45.72 

54.04 

50.82 

49.25 

46.81 

47.44 

47.88 

45.49 

47.65 

51.09 

48.3 



53.79 
53.66 
55.19 
48.89 
58.25 
54.93 
55.32 
69.05 



46.64 

32.7 

31.57 

24. 

30. 



21.01 



Lbs. 
10.7 
9.44 

10.14 
7.7 
9.72 
8.94 
8.29 
8.66 
8.41 
8.2 
7.99 
7.84 
7.67 
7.34 
7.08 
5.72 



10.11 
10.06 

9.S8 
9.79 
9.46 
9.21 
8.93 
10.36 



8.47 
8.63 
8.99 
5.5 
5. 



4.69 



Lbs. 
2.13 
4.53 



3.4 
3.33 
8.82 
3.14 
6.13 
.94 
2.25 
1.S6 
3.86 
1.64 
5.63 



3.03 
.SI 
.6 

1.24 

1.01 

1.08 



5.31 
10.51 
3.55 



No. 

42.3 

41.2 

42.09 

46.37 

42.2 

49. 

41.4 

44. 

45. 

47.8 

47.2 

46.7 

49.2 

47. 

43.3 



41.6 
41.7 

39.8 

45.8 

38.45 

40.7 

40.5 

32.44 



48.3 
68.5 
70.9 
104. 
75. 



106.6 



(Sir H. pe la Bechk and Dr. Lyon Playfair, 1851.) 
(Averages of all Experiments.) 



Fuels. 


Rate of Evap- 
oration or lbs. 
Evaporated 
per Hour. 


Weight 

per 

Cubic Foot. 


Steam from 

Water at 212° 

by 1 lb. of 

Fuel. 


Coal. Welsh 


Lbs. 
448 
411 
448 
431 
433 
458 
484 
409 
419 
549 


Lbs. 
53.1 
49.8 
49.7 
50. 
47.2 
69. 
65.6 
61.1 
65. 
65.3 


Lbs. 
9.05 


Newcastle 

Lancashire 

Scotch 

Derbyshire 


8.37 
7.94 

7.T 

7.58 


Patent Fuel. Warlich's 


10.36 


Livingstone's 


10.03 


Lyon's 


9.5S 


Wylam's 


8.9 


Bell's 


S.53 



10 lba. fresh water have been evaporated in a tubular boiler by 1 lb. of anthracite 
coal. 



568 



FUEL. 



.Mean Relative Evaporating Power of different Fuels 
and. Total Heat of Combustion. 



Fuel. 




W T ater 
Evaporated 
from 21-2°. 


Evaporate 
Power. 


Total Heat 

in 

Thermal Units. 




Lbs. 
9.5 

8.75 

9. 

8.5 
4.35 
5.5 
10.36 
8.53 


1. 

.92 

.£5 
.80 
.45 
.5S 
1.09 
.89 


15225 
14700 
15S37 
15080 
13 620 
12 760 
7 215 
9 660 


Bituminous coal - 


u caking 
u cannel 






Coke, natural ......... 




u artificial 

Pine wood 




Peat 




Patent fuel, AYarlich's, maximum 

u Bell's, minimum 



Irtelati 

Description of Coal, etc. 


ve "Val 

ff'llfc 


Relative Evapo- Z 
rative Power for ^ 
equal Weights 
of Coal. 


differe 

* £ » 


;nt Fw 

~ c 

S3 

P5~ 


els. 

i 

> § 
W 


o c = 


■% to 


Anthracites. 
Peach Mountain, Pa. . . . 
Beaver Meadow, No. 5 . . 

Bituminous. 


10.7 

9.S3 

S.66 

8.4S 
7.S4 
7.34 
6.95 

4.69 


1. 

.923 

.809 
.792 
.733 
.6S6 
.649 

.436 


l. 

.9S2 

.776 
.73S 
.6(53 
.616 
.625 

.175 


.505 
.207 

.505 
.5SS 
.5S1 
1. 
.521 


.633 

.74S 

.SS7 
.41S 

i. 

.0S4 
.499 

16.417 


.725 
.6 

.346 
1. 
.333 
.578 
.649 


.945 
1. 

.904 


Pictou (Cunard's) 


.S76 
.S52 




.84S 




.909 


Pine icood, dry 





Destructive Distillation, of various Coals. 



Coal. 


Coke. 


Tar. 


Water. 


Ammo- 
nia. 


Carbon. 
Acid. 


Sulph. 
Hydro- 
gen. 


defiant 
Gas and 
Hydro- 
carbon. 


Other 
Gases 
Inflam- 
mable. 


Anthracite 

Oldcastle Fiery Vein 

Binea Coal 

Llangennach 


92.9 

70.8 
8S.1 
83.69 


5.86 
2. OS 
1.22 


2.S7 
3 39 
3.58 
4.07 


.2 

.35 
.08 
.08 


.06 

.44 

1.6S 

3.21 


.04 
.12 
.09 
.02 


.27 
.31 
.43 


3.93 
9.77 
4.08 
7.28 



!M!iscellaneoxxs. 

One pound of anthracite coal in a cupola furnace will melt from 5 to 10 lbs. of cast 
iron; 8 bushels bituminous coal in an air furnace will melt 1 ton of cast iron. 

Small coal produces about % the effect of large coal of the same description. 

Experiments by Messrs. Stevens at Bordentown, N. J., gave the following results : 

Under a pressure of 30 lbs., 1 lb. pine wood evaporated 3.5 to 4.75 lbs. water. 
1 lb. Lehigh coal, 7.25 to 8.75 lbs. 

Bituminous coal is 13 per cent, more effective than coke for equal weights; and in 
England the effects are alike for equal coste. 

Radiation from Fuel.— The proportion which the heat radiated from incandes- 
cent fuel bears to the total heat of combustion is, 

From Wood 29 | From Charcoal and Peat 5 

The least consumption of coal yet attained is \% lbs. per indicated horse-power. 
It usually varies in different engines from 2 to 8 lbs. 

The hulk of pine wood is about 5>j times as great as its equivalent bulk of bitu* 
minous coal. 



FUEL. COMBUSTION. 



569 



Experiments undertaken by the Baltimore and Ohio R. R. Co. determined the 
evaporating effect of 1 ton of Cumberland coal (2240 lbs.) equal to 1.25 tons of anthra- 
cite, and 1 ton of anthracite to be equal to 1.75 cords pine wood ; also that 2000 lbs. 
Lackawanna coal were equal to 4500 lbs. best pine wood. 

Relative Evaporation of several CoralDiisti"bles in Pounds 
ofWater, Heated 1° toy 1 Lb. ofthe ^Material. 
Combustible. Composition Water Combustible. Composition. Water. 



Alcohol 812 

Bituminous Coal . . 

Carbon 

Coke 

Hydrogen (mean) . 

Oak wood, dry... . 



JHyd. .12) 
\Carb. .45) 
(Hyd. .04) 
(Carb. .75) 

Carb. .84 

/Hyd. .06.) 
\Carb. .53/ 



Lbs. 

8120 

9830 

14220 

9028 
50854 

6018 



Oak wood, green . . . 

Olive Oil 

Peat, charred 

" dry 

Pine wood, dry 

Sulphuric Ether. .7 

Tallow 



/Hyd. .13) 

\Carb. .77/ 

Carb. .4 



(Hyd. 
\Carb. 



13) 



Lbs. 

5662 

J 4560 

5620 

3900 
3618 

8680 

14560 



1 lb. Hydrogen will evaporate 62.6 lbs. water from 212°=60.509 lbs. heated 1°. 

1 lb. Carbon " 14.6 lbs. " 212°, or raise 12 lbs. water at 

60° to steam at 120 lbs. pressure,. 

A pound of Oxygen will generate the same quantity of heat whether in combus- 
tion with hydrogen, carbon, alcohol, or other combustible. 

Areas and 3?rod\ictiori.s of Coal Fields. 



State. 


Sq. Miles. 


State 


Sq. Miles. 


State. 


Sq. Miles. 


Illinois 


44000 
21000 
15437 
13500 


Ohio 

Indiana 

Missourif 

Michigan! 


11000 
7700 
6000 
5000 


Tennessee 

Alabama 

Maryland 

Georgia 


4300 


Virginia 

Pennsylvania*. . . 
Kentucky . , 


3400 
550 
150 



Countries. 


Area 


Coal raised 1 
in 1845 


Countries 


Area. 


Coal raised 
in 1845. 


Great Britain .... 


Sq. Miles 

11 S5) 

529 

133 132 


Tons 1 
31 500 000 

4960977 
4400 000 


France 


Sq. r.Iiles. 


Tons. 

41 1 617 


P» Igiuiri 


Prussia 


\i 500 000 


United States 


Austria 


659 340 



COMBUSTION. 

Combustion is one of the man}' sources of heat, and denotes the com- 
bination of a body with any of the substances termed Supporters of Com- 
bustion ; with reference to the generation of steam, we are restricted to 
but one of these combinations, and that is Oxygen. 

All bodies, when intensely heated, become luminous. When this heat 
is produced b) T combination with oxygen, they are said to be ignited ; and 
when the body heated is in a gaseous state, it forms what is termed Flame. 

Carbon exists in nearly a pure state in charcoal and in soot. It com* 
bines with no more than 2% of its weight of ox} T gen. In its combustion, 
1 lb. of it produces sufficient heat to increase the temperature of 14 500 lbs. 
of water 1°. 

Hydrogen exists in a gaseous state, and combines with 8 times its weight 
of oxygen, and 1 lb. of it, in burning, raises the heat of 50000 lbs. of water l c $ 

An increase in the rapidity of combustion is accompanied b} r a diminu- 
tion in the evaporative efficiency of the combustible. 



* Bituminous and Anthracite. 



X Mean effect. 



! 



5*70 COMBUSTION. 

COMBUSTION OF FUEL. 

The constituents of coal are Carbon, Hydrogen, Azote, and Oxygen. 

The volatile products of the combustion of coal are hydrogen and car 
bon, the unions of which (relating to combustion in a furnace) are Car 
buretted hydrogen and Bi-carburetted hydroyen or defiant gas, which, upon 
combining with atmospheric air, becomes Carbonic acid or Carbonic Oxide, 
Steam, and uncombined Nitrogen. 

Carbonic oxide is the result of imperfect combustion, and Carbonic acid 
that of perfect combustion. 

The perfect combustion of carbon evolves heat as 15 to 4.55 compared 
with the imperfect combustion of it, as when carbonic oxide is produced. 

1 lb. carbon combines with 2.66 lbs. of oxygen, and produces 3.66 lbs. 
of carbonic acid. 

Smoke is the combustible and incombustible products evolved in the combustion 
of fuel, which pass off by the flues of a furnace, and it is composed of such portions 
of the Hydrogen and Carbon of the fuel gas as have not been supplied or combined 
with oxygen, and consequently have not been converted either into Steam or Car- 
bonic acid ; the Hydrogen so passing away is invisible, but the Carbon, upon being 
separated from the Hydrogen, loses its gaseous character, and returns to its element- 
al-;-: stale of a black pulverulent body, and as such it becomes visible. 

The bituminous portion of coal is converted into the gaseous state alone, the car- 
bonaceous portion only into the solid state. It is partly combustible and partly in- 
combustible. 

To effect the combustion of 1 cubic foot of coal gas, 2 cubic feet of oxygen are re- 
quired; an>"_ as 10 cubic feet of atmospheric air are necessary to supply this volume 
of oxygen, 1 a*g>ic foot of gas requires the oxygen of 10 cubic feet of air. 

In furnace^- with a natural draught, the volume of air required exceeds 
that when th* -draught is produced artificially. 

An insufficient supply of air causes imperfect combustion ; an excessive 
supply, a waste of heat. 

The quantit} r of atmospheric air that is chemically required for the com- 
bustion of 1 lb. of bituminous coal is 150.35 cubic feet. Of this, 44.64* 
cubic feet combine with the gases evolved from the coal, and the remain- 
ing 105.71 cubic feet combine with the carbon of the coal. 

The combination of the gases evolved b} T combustion gives a resulting 
volume proportionate to the volume of atmospheric air required to furnish 
the oxygen, as 11 to 10. Hence the 44.64 cubic feet must be increased in 
this proportion, and it becomes 44.64 -}- 4.46 = 49.1. 

The gases resulting from the combustion of the carbon of the coal and 
the oxygen of the atmosphere, are of the same bulk as that, of the atmos- 
pheric air required to furnish the oxygen, viz., 105.71 cubic feet. The 
total volume, then, of the atmospheric air and gases at the bridge wall, 
flues, or tubes, becomes 105.71 + 49.1 = 154.81 cubic feet, assuming the 
temperature to be that of the external air. Consequently, the augment- 
ation of volume due to the increase of the temperature of a furnace is to 
be considered and added to this volume in the consideration of the capaci- 
ty of the flue or the calorimeter of a furnace. 

There is required, then, to be admitted through the grate of a furnace 
for the combustion of 1 pound of bituminous coal as follows : 

Coal containing 80 per cent, of carbon, or .7047 per cent, of coke. 

1 lb. coal x 44. 6-4 cubic f.-et of gas =z 44.04 

.7047 lb. carbonxl50 cubic feet of air = 105.71 

150.35 cubic feet. 

* P»y experiment, 4.464 cubic feet of gas are evolved from 1 lb. of bituminous coal, 
requiring 44.64 cubic feet of air. 



COMBUSTION. 



571 



For anthracite, by the observations of W. R. Johnston, an increase of 30 
per cent, over that for bituminous coal is required — 195.45 cubic feet. 

Coke does not require as much air as coal, usually not to exceed 108 
cubic feet, depending upon its purity. 

The heat of an ordinary furnace may be safely considered at 1000° ; 
hence the air entering the ash-pit and the gases "evolved in the furnace 
under the general law of the expansion of permanent^ elastic fluids of 
5Ygths of its volume (or .002087) for each degree of heat imparted to it, 
the 154.81 is increased in volume from 100° (the assumed ordinary tem- 
perature of the air at the ash-pit) to 1000° = 900° ; then 900 X .002087 == 

1.8783 times, or 154.81 + 154.81 x 1.8783 = 445.59 cubic feet. 

If the combustion of the gases evolved from the coal and the air was 
complete, there would be required to give passage to the volume of but 
445.59 cubic feet over the bridge wall or through the flues of a furnace ; 
but b}' experiments it appears that about one half of the oxj^gen admitted 
beneath the grates of a furnace passes off uncombined, the area of the 
bridge wall, or the flues or tubes, must consequently be increased in this 
proportion, hence the 445.59 becomes 891.18. 

The velocity of the gases passing from the furnace of a proper-propor- 
tioned boiler "may be estimated at from 30 to 36 feet per second. Then 

891 18 
r(v — rk^' — o£ = ..00687 square feet, or .99 square inches, of area at the 

bridge wall for each pound of coal consumed per hour. 

A limit, then, is here obtained for the area at the bridge W£_'', or of the 
flues or tubes immediately behind it, below which it mr.. not be de- 
creased, or the combustion will be imperfect. In ordinary. . actice it will 
be found advantageous to make this area .014 square fej--, or 2 square 
inches for every pound of bituminous coal consumed per square foot of 
grate per hour,"and so on in proportion for any other quantity. 

The quantities of heat evolved are very nearly the same for the same 
substance, whatever the temperature of the combustible. 



Relative Volumes of Air required for Combustion of Fuels. 



Lbs. 

Charcoal 11 . 16 

Coke 11.28 



Anthracite Coal. 
Bituminous fck . 



Lbs. 
12.13 

10.98 



Peat, dry T.08 

Wood, dry 6. 



The volume of air chemically required for the combustion of different 
woods in cubic feet is as follows : 

Pine 15S | Birch 153 1 Beech 152.9 [ Oak 154.4 



Relative Volumes of Gases or Products of Combustion per Pound of FueL 





Supply of Air per lb. 


of Fuel. . 


Temp. 
Air. 


Supply of Air per lb. 


of Fuel 


Temp. 

Air. 


12 lb?. 

Volume 
per lb. 


IS lbs. 

Volume 
per lb. 


24 lbs. 

Volume 
per lb. 


12 lbs. 

Volume 
per lb. 


IS lbs. 

Volume 
per lb. 


24 lbs. 

Volume 
per lb. 


32° 
CS° 

104° 


Cub. Feet. 
150 
161 
1T2 


Cub. Feet. 
225 
241 
258 


Cub. Feet. 
300 
322 
344 


212° 
392° 

572° 


Cub. Feet. 
205 
259 
314 


Cub. Feet. 
307 
389 
471 


Cub. Feet. 
409 
519 
628 



The perfect combustion of 1 lb. of carbon requires 12 lbs. air ; hence the 
weight == 12 x 1. The total heat of combustion of 1 lb. carbon or char- 
coal is 14500 thermal units; the mean specific heat of the products of 
combustion is .238, which, multiplied by 13 as above = 3094, and 14500* 

* Mean of all experiments 13 964. 



572 



COMBUSTION. 



-4-3.094 — 4689° temperature of a furnace, assuming ever} T atom of oxygen 
that was ignited in the furnace entered into combination. 

If, however, as in the case in ordinary furnaces, twice the volume of air 
enters, then the products of combustion of 1 lb. of coal will be 25 lbs., 
which, multiplied by its specific heat as before, and if divided into 14 500, 
the quotient will be 2437°, which is the temperature of an ordinary furnace. 

If 18 lbs. air per lb. of coal are furnished, as per blast or artificial 
draught, then 3207° is the resultant temperature. 

Volumes of JProd.xxcts of Combustion at different Tem- 
peratures of Combustion. 





Supply of Air per lb. of Carbon 




Supply of Air per Jb of Carbon 




12 lbs 


18 lbs. 


24 lbs. 


12 lbs. 


IS lbs. 


24 lbs. 


Tempera- 








Tempera- 








ture. 


Volume- 


Volume- 


Volume 


ture. 


Volume. 


Volume. 


Volume. 




Cub feet. 


Cub feet. 


Cub feet 




Cub feet. 


Cub feet. 


Cub feet 


32° 


150 


225 


300 


1112° 


479 


718 


957 


C8° 


161 


241 


322 


1472° 


5S8 


882 


1176 


104° 


172 


258 


344 


1832° 


697 


1046 


1395 


212° 


205 


307 


409 


2500° 


906 


1359 


1812 


392° 


259 


389 


519 


3275° 


1136 


1704 


— 


752° 


369 


553 


738 


4640° 


1551 


— 


— 



Ratio of Combustion. — The quantity of fuel burned per hour per square 
feet of grate varies very much in different classes of boilers. In Cornish 
boilers it is 3% lbs. per square foot ; in Land boilers, 10 to 20 lbs. ; (En- 
glish) 13 to 14 lbs. ; in Marine boilers (natural draught), 10 to 18 lbs. ; 
(blast) 30 to 60 lbs. ; and in Locomotive boilers, 80 to 120 lbs. 

The volumes of air and smoke for each cubic foot of water converted 
into steam, is for coal and coke 2000 cubic feet, and for wood 4000 cubic 
feet ; and for each lb. of fuel as follows : 

Coal .... 207. Cannel coal . . 315. Coke .... 216. Wood .... 173. 

To Compute tlie Consumption, of Fuel in a Steam-Engine. 

Rule. — Compute the volume of the cylinder to the point of cutting off the steam. 
Multiply the result by the number of cylinders, by twice the number of revolutions 
of the engine and by 60 (minutes), and divide the product by the density of the 
steam at its pressure in the cylinder, and the quotient will give the number of cubic 
feet of water expended in steam. 

Multiply the number of cubic feet by 64.3125, divide the product by the evapora- 
tion of the boiler per lb. of fuel consumed, and the quotient will give the consump- 
tion in pounds per hour. 

Note In computing the evaporation of water by the boiler of an engine, or the 

volume of steam used, the space displacement of the piston in its course is alone con- 
sidered. 

Example. — The cylinder of a marine engine is 95 ins. in diameter by 10 feet stroke 
cf pi-ton; the pressure of the steam in the steam chest is 15.3 lbs. per square inch, 
cut off at X stroke ; the number of revolutions 14^ and the evaporation estimate4 
at 8 lbs. of salt water per lb. of coal ; what is the consumption of coal per hour ? 

Volume of steam at above pressure, compared with water (15.3 -4- 14. T) = SS3. 

Area of 95 ins.=7083.2, which -^- 144 = 49. 22 cubic feet. 

Point of cutting off 5 feet, and 49.22x5X14.5x2 (strokes ofpiston)X60 (minutes] 
= 428214 cubic feet = volume of steam per hour. 

Hence, 428214-=- SS3 = 484.9*5 cubic feet of water evaporated per hour, and 434.95 
X64.3125 lbs.= 3118S, which -^ 8 = 3398.5 lbs. coal per hour. 

Note. — The elements given are those of one engine of the Steamer Arctic, and 
thtf consumption of fuel for a run of 12 days (one engine) was 3S10 lbs. per hour. 



STEAM. 573 

STEAM. 

Steam, arising from water at its boiling point, is equal to the press- 
ure of the atmosphere, which is 14.72322 lbs. at 60° upon a square 
inch. 

In all calculations concerning steam,, it is necessary to have some 
or all of the following elements, viz. : 

Its Pressure, which is termed its tension or elastic force, and is ex- 
pressed in pounds per square inch. 

Its Temperature, which is the number of degrees of heat indicated 
by a thermometer immersed in it. 

Its Density, which is the weight of a unit of its volume compared 
with that of water. 

Its Relative volume, which is the space occupied by a given weight 
or volume of steam, compared with the weight or volume of the water 
that produced it. 

Steam in contact with water is at its maximum density. 

Each augmentation of 1 degree of Fahrenheit in the temperature 
of steam will produce an increase of .00202 of the volume occupied by 
the fluid at the temperature of 32°. 

Under the pressure of the atmosphere alone, the temperature of 
water can not be raised above its boiling point. 

The Expansive force of the steam of all fluids is the same at their 
boiling point. 

A cubic inch of water, evaporated under the ordinary atmospheric 
pressure, is converted into 1700* cubic inches of steam, or, in a unit 
of measure, very nearly 1 cubic foot, and it exerts a mechanical force, 
equal to the raising of 2120. 14 lbs. 1 foot high. 

27.2222 cubic feet of steam at the pressure of the atmosphere weigh 
1 lb. avoirdupois. 

A pressure of 1 lb. upon a square inch will support a column of 
mercury at a temperature of 60° 2.0376 inches in height ; hence it 
will raise a mercurial siphon gauge one half of this r or 1.0188 inches. 

A column of mercury 1 inch in height will counterbalance a press- 
ure of .490774 lbs. upon a square inch. 

The Velocity of steam, when flowing into a vacuum, is about 1550 
feet per second when at an expansive power equal to the atmosphere, 
when at 10 atmospheres the velocity is increased to but 1780 feet; 
and when flowing into the air under a similar pressure it is about 650 
feet per second, increasing to 1600 feet for a pressure of 20 atmos- 
pheres. 

The Boiling Points of Water, corresponding to different heights of the 
barometer, is given under Heat, page 530. 

■ The elasticity of the vapor of alcohol, at all temperatures, is about 2.125 
times that of steam. 

Thus, the volume of a cubic foot of water evaporated into steam is 1700 
cubic feet ; hence 1 -4- 1700 = .00058823, which represents the density or 
specific gravity of steam at the pressure of the atmosphere. 

* Pole's Formula makes it 1712. 

3C 



574 



STEAM. 



The Specific Gravity of steam, compared with air, is as the weight of a 
cubic foot of it compared with an equal volume of air. Thus the weight 
of a cubic foot of steam at the pressure of the atmosphere is 257.353 grains, 
and the weight of a like quantity of air at 34° is 527.04 grains. Hence 
257.353 -4- 527.04 = .4883, the specific gravity of steam compared with air, 
and with water it is .00058823. 

The volumes here given are from the results of experiments, and accord 
very nearly with the results deduced by the formulae of Pambour, and 
also with that of Pole and Tate, which is, 

m + n P<* = V. m = 12.5, n — 20570, and a = — .9301. 

In the following table and calculations, the unit of measure is 1700 
cubic inches. 

Elastic Force, Temperature, Volume, and. Density 

of Steam. 

From a Temperature of 32° to 387.3°, and from a Pressure of .2 to 408 
Inches of Mercury. 





Elastic Force per 








Elastic 


Force 






Temper- 


Square 


Inch. 


Volume. 


Density. 


Temper- 
ature. 


per Square Inch. 


Volume. 


Density. 


ature. 


In Mer- 


In 


In Mer- 


In 




cury. 


Pounds. 








cury. 


Pounds. 






Deg. 


Ins. 


Lbs. 


Cub. Ft. 




Deg. 


Ins. 


Lbs. 


Cub. Ft. 




32 


.2 


.09S 


187407 


.0000053 


214.5 


31.62 


15.5 


1618 


.000617 


85 


.221 


.108 


170267 


.0000058 


216.3 


32.64 


16. 


1573 


.000635 


40 


.263 


.129 


144529 


.0000069 


218. 


33.66 


16.5 


1530 


.000653 


45 


.316 


.155 


121483 


.0000082 


219.6 


34.68 


17. 


1488 


.000672 


50 


.375 


.184 


103350 


.0000096 


221.2 


35.7 


17.5 


1440 


.000694 


55 


.443 


.217 


8S3S8 


.0000113 


222.7 


36.72 


18. 


1411 


.000708 


60 


.524 


.257 


75421 


.0000132 


224.2 


37.74 


18.5 


1377 


.000726 


65 


.616 


.302 


64762 


.0000154 


225.6 


38.76 


19. 


1343 


.000744 


70 


.721 


.353 


55862 


.0000179 


227.1 


39.78 


19.5 


1312 


.000762 


75 


.851 


.417 


47771 


.0000209 


228.5 


40.8 


20. 


1281 


.00078 


80 


1. 


.49 


41031 


.0000244 


229.9 


41.82 


20.5 


1253 


.000798 


85 


1.17 


573 


35393 


.0000282 


231.2 


42.84 


21. 


1225 


.00081 


90 


1.36 


.666 


30425 


.0000329 


232.5 


43.86 


21.5 


1199 


.000834 


95 


1.58 


.774 


266S6 


.0000375 


233.S 


44. SS 


22. 


1174 


.000S51 


100 


1.86 


.911 


22S73 


.0000437 


235.1 


45.9 


22.5 


1150 


.000S69 


103 


2.04 


1. 


20958 


.0000477 


236.3 


46.92 


23. 


1127 


.000836 


105 


2.18 


1.068 


19693 


.00005 


237.5 


46.94 


23.5 


1105 


.000904 


110 


2.53 


124 


1666T 


.000059 


238.7 


48.96 


24. 


1184 


.000922 


115 


2.93 


1.431 


14942 


.000066 


239.9 


49.98 


24.5 


1064 


.090939 


120 


3.33 


1.632 


13215 


.000075 


241. 


51. 


25. 


1044 


.000957 


125 


3.79 


1.857 


11723 


.000085 


243.3 


53.04 


26. 


1007 


.000993 


130 


4.34 


2.129 


10328 


.000096 


245.5 


55.08 


27. 


973 


.001027 


135 


5. 


2.45 


9036 


.00011 


247.6 


57.12 


28. 


941 


.001062 


140 


5.74 


2.813 


7933 


.000125 


249.6 


59.16 


29. 


911 


.001097 


145 


6.53 


3.1 


7040 


.000142 


251.6 


61.2 


30. 


883 


.001132 


150 


7.42 


3.636 


6243 


.00016 


253.6 


63.24 


31. 


S57 


.001166 


155 


8.4 


4.116 


5559 


.000179 


255.5 


65.28 


32. 


S33 


.0012 


160 


9.46 


4.635 


49T6 


.0002 


257.3 


67.32 


33. 


810 


.001234 


165 


10. 6S 


5.23 


4443 


.000225 


259.1 


69.36 


34. 


788 


.001269 


170 


12.13 


5.94 


3943 


.000253 


260.9 


71.4 


35. 


767 


.001304 


175 


13.62 


6.67 


3538 


.000282 


262.6 


73.44 


36. 


74S 


.001337 


ISO 


15.15 


7.42 


3208 


.000311 


264.3 


75. 4S 


37. 


729 


.001371 


185 


17. 


8.33 


2879 


.000347 


265.9 


77.52 


38. 


712 


.001404 


190 


19. 


9.31 


2595 


.000385 


267.5 


79.56 


39. 


695 


.00143S 


195 


21.22 


10.4 


2342 


.000426 


269.1 


81.6 


40. 


679 


.001472 


200 


23.64 


11.58 


2118 


.000472 


270.6 


83.64 


41. 


664 


.001506 


205 


26.13 


12.8 


1032 


.000517 


272.1 


85.68 


42. 


649 


.00154 


210 


2S.S4 


14.13 


1763 


.000567 


273.6 


87.72 


43. 


635 


.001574 


211 


29.41 


14.41 


1730 


.000578 


275. 


89.76 


44. 


622 


.001007 


212 


30. 


14.7 


1700 


.000583 


276.4 


91.8 


45. 


610 


.001639 


212.S 


30.6 


15. 


1669 


.0005J9 


277.8 


93.84 


4G. 


598 


.001672 



STEAM. 



575 





Elastic Force 




Temper- 


per Square Inch. 


Volume. 


ature. 


In 


In 




Mercury. 


Pounds. 




Deg. 


Ins. 


Lbs. 


Cub. Ft. 


279. 2 


95.S8 


47. 


585 


280.5 


07.92 


48. 


575 


281.9 


99.96 


49. 


564 


283.2 


102. 


50. 


554 


284.4 


104.04 


51. 


544 


285.7 


106.08 


52. 


534 


286.9 


108.12 


53. 


525 


288.1 


110.16 


54. 


516 


2S9.3 


112.2 


55. 


508 


290.5 


114.24 


56. 


500 


291.7 


116.28 


57. 


492 


292.9 


118.32 


58. 


484 


294.2 


120.36 


59. 


477 


295.6 


122.4 


60. 


470 


296.9 


124.44 


61. 


463 


298.1 


126.48 


62. 


456 


299.2 


128.52 


63. 


449 


300.3 


130.56 


64. 


443 


301.3 


132.6 


65. 


437 


302.4 


134.64 


66. 


431 


303.4 


136.68 


67. 


425 


304.4 


138.72 


6S. 


419 


305.4 


140.76 


69. 


414 


30o.4 


142.8 


70. 


408 


307.4 


144.84 


71. 


403 


308.4 


146.88 


72. 


398 


309.3 


148.92 


73. 


393 


310.3 


150.96 


74. 


388 


311.2 


153.02 


75. 


383 


312.2 


155.06 


76. 


379 


313.1 


157.1 


77. 


374 







Elastic Force 






Dcnsitv* 


Temper- 
ature. 


per Square Inch. 


Volume. 


Density. 




In 


In 






Mercury. 


Pounds. 








Deg. 


Ins. 


Lbs. 


Cub Ft. 




.001706 


314. 


159.14 


78. 


370 


.002703 


.001739 


314.9 


161.18 


79. 


366 


.002732 


.001773 


315.8 


163.22 


80. 


362 


.002762 


.001805 


316.7 


165.26 


81. 


358 


.002793 


.001838 


317.6 


167.3 


82. 


354 


.002824 


.001872 


318.4 


169.34 


83. 


350 


.002828 


.001904 


319.3 


171.38 


84. 


346 


.00289 


.001937 


320.1 


173.42 


85. 


342 


.002923 


.00196S 


324.3 


183.62 


90. 


325 


.003076 


.002 


328.2 


193.82 


95. 


310 


.003225 


.002032 


332. 


203.99 


100. 


292 


.003389 


.002066 


335.8 


214.19 


105. 


282 


.003546 


.002096 


339.2 


224 39 


110. 


271 


.00369 


.002127 


342.7 


234.59 


115. 


259 


.003861 


.002159 


345.8 


244.79 


120. 


251 


.003984 


.002192 


349.1 


254.99 


125. 


240 


.004166 


.002227 


352.1 


265.19 


130. 


233 


.004291 


.002257 


355. 


275.39 


135. 


224 


.004464 


.002288 


357.9 


2S5.59 


140. 


218 


.0045S7 


.00232 


360.6 


295.79 


145. 


210 


.004761 


.002352 


363.4 


306. 


150. 


205 


.004ST9 


.0023S6 


366. 


316.19 


155. 


198 


.00505 


.002415 


368.7 


326.39 


160. 


193 


.005181 


.002451 


371.1 


336.59 


165. 


187 


.005347 


.002481 


373.6 


346.79 


170. 


1S3 


.005464 


.002512 


376. 


357. 


175. 


178 


.005617 


.002544 


378.4 


367.2 


180. 


174 


.005747 


.002577 


380.6 


377.1 


185. 


169 


.005919 


.00261 


382.9 


387.6 


190. 


166 


.006024 


.002638 


384.1 


S97.8 


If5. 


161 


.006211 


.002673 


387.3 


408. 


200. 


158 


.006329 



The Temperatures of steam, as deduced from the experiments of the Comm. of 
the Frankliu Institute, range somewhat different from those given in the above table. 

Thus, at the pressure of the atmosphere, the temperatures are alike, viz., 212° ; but 
at 6.4 atmospheres = 192 inches of mercury, they are 320° and 327.5°. 

To Compute tlie ^Pressure of Steam. 
When the Height of the Column of Mercury it will Support is given. 
Rule. — Divide the height of the column of mercury in inches by 2.0376, 
and the quotient will give the pressure per square inch in pounds. 

Exajuple. — The height of a column of mercury is 203.76 inches; what pressure 
per square inch will it contain ? 

203.76-^2.0376 = 100 lbs. 

To Compute tlie Temperature of Steam. 
Bulk. — Multiply the 6th root of its force in inches of mercury bj T 177, 
and subtract 100 from the product, the remainder will give the tempera- 
ture in degrees. 

Example.— When the elastic force of steam is equal to a pressure of 49 inches of 
mercury, what is its temperature ? 

Note.— To extract the 6th root of a number, ascertain the cube root of its squaw 
root. 

V of 49 = 7, and fy of 7 = 1.9129. Hence 1.9129x177 — 100 = 23S°.5S. 

To Compute tlie Pressure of Steam iix Indies of 

Mercury. 

j When the Temperature is given. Rule. — Add 100 to the temperature, 

divide the sum by 177, and the 6th power of the quotient will give the 

pressure in inches of mercury. 



574 



STEAM. 



The Specific Gravity of steam, compared with air, is as the weight of a 
cubic foot of it compared with an equal volume of air. Thus the weight 
of a cubic foot of steam at the pressure of the atmosphere is 257.353 grains, 
and the weight of a like quantity of air at 34° is 527.04 grains. Hence 
257.353 -h 527.04 = .4883, the specific gravity of steam compared with air, 
and with water it is .00058823. 

The volumes here given are from the results of experiments, and accord 
very nearly with the results deduced by the formulae of Pambour, and 
also with that of Pole and Tate, which is, 

m + n P« = V. to = 12.5, n = 20570, and a = — .9301. 

In the following table and calculations, the unit of measure is 1700 
cubic inches. 

Elastic Force, Temperature, Volume, and. Density 

of Steam. 

From a Temperature of 32° to 387.3°, and from a Pressure of .2 to 408 
Inches of Mercury. 





Elastic Force per 








Elastic 


Force 






Temper- 


Square 


Inch. 


Volume. 


Density. 


Temper- 
ature. 


per Square Inch. 


Volume. 


Density. 


ature. 


In Mer- 


In 


In Mer- 


In 




cury. 


Pounds. 








cury. 


Pounds. 






Deg. 


Ins. 


Lbs. 


Cub. Ft. 




Deg. 


Ins. 


Lbs. 


Cub. Ft. 




32 


.2 


.098 


187407 


.0000053 


214.5 


31.62 


15.5 


1618 


.000617 


85 


.221 


.10S 


170267 


.0000058 


216.3 


32.64 


16. 


1573 


.000635 


40 


.263 


.129 


144529 


.0000069 


218. 


33.66 


16.5 


1530 


.000653 


45 


.316 


.155 


1214S3 


.0000082 


219.6 


34.68 


17. 


14SS 


.000672 


50 


.375 


.184 


103350 


.0000096 


221.2 


35.7 


17.5 


1440 


.000694 


55 


.443 


.217 


833S8 


.0000113 


222.7 


36.72 


18. 


1411 


.000708 


60 


.524 


.257 


75421 


.0000132 


224.2 


37.74 


1S.5 


1377 


.000726 


65 


.616 


.302 


64762 


.0000154 


225.6 


38.76 


19. 


1343 


.000744 


70 


.721 


.353 


55862 


.0000179 


227.1 


39.78 


19.5 


1312 


.000762 


75 


.851 


.417 


47771 


.0000209 


22S.5 


40.8 


20. 


1281 


.00078 


80 


1. 


.49 


41031 


.0000244 


229.9 


41.82 


20.5 


1253 


.000798 


85 


1.17 


.573 


35393 


.0000282 


231.2 


42.84 


21. 


1225 


.000S1 


90 


1.36 


.666 


30425 


.0000329 


232.5 


43.86 


21.5 


1199 


.000334 


95 


1.58 


.774 


266S6 


.0000375 


233.S 


44 S8 


22. 


1174 


.000S51 


100 


1.S6 


.911 


22S73 


.0000437 


235.1 


45.9 


22.5 


1150 


.OO0S69 


103 


2.04 


1. 


20958 


.0000477 


236.3 


46.92 


23. 


1127 


.000S36 


105 


2.18 


1.068 


19693 


.00005 


237.5 


46.94 


23.5 


1105 


.000904 


110 


2.53 


124 


16667 


.000059 


238.7 


48.96 


24. 


1184 


.000922 


115 


2.93 


1431 


14942 


.000066 


239.9 


49.98 


24.5 


1064 


.090939 


120 


3.33 


1.632 


13215 


.000:175 


241. 


51. 


25. 


1044 


.000957 


125 


3.79 


1.857 


11723 


.000085 


243.3 


53.04 


26. 


1007 


.000993 


130 


4.34 


2 129 


10328 


.000096 


245.5 


55.08 


27. 


973 


.001027 


135 


5. 


2.45 


9036 


.00011 


247.6 


57.12 


28. 


941 


.001062 


140 


5.74 


2.813 


7933 


.000125 


249.6 


59.16 


29. 


911 


.001097 


145 


6.53 


3.1 


7040 


.000142 


251.6 


61.2 


30. 


8S3 


.001132 


150 


7.42 


3.636 


6243 


.00016 


253.6 


63.24 


31. 


857 


.001166 


155 


8.4 


4.116 


5559 


.000179 


255.5 


65.28 


32. 


S33 


.0012 


160 


9.46 


4.635 


4976 


.0002 


257.3 


67.32 


33. 


810 


.001234 


165 


10. 6S 


5.23 


4443 


.000225 


259.1 


69.36 


34. 


788 


.001269 


170 


12.13 


5.94 


3943 


.000253 


260.9 


71.4 


35. 


767 


.001304 


175 


13.62 


6.67 


3538 


.000282 


262.6 


73.44 


36. 


74S 


.001337 


ISO 


15.15 


7.42 


3208 


.000311 


264.3 


75.48 


37. 


729 


.001371 


185 


17. 


8.33 


2879 


.000347 


265.9 


77.52 


38. 


712 


.001404 


190 


19. 


9.31 


2595 


.000385 


267.5 


79.56 


39. 


695 


.001433 


195 


21.22 


10.4 


2342 


.000426 


260.1 


81.6 


40. 


679 


.001472 


200 


23.64 


11.58 


2118 


.000472 


270.6 


83.64 


41. 


664 


.001506 


205 


26.13 


12.8 


1932 


.000517 


272.1 


85.68 


42. 


649 


.00154 


210 


2S.S4 


14.13 


1763 


.000567 


273.6 


87.72 


43. 


635 


.001574 


211 


29.41 


14.41 


1730 


.000573 


275. 


89.76 


44. 


622 


.001607 


212 


30. 


14.7 


1700 


.0005S3 


276.4 


91.8 


45. 


610 


.001639 


212.S 


30.6 


15. 


1669 


.000509 


277.8 


93.84 


46. 


598 


.001672 



STEAM. 



575 





Elastic Force 








Elastic Force 






Temper- 


per Square Inch. 


Volume. 


Density. 


Temper- 
ature. 


per Square Inch. 


Volume. 


Density. 


ature. 


In 


In 


In 


In 




Mercury. 


Pounds. 








Mercury. 


Pounds. 






Deg. 


Ins. 


Lbs. 


Cub. Ft. 




Deg. 


Ins. 


Lbs. 


Cub Ft. 




279. 2 


95.S8 


47. 


585 


.001706 


314. 


159.14 


78. 


370 


.002703 


280.5 


07.92 


48. 


575 


.001739 


314.9 


161.18 


79. 


366 


.002732 


281.9 


99.96 


49. 


564 


.001773 


315.8 


163.22 


80. 


362 


.002762 


283.2 


102. 


50. 


554 


.001805 


316.7 


165.26 


SI. 


358 


.002793 


284.4 


104.04 


51. 


544 


.001S38 


317.6 


167.3 


82. 


354 


.002824 


285.7 


106.08 


52. 


534 


.001872 


318.4 


169.34 


83. 


350 


.002828 


286.9 


108.12 


53. 


525 


.001904 


319.3 


171.38 


84. 


346 


.00289 


288.1 


110.16 


54. 


516 


.001937 


320.1 


173.42 


85. 


342 


.002923 


2S9.3 


112.2 


55. 


508 


.00196S 


324.3 


183.62 


90. 


325 


.003076 


290.5 


114.24 


56. 


500 


.002 


328.2 


193.82 


95. 


310 


.003225 


291.7 


116.28 


57. 


492 


.002032 


332. 


203.99 


100. 


292 


.003389 


292.9 


118.32 


58. 


484 


.002066 


335.8 


214.19 


105. 


282 


.003546 


294.2 


120.36 


59. 


477 


.002096 


339.2 


224 39 


110. 


271 


.00369 


295.6 


122.4 


60. 


470 


.002127 


342.7 


234.59 


115. 


259 


.003861 


296.9 


124.44 


61. 


463 


.002159 


345.8 


244.79 


120. 


251 


.003984 


298.1 


126.48 


62. 


456 


.002192 


349.1 


254.99 


125. 


240 


.004166 


299.2 


128.52 


63. 


449 


.002227 


352.1 


265.19 


130. 


233 


.004291 


300.3 


130.56 


64. 


443 


.002257 


355. 


275.39 


135. 


224 


.004464 


301.3 


132.6 


65. 


437 


.002288 


357.9 


285.59 


140. 


218 


.0045S7 


302.4 


134.64 


66. 


431 


.00232 


360.6 


295.79 


145. 


210 


.004761 


303.4 


136.68 


67. 


425 


.002352 


363.4 


306. 


150. 


205 


.004S79 


304.4 


13S.72 


68. 


419 


.0023S6 


366. 


316.19 


155. 


198 


.00505 


305.4 


140.76 


69. 


414 


.002415 


368.7 


326.39 


160. 


193 


.005181 


303.4 


142.8 


70. 


408 


.002451 


371.1 


336.59 


165. 


187 


.005347 


307.4 


144.84 


71. 


403 


.0024S1 


373.6 


346.79 


170. 


1S3 


.005464 


308.4 


146.88 


72. 


398 


.002512 


376. 


357. 


175. 


178 


.005617 


309.3 


148.92 


73. 


393 


.002544 


3T8.4 


367.2 


180. 


174 


.005747 


310.3 


150.96 


74. 


3S8 


.002577 


380.6 


377.1 


185. 


169 


.005919 


311.2 


153.02 


75. 


383 


.00261 


382.9 


387.6 


190. 


166 


.€06024 


312.2 


155.06 


76. 


379 


.002638 


384.1 


897.8 


If5. 


161 


.006211 


313.1 


157.1 


77. 


374 


.002673 


387.3 


408. 


200. 


158 


.006329 



The Temperatures of steam, as deduced from the experiments of the Comm. of 
the Franklin Institute, range somewhat different from those given in the above table. 

Thus, at the pressure of the atmosphere, the temperatures are alike, viz. , 212° ; but 
at 6.4 atmospheres = 192 inches of mercury, they are 320° and 327.5°. 

To Compute tlie Pressure of* Steam. 

When the Height of the Column of Mercury it will Support is given. 
Rule. — Divide the height of the column of mercury in inches by 2.0376, 
and the quotient will give the pressure per square inch in pounds. 

Example. — The height of a column of mercury is 203.76 inches; what pressure 
per square inch will it contain ? 

203.76-^2.0376 = 100 lbs. 

To Compute tlie Temperature of Steam. 
Bulk. — Multipty the 6th root of its force in inches of mercury b}' 177, 
and subtract 100 from the product, the remainder will give the tempera- 
ture in degrees. 

Example.— When the elastic force of steam is equal to a pressure of 49 inches of 
Enercury, what is its temperature ? 

Note.— To extract the 6th root of a number, ascertain the cube root of its squaw 
foot. 

V of 49 = 7, and V of 7 = 1.9129. Hence 1.9129x177 — 100 = 238°.58. 

To Compute tlie Pressure of Steam iii Inches of 

Mercury. 

, When the Temperature is given. Rule. — Add 100 to the temperature, 

divide the sum b^ 177, and the 6th power of the quotient will give the 

pressure in inches of mercury. 



576 STEAM. 

m 
Example. — The temperature of steam is 312° ; what is its pressure? 

100 t 3l2 = 2.3277, and 2.3277 6 = 159 vis. 
1(7 

Note. — To involve the 6th power of a number, square its cube. 

To Compute tlie Specific Gravity of Steam, compared 
-with. -A.ir. 

Rule.— Divide the constant number 830.11 (1 700 x. 4883) by the volume 
of the steam at the temperature of pressure at which the gravity is re- 
quired. 

Example. — The pressure of steam is 60 lbs., and the volume of it is 470; what is 
its specific gravity? 

830.11-=- 470 = 1.766. 

Note.— The specific gravity of steam compared with water = .0005SS23. 

To Compute the Volume a Cuoic Foot of Water occupies 

in. Steam. 

When the Elastic Force and -Temperature of the Steam are given. 
Rulk. — To 459 add the temperature in degrees, and multiply the sum by 
76.5 ; divide the product by the elastic force of the steam in inches of mer- 
cury, and the quotient will give the volume required. 

Note. — "When the force in inches of mercury is not given, multiply the pressure 
in pounds per square inch by 2.0376. 

Or, '■ — — * 18329 == volume ; p representing the pressure of the 

steam per square inch, and t the temperature. 

This formula is based upon the discovery of Gay-Lussac. viz., that if the tempera- 
ture of a given weight of any elastic fluid be made to vary, its tension being the 
same, it will receive augmentation of volume exacrly proportional to the augmenta- 
tion of temperature, and for every increase of 1°, will be produced an increase of 
.00202 of the volume occupied by the fluid at a temperature of 32°. 

Example — The temperature of a cubic foot of water evaporated into steam is 376°, 
and the elastic force is 357 inches; what is its volume? 

459 4-376X76.5 63877.5 ^ An t m 
357 =-W-^ l4S - 93 CUbl °S eet ' 

To Compute tlie Density or Specific Gravity of Steam 
■under different Pressures. 

When the Volume is given. Rule.— Divide 1 by the volume in cubic 
feet, and the quotient will give the density required. 
Example. — The volume is 210 ; what is the densitv? 
1^-210 = .004761. 

When the Pressure is given. — Take the temperature due to the press- 
ure, and proceed as by the foregoing rule to compute the volume, which, 
when obtained, proceeds as above. 

To Compute tlie Volume of Water contained in a given 
Volume of Steam.. 

When its Density is given. Rule. — Multiply the volume of the steam 
f >v its density, and the product will give tlie volume of the water in cubic 
feet. 

Hence, To Compute the weight of the water m pounds, Multiply the product by 
C2.5. 

Example.— The density of a volume of 17000 cubic feet of steam 13 .0011765; what 
is the weight of it in pounds? 

17000 x .0011765 = 20 = volume of water, and 20 X 62.5 = 1250 pounds. 






STEAM. 577 

To Compute tlie "Volume of Steam required, to raise a 
Griven Volume of Water to any Given Temperature. 

Rule, — Multiply the water to be heated bj' the difference of tempera- 
tures between it and that to which it is to be raised, for a dividend ; then 
to the temperature of the steam add 966°.6, and from that sum take the 
required temperature of the water for a divisor ; the quotient will give the 
volume of steam in the same terms as the water. 

Example What volume of steam at 212° will raise 100 cubic feet of water at 

80° to 212°? 



100x212° — 80° 13200 ,. / ;<> • 'j * 

212° + 966°.6-212 q : =9ooT6 cuhlc f eet of water formed into steam, oc 

cupying (13.66x1700) 232 22 'cubic feet. 

To Compute the "Velocity- -with Avhich Steam Flo-ws into 
a Vacuum. 

Rule. — To the temperature of the steam add the constant 459, and 
multiply the square root of the sum by 60.2 ; the quotient will give the 
velocity in feet per second. 

To Compute the Number of Cubic Inches of Water, at 
any Griven Temperature, that must he mixed, v^ith a 
Cubic Foot* of Steam to raise or reduce the jVtixture to 
any Required Temperature. 

Rule. — From the required temperature subtract the temperature of the 
water ; then ascertain how often the remainder is contained in the re- 
quired temperature subtracted from 1178°. 6, and the quotient will give the 
quantity required. 

The sum of the Sensible and Latent Heats for several temperatures will be found 
under Heat, p. 524. 

Example. — The temperature of the condensing water of an engine is 80°, and 
the required temperature 100° ; what is the proportion of condensing water to that 
evaporated ? 

100 — 80 = 20. Then, 11T8 ^~" 100 _ 53.93 cuoic mches to 1# 

Or, let w represent temperature of condensing water, t the required temperatures 
and S the sum of sensible and latent heats. 

r^ S— t 

Then = volume. 

t—w 

When the Temperature of the Steam is given. 

l-L-T t 

— — = volume ; I representing the latent heat, and T the temperature of the 

steam in the cylinder. 

Example.— The temperature of the steam in a cylinder is 230°, and the other ele- 
ments the same as in the preceding example ; required the volumes of injection water. 

Sum of sensible and latent heat of steam at 230° = 1182.2. 



1182.2 — 230 + 230 — 100 1082.2 „_•• , 

iooH =-^o-= 5411 volumes ' 

To Compute the Temperature of Water in the Condenser 
or Reservoir of a Steam-engine. 

/+ T + V xw 

vXi = t; Y representing volume of injection water. 

Example. — The elements the same as in the preceding example. 
952°.2+23 + 54.UXS0 _ 5511 _ t(H)09 
54.11+1 ~~ 55.11 * 

* Tlie exact volume is assumed to be 1700 cubic ins. Hence, when accuracy is 
required, the result, as determined by the rule, must be X 17*28, and -r-1700. 

3C* 



578 



STEAM. 



Elastic Force and. Temperatures of* tlie "Vapors of Al- 
cohol, Sulphuric Ether, Sulphuret of Carbon, Petro- 
leum, and. Turpentine. 





Force in 




Force in 




Force in 




Force in 


Temp. 


Inches of 


Temp. 


Inches of 


Temp. 


Inches of 


Temp. 


Inches of 




Mercury. 




Mercury. 




Mercury. 




Mercury. 


Alcohol. 


173° 


30. 


94° 


24.70 


279.5° 


300. 


32° 


.4 


180 


34.73 


96) 

104/ 
120 


30. 


347. 


606. 


50 


.86 


200 


53. 




60 


1.23 


212 


675 


39.47 


Peteoleum. 


TO 


1.T6 


220 


7S.5 


150 


67.6 


316° 


30. 


SO 


2.45 


240 


111.24 


212 


178. 


345 


44.1 


90 


3.4 


264 


16G.1 


SULPHTJRET OF 


375 


64. 


100 


4.5 






Caruox. 




120 


8.1 


Ether. 


53.5° 


7.4 


Oil of Turpentine. 


130 


10.6 


34° 


6.2 


72.5 


12.55 


304° 


30. 


140 


13.9 


54 


15.3 


110. 


80. 


340 


47.78 


160 


22.6 


74 


16.2 


212. 


126. 


362 


62.4 



Loss of Pressure of Steam in an Engine by Imperfect Condensation. 



Temp. 



60° 
80° 



Square Inch. 


Temp. 


Square Inch. 


Temp. 


Square Inch. 


Lbs. 

.26 
.5 


100° 
120° 


Lbs. 

.93 

1.65 


140° 


Lbs. 
2.SS 



Pressure above 
Atmosphere. 



Velocity of Flow of Steam into the Atmosphere per Second. 

-it,,!™;*, 1 Pressure above -ir„i„„:f„ Pressure above v«i ««;♦«. 

Velocity. Atmosnhere. Velocity. Atmosphere. Velocity. 



Lbs. 
1 
3 
5 



Feet. 
4S2 
791 
973 



Lbs. 
10 
20 
30 



Feet. 
1241 
1504 
1643 



Lbs. 
50 
70 

100 



Feet. 
1791 
1S77 
1957 



Mechanical Equivalent of Heat contained in Steam. 

1 lb. water heated from 32° to 212°, requires as much heat as 
would raise 180 lbs. 1°. Hence 180" 

1 lb. water at 212°, converted into steam at 212° (14.7 lbs.), ab- 
sorbs as much heat for its conversion as would raise 966.6 lbs. 

water 1°. Hence 966°. 6 

1146°. e 

This number, 1146.6, is a constant, and expresses the units of heat in 
1 lb. of steam from 32° up to the temperature at which the conversion 
takes place. 

Thus, 1 lb. water heated from 32° to 332°, requires as much heat 

as would raise 300 lbs. 1°. Hence 300° 

And 1 lb. water at 332°, converted into steam at 332° (100 lbs.), 
absorbs as much heat for its conversion as would raise 846.6 
lbs. water 1°. Hence 846° .6 

1146°. 6 

The Mechanical Equivalent, or maximum theoretical dut} r of this quan- 
tity of heat, as contained in 1 lb. of steam, is 772 lbs. x 11-16.6 units of heat 
= 885175.2 lbs. raised lfoot higr.. 

The amount of duty realized in the production and use of lib. of steam 
falls far short of this theoretical computation. 



STEAM. 



579 



MIXTTTEE OF AIR AND STEAM. 

' Water contains a portion of air or other uncondensable gaseous matter, and when it 
is converted into steam, this air is mixed with it, and when the steam is condensed 
it is left in a gaseous state. If means were not taken to remove this air or gaseous 
matter from the condenser of a steam-engine, it would fill it and the cylinder, and 
obstruct their operation ; but, notwithstanding the ordinary means of removing it 
(by the air-pump), a certain quantity of it always remains in the condenser. 
20 volumes of water absorb 1 volume of air. 

STEAM ACTING EXPANSIVELY. 

To Compute tlie mean. Pressure of Steam upon a IPiston 

"by- Hyperbolic Logarithms. 

Rulk. — Divide the length of the stroke of a piston, added to the clear- 
ance in the cj'linder at one end, by the length of the stroke at which the 
steam is cut off, added to the clearance at that end, and the- quotient will 
express the relative expansion of the steam or number. 

Find in the table the logarithm of the number nearest to that of the 
quotient, to which add 1. The sum is the ratio of the gain. 

Multiply the ratio thus obtained by the pressure of the steam (including 
the atmosphere) as it enters the cylinder, divide the product by the relative 
expansion, and the quotient will give the mean pressure required. 

Ta"ble of Hyperbolic Logarithms. 



No. 


Log. 


No. 


Log. 


No. 


Log. 


No. 


Log. 


No. 


Log. 


1.05 


.049 


2.65 


.975 


4.25 


1.447 


5.8 


1.758 


7.4 


2.001 


1.1 


.095 


2.66 


.978 


4.3 


1.459 


5.85 


1.766 


7.45 


2.008 


1.15 


.14 


2.7 


.993 


4.33 


1.465 


5.9 


1.775 


7.5 


2.015 


1.2 


.182 


2.75 


1.012 


4.35 


1.47 


5.95 


1.783 


7.55 


2.022 


1.25 


.223 


2.8 


1.03 


4.4 


1.482 


6. 


1.792 


7.6 


2.028 


1.3 


.262 


2.85 


1.047 


4.45 


1.493 


6.05 


1.8 


7.65 


2.035 


1.33 


.285 


2.9 


1.065 


4.5 


1.504 


6.1 


1.808 


7.66 


2.036 


1.35 


.3 


2.95 


1.082 


4.55 


1.515 


6.15 


1.816 


7.7 


2.041 


1.4 


.336 


3. 


1.099 


4.6 


1.526 


6.2 


1.824 


7.75 


2.048 


1.45 


.372 


3.05 


1.115 


4.65 


1.537 


6.25 


1.833 


7.8 


2.054 


1.5 


.405 


3.1 


1.131 


4.66 


1.54 


6.3 


1.841 


7.85 


2.061 


1.55 


.438 


3.15 


1.147 


4.7 


1.548 


6.33 


1.845 


7.9 


2.067 


1.6 


.47 


3.2 


1.163 


4.75 


1.558 


6.35 


1.848 


7.95 


2.073 


1.65 


.5 


3.25 


1.179 


4.8 


1.569 


6.4 


1.856 


8. 


2.079 


1.66 


.506 


3.3 


1.194 


4.85 


1.579 


6.45 


1.864 


8.05 


2.086 


1.7 


.531 


3.33 


1.202 


4.9 


1.589 


6.5 


1.872 


8.1 


2.092 


1.75 


.56 


3.35 


1.209 


4.95 


1.599 


6.55 


1.879 


8.15 


2.098 


1.8 


.588 


3.4 


1.224 


5. 


1.609 


6.6 


1.887 


8.2 


2.104 


1.85 


.612 


3.45 


1.238 


5.05 


1.619 


6.65 


1.895 


8.25 


2.11 


1.9 


.642 


3.5 


1.253 


5.1 


1.629 


6.66 


1.896 


8.3 


2.116 


1.95 


.668 


3.55 


1.267 


5.15 


1.639 


6.7 


1.902 


8.33 


2.119 


2. 


.693 


3.6 


1.281 


5.2 


1.649 


6.75 


1.91 


8.35 


2.122 


2.05 


.718 


3.65 


1.295 


5.25 


1.658 


6.8 


1.917 


8.4 


2.128 


2.1 


.742 


3.66 


1.297 


5.3 


1.668 


6.85 


1.924 


8.45 


2.134 


2.15 


.765 


3.7 


1.308 


5.33 


1.673 


6.9 


1.931 


8.5 


2.14 


2.2 


.788 


3.75 


1.322 


5.35 


1.677 


6.95 


1.939 


8.55 


2.146 


2.25 


.811 


3.8 


.1.335 


5.4 


1.686 


7. 


1.946 


8.6 


2.152 


2.3 


.833 


3.85 


1.348 


5.45 


1.696 


7.05 


1.953 


8.65 


2.158 


2.33 


.845 


3.9 


1.361 


5.5 


1.705 


7.1 


1.96 


8.66 


2.159 


2.35 


.854 


3.95 


1.374 


5.55 


1.714 


7.15 


1.967 


8.7 


2.163 


2.4 


.875 


4. 


1.386 


5.6 


1.723 


7.2 


1.974 


8.75 


2.169 


2.45 


.896 


4.05 


1.399 


5.65 


1.732 


7.25 


1.981 


8.8 


2.175 


2.5 


.916 


4.1 


1.411 


5.66 


1.733 


7.3 


1.988 


8.85 


2.18 


2.55 


.936 


4.15 


1.423 


5.7 


1.74 


7.33 


1.991 


8.9 


2.186 


2.6 


I .956 


4.2 


1.435 


5.75 


1.749 


7.35 


1.995 


8.95 


2.192 



5S2 STEAM-ENGIXE. 

STEAM-ENGINE. 

The extremes of proportions here given are for the particular require- 
ments of variations in speed, pressures, and differences in draughts of 
water, etc., in the varied purposes of Marine, River, and Land practice. 

CONDENSING ENGINE. 

For a Range of Pressures of from lO to 60 l"bs. (IVIercuriai 
Grange) per Square Inch, Cut OfFat One Halfthe Stroke, 
and. for from 15 to 60 Revolutions per jVIinute. 

Piston Rod Its diameter .1 that of the cylinder or air-pump for which 
it is designed. If of steel, .8 the diameter of wrought iron ; and if an air- 
pump rod is of copper or brass, .11 and .125 the diameter of pump. 

Condenser {Jet). The capacity of it should be from .35 to .45 that of 
the steam cylinder. (Surface.) The area of its condensing surfaces 
should be .55 to .65 of that of the evaporating surface of the boiler, when 
a natural draught is employed ; but with a blower, or forced draught, this 
proportion should be increased to .6 and .75. 

When a Circulating pump is used, the area may be reduced to .45 and .5. 

Air-pump (Single acting and direct connection). The capacity of it 
should be from .1 to .2 that of the steam cylinder. 

Foot Valve. The dimensions of it should give an area of from .25 to .5 
of the area of the air-pump in inches, the mean being .375 for 37^" revo- 
lutions. 

Delivery Valve. When a solid piston is used in the air-pump, its area 
should correspond with that of the foot valve ; but when an open piston 
alone is used, this proportion may not be obtained. 

Out-board Delivery Valve. The area of it should be from .5 to .8 that 
of the foot valve. 

a X s x r 

Steam and Exhaust Valves (Puppet), 9 i non = area for steam, and 

fl X s X ?* fl X S X 7* 

20 000 ~ area f° r €x ^ aust ' (Slide), 30 OOQ = ar6a ^* or steam > an( * 

X 8 X T 

.).-)—(* — area for exhaust, a representing area of steam cylinder in 
sq. ins., s stroke of pistons in ins., and r number of revolutions per minute. 

Injection CocJcs. There should be two to each condenser, the area of 
each sufficient to supply 70 times the quantity of water evaporated when 
the engine is working at its maximum ; and in all marine and river en- 
gines, there should be three, viz., a Bottom, Side, and Bilge. 

The Bilge injection is properly a branch of the bottom injection pipe, 
and may be of less capacity. 

The proportions here given will admit of a sufficient volume of water when the 
engine is in operation in the Gulf Stream, where the water is at times nt the tem- 
perature of 84°, and the volume required to give the water of condensation a tem- 
perature of 100° is TO times that of the quantity evaporated. 

Feed Pump (Single acting, Marine). Its volume should be .007 to .01 
that of the steam cvlinder. (River and Land), or when fresh water alone 
is used, .0035 to .004. 

NON-CONDENSING ENGINE. 

For a Range of* Pressures of from SO to ISO lbs. (Mer- 
curial Grange) per Square Inch, Cut Off at One Half* the 
Stroke, and. for 30 to IOO Revolutions per Minute. 

Piston Rod. Its diameter should be from .125 to .2 that of the steam 
cylinder. If of steel, .8 the diameter of wrought iron. 



STEAM-ENGINE. 583 

Steam and Exhaust Valves (Puppet). Their area is determined by the 
rule given for them in a condensing engine, using for divisors 30 000 and 
22 750. 

A decrease in the capacity of the cylinder is not attended with a proportionate de- 
crease of their area. A 12-inch cylinder by 4 feet stroke has 9 ins. area of valve, 
which is 1 inch in every 600 ins. capacity ; and a 6-inch cylinder by 1 foot stroke 
has 2 ins. area, which is 1 inch in every 125 ins. capacity. 

Feed Pump (Single acting, Marine). Its volume should be from .025 
to .033 that of the steam cylinder. (River or Land), or where fresh water 
alone is used .0133 to .014. 

GENERAL RULES. 

Engines. 

Steam Cylinder, Thickness. — (Vertical.) Multipl}' its diameter by the 
extreme pressure of steam in pounds per square inch that it may be sub- 
jected to, and divide the product by 2400 ; the result will give the thick- 
ness in inches. (Horizontal), divide by 2000. (Inclined), divide as above, 
in a ratio inversely as the sine of the angle of inclination. 

Shafts, Gudgeons, etc. — To resist torsion. See Rules for Torsional 
strength, pp. 485, 487. 

Journals of Shafts. — Their length should be from 1.15 to 1.25 times 
their diameter, and in main centres the)' should be 1.5 times. 

The bearings of driving and propeller shafts are not here considered as journals. 

Cross Heads, Wrought iron (Cylinder). Its section in the centre should 
be determined by the following formulae : 
axpxl . .s 

700 ='.^V r u jW(j| . 

ins., p the extreme pressure in pounds per square inch that it may be sub- 
jected to, I the length of the cross head between the centres of its journals in 
feet, and s the product of the square of the depth d, and the breadth, b, of the 

section. (Air-pump), -jg— = s, and as above for d and b. 

If the section of either of these cross heads is cylindrical, for s put 
Vs X 1.7. 

Diameter of boss twice, and diameter of end journals the same as that 
of the piston rod. Section at ends .5 that of their centre. 

Steam Pipe.— Its area should exceed that of the steam valve. 

Connecting Rod. — Its length should be 2.25 times the stroke of the piston 
when it is at all practicable to afford the space. When, however, it is im- 
perative to reduce this proportion, it may be a little less than twice the 
stroke. 

The comparative friction of long and short connecting rods is, for once the length 
of stroke of piston, 12 per cent, additional ; twice, 3 per cent. ; and thrice, 1.33. 

Neck. — Its diameter should be from 1 to 1.1 that of the piston rod. 

Centre of the bod}' (Horizontal), its diameter is ascertained in the follow- 
ing manner : 

Multiply the length of the body (between the necks) in feet by the area 
of a neck in inches, divide the product by % the stroke of the piston or 
the throw of the rod in feet, and the quotient is the area of the centre in 
square inches, from which the diameter may be determined. (Vertical). 
The area deduced Iry this rule may be somewhat less. 

Connecting Links. — Their length should be .5 of that of the stroke or 
of the throw of their attachments. 



584 STEAM-ENGIXE. 

Where a pair of connecting rods is used, as in some descriptions of en* 
gines, and with a pair of connecting links, their necks should have an area 
of .7 to .75 of that of the attached rod. 

When a second set of connecting rods or links are used, as with some 
air-pump connections, etc., their necko should have an area, in a ratio, in- 
versely as their throw to that of the first set. 

Strajjs of Connecting Rods, Links, etc. — The area of the Strap at its 
least section should be .65 that of the neck of the attached rod. The Key 
should be in thickness .3 the diameter of the neck, the width of Gib and 
Key combined should be 1.25 times, and the Slot should be 1.35 times 
that of the same diameter. The Draft of keys should be from .5 to .6 
of an inch per foot. Distance of Slot from end of rod .5 diameter of pin. 

Pins. For Cranks, Beams, etc., their area at their journals should be 1.6 
to 1.75 times ; and for Air-pump connections their area should be 1.5 times 
that of the attached rod; their length 1.3 to 1.5 times their diameter. 

Cranks (Wrought Iron). — The Hub, compared with the neck of the shaft, 
should be 1.75 the diameter and 1 the depth. The Eye, compared with 
the pin, should be twice its diameter, and 1.5 the depth. The Web, at the 
periphery of the hub, should be, in width, .7 the width, and in depth, .5 
the depth of the hub ; and at the periphery of the eve it should be, in width, 
.8 the width, and in depth, .6 the depth of the eye". 

(Cast Iron.) The diameters of the Hub and Eye should be, respectively, 
twice the diameter of the neck of the shaft, and 2.25 times that of the 
crank pin. 

The Radii for the fillets of the sides of the -web should be one half of the width 
of the web at the end for which the fillet is designed ; for the fillets at the back of 
the web, they should be one half of that at the side3 of their respective ends. 

Beams, Open or Trussed. Their length from centres should be 1.8 to 2 
the stroke of the piston, and their depth .5 their length. If strapped, the 
Strap at its smallest dimensions should have at least .9 the area of the 
piston rod, its depth being equal to .5 its breadth. The end centre jour- 
nals should have each 1, and the main centre journals 2.5 times the area 
of the piston or driving rod. 

This proportion for the strap is when the depth of the beam is half of it3 length, 
as above; consequently, when its depth is less, the area of the strap must be in- 
creased ; and when the depth of the strap is greater or less than half its width, its 
area is determined by the product of its b d 2 , being the same as if its depth was half 
its width. 

(Cast Iron). Area of Section at Centre. Multiply the extreme press- 
ure upon the piston in pounds by half the length of the beam, and divide 
the product by 500 times the depth of the centre in inches. 

Their depth at their centre should be .5 to .75 the diameter of the cylin- 
der, and, when of uniform thickness, should have a thickness of not less 
than .1 of their depth. 

Vibration of End Centres. 1+2— "/(7-f-2) 2 — (s -4- 2) 2 = vibration at each 
end ; I representing the length of beam, and s the stroke of the piston in feet. 

Plumber Blocks (Shaft). Bottom .4, and Binder .45, the diameter of 
the shaft journal. Hol'ding -down Bolts. If two are used .3 to .33, and if 
four, .22 to .25, the diameter of the shaft. 

Cocks. The angles of the sides of their plug should be from 7 C to 8 3 
from the plane of it. 

Pumps. The velocity of water in pump openings should not exceed 
500 feet per minute. 

Fly Wheels and Governors. See Rules, p. 423 and 598. 
Smoke Pipes and Chimneys. Their area at their base should exceed 
that of the extremity of the flue or flues with which they are connected. 
The intensity of their draught is as the square root of their height. 



STEAM-ENGINE. 



585 



The relative volumes of their draught is determined by the formula : 

Vh .1 a = volume in square feet; h representing the height of the pipe or 
chimney in feet, and a its area in square feet. 

When wood is consumed their area should be 1.6 times that for coal. 

The less the height of a chimney the higher the temperature of its air is 
required. 

Chimneys. The diameter at their base should not be less than from .1 
to .11 of their height. 

The batter or inclination of their external surface should be .35 of an 
inch to the foot, which is about equal to 1 brick (% brick each side) in 
25 feet. 

The diameter of the base should be determined by the internal diameter 
at the top, and the necessarj' batter due to the height. 

The thickness of the walls should be determined by the internal diame- 
ter at the top ; thus, for a diameter of 4 feet and less, the thickness may 
be 1 brick, but for a diameter in excess of that it should be 1.5 bricks. 

Velocities of the Current of heated Air in a Chimney 100 Feet in Height in Feet 
per Second. 





Air at Base of Chimney 




Air at Base of Chimney. 


















150° 


250° 


350° 


450° 




150° 


250° 


350° 


450° 




Feet. 


Feet. 


Feet. 


Feet 




Feet. 


Feet. 


Feet. 


Feet. 


10° 


24 


30 


33 


35 


60° 


19 


26 


29 


33 


32° 


22 


28 


31 


34 


70° 


18 


25 


29 


32 


50° 


20 


27 


30 


33 


80° 


17 


24 


28 


32 



When the Height of the Chimney is less than 100 feet, — Multiply the ve- 
locity as obtained for the temperature by one tenth the square root of the 
height of the chimney in feet. 

The draught consequent upon a steam jet in a chimney or smoke-pipe 
Is nearly equal to that of a moderate blast. 

Water Wheels (Arms). — Their number should be from .75 to .8 the 
diameter of the wheel in feet. (Blades) Wood. — For a distance of from 
5 to 5.5 feet between the arms, their thickness should be from .09 to .1 of 
an inch for each foot of diameter of wheel. 

A wrought iron blade .625 inch thick bent at a stress withstood by an oak blade 
3.5 inches thick. 

Blades (Area). — The area of the blades, compared with the area of im- 
mersed amidship section of a vessel, depend upon the dip of the wheels, 
their distance apart, the model and rig of the vessel. 

In River service, the area of a single line of blade surface varies from 
.3 to .4 that of the immersed section ; in Bay or Sound service, the area 
varies from .15 to .2 ; and in Sea service, the area varies from .07 to .1. 

Propellers (Screw). — The Pitch of it should vary with the area of the 
circle described by the screw to the area of the midship section of the vessel. 

Area, Two-Bladed. 

Area of disc of propeller to midship 
section being 1 to 

Ratio of pitch to the diameter of 
propeller is 1 to 

For Four-bladed screws, multiply the ratio of the pitch to the diameter 
as given above by 1.35. Length, J.66 diameter. 

Slip. — The slip of a screw propeller is directly as its pitch. 

The economical effect of a screw is inversely as its pitch, the greater 
the pitch the less effect. 

3D 






5 


4^ 


4 


3^ 


.8 


1.02 


1.11 


1.2 


1.27 



o 


2^ 


2 


1.31 


1.4 


1.47 



586 STEAM-ENGINE. 

An expanding pitch has less slip than a uniform pitch, and, consequent* 
ly, is more effective. 

Safety Valve, — Area in square inches .7 to .8 area of grate surface in 
square feet. 

Act of Congress (U. S.). — For boilers having flat or stayed surfaces, 30 ins. for 
every 500 square feet of effective* heating surface ; for cylindrical boilers, or cylin- 
drical flued, 24 square inches. 

Locked Safety Valves. — Effective heating surface, less than 700 square feet, valve 

2 ins. in diameter; less than 1500, 3 ins. in diameter; less than 2000, 4 ins. in diam- 
eter ; less than 2500, 5 ins. in diameter ; and above 2500, 6 ins. in diameter. 

Memoranda. 

The saving of fuel by a fresh-water condenser may be safely estimated 
at from 15 to 20 per cent., added to an increased speed of engine of from 

3 to 4 per cent. 

The loss of power in condensing engines by the cooling of the cylinder 
from exposure to the air, is .01 ; and in non-condensing engines, about .015. 

Friction Clutch, 5 ins. face, driven by a belt 14 ins. wide, broke a cast- 
iron shaft 6)4 ins. in diameter. 

A condensing steam-engine in river service, 43?4 inches in diameter of cylinder, with a stroke of 
piston of 10 feet, pressure of steam 25 lbs., revolutions 25, broke one of a pair of cast-iron water-wheel 
shafts 11 ins. in diameter. — Steam-boat Utica. 

A condensing steam-engine in river service, 55 ins. in diameter of cylinder, with a stroke of pistoD 
of 10 feet, pressure of steam 25 lbs., revolutions 26, broke one of a pair of cast-iron water-wheel shafts 
11.38 ins. in diameter. — Steam-boat New Philadelphia. 

A non-condensing engine, driving a rolling mill, cylinder 13 ins. in diameter, with a stroke of piston 
of 6 feet, pressure of steam 75 lbs., revolutions 30, broke a single cast-iron shaft of 11 ins. in diameter. 
Soft Iron. 

A condensing steam-engine in river service, 65 ins. in diameter of cylinder, with a stroke of piston 
of 10 feet, pressure of steam 12 lbs., revolutions 25, broke one of a pair of cast-iron water-wheel shafts, 
12k ins. in diameter. — Steam-boat De Witt Clinton. 

Two steam cylinders, 50 ins. in diameter, by 10 feet stroke of piston, having one air pump disabled, 
both engines worked by the remaining one, 30 ins. in diameter by 4.75 feet stroke , pressure of steam 
25 lbs., cut off at half the stroke ; vacuum 25^ ins. — Steamer Sonora. 

Radial and. TT'eath.erin.g "Water Wheels. 

Radial. — The loss of effect is the sum of the loss by the oblique action of the wheel 
blades upon the water, their slip, and the thrust and drag of the aims and blades as 
they enter and leave the water. 

The loss by oblique action is computed by taking the* mean of the square of the 
sines of the angles of the blades when fully immersed in the water. 

The loss by the oblique action of the blades of the water-wheels of the steamer 
Arctic (for details of which see p. 638), when her wheels were immersed 7 feet 9 ins. 
and 5 feet 9 ins., was 25^" and lS^ per cent., which was the loss of usefid effect of the 
portion of the total power developed by the engines, which was applied to the wheels. 

Feathering.— The loss of effect is confined to the thrust and drag of the arms and 
blades as they enter and leave the water. 

Comparative Effects. 

In two wheels of a like diameter (26 feet, and 6 feet immersion), like number and 
depth of blades, etc., the losses are as follows: 
Radial 25.6 per cent. | Feathering 15.4 per cent. 

The loss of effect by thrust and drag in a feathering wheel, having these elements 
and included in the above given loss, is computed at 2 per cent. 

The relative loss of effect of the two wheels is, approximately, for ordinary immer» 
sions, 20 and 15 per cent, from the circumference of the wheel. 

The Centre of Pressure of an immersed wheel blade, when the upper or inner" 
edge is level with the surface of the water, is % from the bottom or lower edge. 

* By a rule of the Inspectors, the fire surface from the grate bars to the water line (deducting half 
the flue or tube surface oojy ) is alone computed as effective. This construction of the law is altogethai 
arbitrary upon the part of the Inspectors, as all surfaces are more or less effective. 



STEAM-ENGINE. — SLIDE VALVES. 



'587 



To Compute the Centre of* Pressure of ^Water-wheel 
Blade Allien Immersed. 

Rule.— Divide the difference of the cubes of the depths of the blade be- 
low the surface of the water by the difference of their squares, and % the 
quotient will give the distance of the centre of pressure below the sur- 
face, from which subtract the depth of the upper edge of the blade, and 
the remainder will give the position of the centre of pressure required. 

In the cases here given, the centres of pressure are as follows : 

Radial wheel . 6.4 ins. from the bottom edge. 

Feathering wheel 8.5 " " 

SLIDE VALVES. 

All Dimensions in Inches. 

To Compute how much Lap must be given on the Steam 
Side of a Slide "Valve, to cut off* the Steam at any given 
I?art of the Stroke of the Piston. 

Rule. — From the length of stroke of piston subtract the length of the 
stroke that is to be made before the steam is cut off ; divide the remain- 
der by the stroke of the piston, and extract the square root of the quotientc 
Multiply this root by half the throw of the valve, from the product sub- 
tract half the lead, and the remainder will give the lap required. 

Example. — Having stroke of piston 60 ins. ; stroke of valve 16 ins., lap upon ex- 
haust side % in. = %, of valve stroke, lap upon steam side $}£ ins., lead 2 ins., steam 
to he cut off at % the stroke ; what is the lap ? 

60 — 5 of 60 = 10. ^ = .166. t/.16G=A0S. .40Sx^ = 3.264, and 3.264-|r^ 

2.264 ins. or the lap — half the lead. 

r J?o Compute the Xjap required on the Steam Side of a 
"Valve, to Cut the Steam off at various ^Portions of the 
Strolie of the Piston. 

Valve without Lead. 

Distance of the piston from the end of its stroke when the steam is 
cut off, in parts of the length of its stroke. 



Lap in parts of) 
the stroke . . J 



.354 



5_ 

12 



.323 



.286 



.27 



5 

24 



.25 . 228 



.204 



.177 



.144 



.102 



Illusteation. — Take the elements of the preceding case. 

Under % is .204, and .204 X 16 = 3.264 ins. lap. 

When the Valve is to have Lead — Subtract half the proposed lead from the lap 
ascertained by the table, and the remainder will be the proper lap to give to the 
valve. 

If, therefore, as in the last case, the valve was to have 2 ins. lead, then 2-^-2 — 
3. 264 = 2.264 ms. 



To Compute at -what Part of the Stroke of the Piston any 
given Lap on the Steam Side will cut off the Steam. 

Rule. — To the lap on the steam side add the lead ; divide the sum by 
half the length of throw of the valve. From a table of natural sines (p. 
301) find the arc, the sine of which is equal to the quotient ; to this arc 
add 90°, and from their sum subtract the arc, the cosine of which is equal 
to the lap on the steam side, divided by half the throw of the valve. Find 
the cosine of the remaining arc, add 1 to it, and multiply the sum by half 



588 



STEAM-ENGINE. — SLIDE VALVES. 



the stroke of the piston, and the product will give the length of that part 
of the stroke that will be made by the piston before the steam is cut off. 
Example. — Take the elements of the preceding case. 

— .53125; sin. .53125 = 32° 5'; 32° 5' -f 90° = 122° 5'; 2.25-hS = .2S125 

1.66371, which X ~ 
2 



16^2 

= cos. of 73° 40' ; then 122° 5' —73° 40' ±= 4S° 25' ; cos. -f 1 



= 50 ins., or %th stroke. 

Portion, of the Stroke ofa Piston, at which, the Exhausting 
Port is closed, and. opened. 

Lap on the Exhaust Side of the Valve in Parts of its throw. 







Portion of Stroke at which the Steam is cut off. 




Lap. 


1 
a 


7 
24 


i 


5 

24 


l 

G 


l 

8 


12 


2^ 


A 


















% 


.173 


.161 


.143 


.126 


.109 


.003 


.074 


.053 


X 


.13 


.113 


.1 


.035 


.071 


.05S 


.043 


.027 


£ 


.113 


.101 


.0S5 


.069 


.053 


.043 


.033 


.024 





.092 


.032 


.067 


.055 


.041 


.033 


.022 


.011 


B 

% 


.033 


.026 


.019 


.012 


.003 


.004 


.001 


.001 


% 


.06 


.052 


.04 


.03 


.022 


.015 


.008 


.002 


2 


.073 


.066 


.051 


.04? 


.0C3 


.023 


.013 


.004 





.u92 


.082 


.067 


.055 


.044 


.033 


.022 


.011 



The units in the columns of the table marked A express the distance of the pis* 
ion, in parts of its stroke, from the end of the stroke when the exhaust p'ort in ad- 
vance of it is closed; and those in the columns of the table marked B express the 
distance of the piston, in parts of its stroke, from the end of its stroke when the ex- 
Wist port behind it is opened. 

. Illustration. — A slide valve is to cut off at % from the end of the stroke of the 
piston, the lap on the exhaust side is }£ of the stroke of the valve (16 ins.), and the 
stroke of the piston is 60 inches. At what point of the stroke of the piston will the 
exhaust port in advance of it be closed and the one behind it opened ? 

Under % in table A, opposite to %^ i 3 -^3, which X 60, the length of the stroke 
— 3.1S ins.; and under % in table B, opposite to V^, is .033, which X 60 = 1.9S ins. 

If the lap on the exhaust side of this valve was increased, the effect would be to 
cause the port in advance of the valve to be closed sooner and the port behind it 
opened later. And if the lap on the exhaust side was removed entirely, the port in 
advance of the piston would be shut, and the one behind it open, at the same time. 

The lap on the steam side should always be greater than that on the exhaust side, 
and the difference greater the higher the velocity of the piston. 

In fast-running engines alike to locomotives, it is necessary to open the exhaust 
valve before the end of the stroke of the piston, in order to give more time for the 
escape of the steam. 

To Ascertain the Breadth of* the Forts. 

Half the throw of the valve should be at least equal to the lap on the steam side, 
added to the breadth of the port. If this breadth does not give the required area of 
port, the throw of the valve must be increased until the required area is attained. 

To Compute the Stroke of a Slide "Valve. 

Eur.E. — To twice the lap add twice the width of a steam port in inches, 
and the sum will give the stroke required. 

Expansion by lap, with a slide valve operated by an eccentric alone, can not be 
extended. beyond % of the stroke of a piston without interfering with the efficient 
operation of the valve : with a link motion, however, this distortion of the valve is 
somewhat compensated. When the lap is increased, the throw of the eccentric 
should also be increased. 

When low expansion is required, a cut-off valve should be resorted to in addition 
to the main valve. 



STEAM-ENGINE. INJECTOR. CALORIC ENGINE. 589 



To Compute the Distance of a ZPiston. from the End. of its 

Stroke,when the Lead, produces its Effect. 

Rule. — Divide the lead by the width of the steam port, both in inches, 
and term the quotient sine ; multiply its corresponding versed sine b} T 
half the stroke, and the product will give the distance of the piston from 
the end of its stroke, when steam is admitted for the return stroke and 
exhaustion ceases. 

Example. — The stroke of a piston is 4S ins., width of port 2% ins., and the lead 
X in- ; what will be the distance of the piston from the end of its stroke when ex- 
haustion commences? 



.5 ~- 2.5 = .2 = sine, and versed sine of .2 = .0202. 



48 
Then .0202 X -r- = .4848 ins. 



To Compute tlie Xjead, When the Distance of a JPiston 
from the End. of its Stroke is given. 

Rule. — Divide the distance in inches by half the stroke in inches, and 
term the quotient versed sine ; multiply the corresponding sine by the 
width of the steam port, and the product%vill give the lead. 

Example — Take the elements of the preceding case. 

.4848 -J- 24 = .0202 = versed sine, and sine of versed sine .0202 — .2. 

Then .2x2.5 = .5 ins. 

To Compute the Distance of aPiston from the End of its 
Stroke, when Steam is admitted for its return Stroke. 

Rule. — Divide the width of the steam port, and also that width less the 
lead, by half the stroke of the slide, and term the quotient versed sines 
first and second. Ascertain their corresponding arcs, and multipty the 
versed sine of the difference between the first and second by half the 
stroke, and the product will give the distance required. 



To Compute the Lap and Hiead of Locomotive "Valves. 

.22 t — lap in ins., and .07 t = lead in ins.; t representing the stroke of the 
valve. 

STEAM-INJECTOR. 

William Sellers Sc Co. {Self-adjusting.) 

Volume of Water Discharged per Hour. 





Pressure of Steam in Lbs. 




Pressure of Steam in Lbs. 


No. 


• 60 


80 


100 I 120 




60 | 80 


100 


120 




Cub. ft. 


Cub. ft. 


Cub. ft. 


Cub. ft. 




Cub. ft. 


Cub. ft. 


Cub. ft. 


Cub. ft. 


3 


28.12 


31.66 


35.2 


38.75 


7 


162.65 


182.1 


201.55 


221 


4 


52.16 


58.44 


64.72 


71 


8 


213.2 


238.8 


264.4 


290 


5 


82.18 


92.02 


101.86 


111.7 


9 


260.97 


302.28 


334.59 


366.9 


6 


119.09 


133.33 


147.57 


161.82 


10 


333.64 


373.57 


413.49 


453.41 



Highest temperature admissible of feed water 135°. 

CALORIC PUMPING ENGINE. 
Ericsson's. For an Elevation of 50 Feet. 



Dimen- 


Space 
occupied. 


Volume 
per 


Pipes, 

Suction 

and 


Fuel 
per Hour. 


Furnace. 


COST. 

Deep Well Pump, 
Extra. 






Hour. 


Dis- 


Nut 






Pipes per Foot. 




Floor. 


Height. 




charge. 


Anthr. 


Gas. 


Gas. 


Coal. 


Pump. 


Plain. 


Galvan. 


Ins. 


Ins. 


Ins. 


Gall. 


Ins. 


lbs. 


Cub. ft. 


% 


| 


$ 


$ 


s 


6 


39x20 


51 


200 


.75 


2.5 


18 


210 


220 






— 


8 


48X21 


63 


350 


1 


3.3 


25 


260 


275 


10 


.64 


.88 


12 


54X27 


63 


800 


1.5 


6 





— 


375 


15 


.SO 


1.15 


12t 


42x52 


65 


1600 


2 


12 


— 


— 


550 


25* 


1.25 


1.25 



* Over 90 feet, 92 cents. 



f Duplex. 



Including engine and pump, oil-can and wrench, complete in all l)ut suction and 



590 



STEAM-ENGINE. PUMPS. 



ELEMENTS AND CAPACITIES OF DIRECT- ACTING STEAM-PUMPS. 

"Worthington's. 
Fire and General Service, having Two Double-acting Plungers. 



Diameter 


Diameter 


Length 
of 


Single Strokes or 


Volume delivered 


Diam. of Plunger 




of 
Steam- 


of 
Water- 


Displacements 
per Minute of one 


per Minute, at 
stated Number of 


in any single 
Cylinder Pump 


PRICE. 


cylinders. 


plungers. 




Plunger. 


Strokes. 


for like Volume 
and Speed. 




Inches. 


Inches. 


Inches. 


No. 


Gallons. 


Inches 




4.5 


2.T5 


4 


75 to 150 


15 to 30 


4 


$110 
190 


6 


4 


6 


75 to 125 


50 to 80 


5.625 


7.5 


4.5 


10 


75 to 100 


100 to 140 


6.375 


300 


9 


5.25 


10 


75 to 100 


140 to 1S5 


7.5 


345 


10 


6 


10 


5') to 100 


125 to 245 


8.5 


375 


12 


7 


10 


50 to 100 


165 to 335 


9.S75 


475 


14 


7 


10 


50 to 100 


165 to 335 


9.875 


525 


12 


8.5 


10 


50 to 100 


245 to 490 


12 


550 


14 


8.5 


10 


50 to 100 


245 to 490 


12 


600 


16 


8.5 


10 


50 to 100 


245 to 490 


12 


650 


18.5 


8.5 


10 


50 to 100 


245 to 490 


12 


700 


12 


10.25 


10 


50 to 100 


355 to 715 


14.25 


625 


14 


10.25 


10 


50 to 100 


355 to 715 


14.25 


675 


16 


10.25 


10 


50 to 100 


355 to 715 


14.25 


725 


18.5 


10.25 


10 


50 to 100 


355 to 715 


14.25 


775 


14 


12 


10 


50 to 100 


490 to 980 


17 


800 


16 


12 


10 


50 to 100 


490 to 9S0 


17 


850 


18.5 


12 


10 


50 to 100 


490 to 980 


17 


900 


18.5 


14 


10 


50 to 100 


665 to 1330 


19.75 


1000 


IT 


10 


15 


50 to 90 


510 to 920 


14 


1450 


.20 


12 


15 


50 to 90 


735 to 1320 


17 


1950 


20 


15 


15 I 


50 to 90 


1145 to 2065 


21 


2200 



Above sizes, or of any desired capacity, can be compounded, resulting in a saving 
of 33 per cent, of fuel for like service, by any non-condensing form. 
Exterior packed plungers, for pumping water or oil against extreme pressure. 



Blak 



Water -pistons, Piston-rods, Stuffing-boxes, Linings, Valve -seats, Valve- 
bolts, etc., of Composition. All parts interchangeable, for Duplicate in 
case of Break or Wear. The Improved Water-piston is packed for Hot 
or Cold Water or other Liquids, adjustable to any Pressure. 





Steam- 


Water- 




Volume 


Strokes 




Ex- 








No. 


cyl- 


cyl- 


Stroke. 


per 


per 


Steam- 


haust- 


Suction- 


Delivery- 


PRICE. 




inder. 


inder. 




Stroke. 


Minute. 


pipe. 


pipe. 


pipe. 


pipe." 






Ins. 


Ins. 


Ins. 


Galls. 


No. 


Ins. 


Ins. 


Ins. 


Ins " 


35 


000 


2.5 


1.5 


3 


.023 


1 to 350 


.25 


.375 


.5 


.375 


00 


3 


1.75 


3 


.031 


1 to 350 


.25 


.375 


.75 


.5 


50 





3.5 


2.125 


3 


.04 


1 to 350 


.375 


.5 


1 


.75 


85* 


1.5 


4 


2.375 


5 


.10 


1 to 350 


.5 


.75 


1 


.75 


125* 


2.5 


4.5 


2.75 


6 


.15 


1 to 350 


.5 


.75 


1.25 


1 


150* 


3 


5.5 


8.25 


7 


.25 


1 to 300 


.5 


.75 


1.5 


1.25 


200* 


4 


6 


3.75 


7 


.33 


1 to 300 


.75 


1 


2 


1.5 


225* 


4.5 


6.5 


4.125 


8 


.46 


1 to 300 


.75 


1.25 


2.5 


2 


275* 


5 


7.25 


4.5 


10 


.69 


1 to 250 


1 . 


1.5 


2.5 


2 


350* 


6 


8 


5 


10 


.85 


1 to 250 


1 


1.5 


3 


2.5 


375* 


8.5 

7 


8 


5 


12 


1.02 


1 to 250 


1 


1.5 


3.5 


3 


400 


10 


6 


12 


1.47 


1 to 250 


1.25 


2 


3.5 


3 


450 


8 


12 


7 


12 


2 


1 to 250 


1.5 


2.5 


4 . 


3.5 


525. 


9 


14 


8 


12 


2.61 


1 to 250 


2 


3 


5 


4 


600 


10 


16 


9 


18 


4.96 


1 to 175 


2 


3 


8 







10.5 


16 


10 


24 


8.16 


1 to 100 


2 


3 


8 


6 




11 


IS 


12 


24 


11.75 


lto 10i> 


2.5 


3.5 


10 


8 




12 


20 


14 


24 


16 


1 to 100 


3 


4 


10 


8 






* Hand-power attachment is for purpose of operating pumps in absence of steam, for filline boilers, 
washing ducks, hru purposes, etc. 

Each pump has suction and delivery openings on both sides. 



STEAM-ENGINE. — BOILERS. 



591 



Kncrwles's. 

Dimensions and Prices given below are for Boiler-fed or Pressure Pumps, 

These Pumps can be safely run at the speed of an efficient Fire-pump. 





Steam- 


Water- 




Volume 


Strokes 




Ex- 




Dis- 




No. 


cyl- 


cyl- 


Stroke. 


per 


per 


Steam- 


haust- 


Suction- 


charge- 


PRICE. 




inder. 


inder. 




Stroke. 


Minute. 


pipe. 


pipe. 


pipe. 


pipe. 






Ins. 


Ins. 


Ins. 


Galls. 


No. 


Ins. 


Ins. 


Ins. 


Ins. 


$ 


000 


2.5 


1.5 


3 


.02 


1 to 350 


.25 


.375 


.5 


.375 


35 


00 


3 


1.75 


3 


.03 


1 to 350 


.25 


.375 


.75 


.5 


50 





3.25 


2 


4 


.05 


1 to 300 


.5 


.75 


1.25 


1 


85 


1 


3.5 


2.25 


4 


.07 


1 to 300 


.5 


.75 


1.25 


1 


125 


2 


4 


2.5 


5 


.11 


1 to 300 


.5 


.75 


1.25 


1 


150 


3 


5 


3.25 


7 


.25 


1 to 275 


.75 


1 


2 


1.5 


200 


4 


5.5 


3.75 


7 


.34 


1 to 275 


.75 


1 


2 


1.5 


225 


4.5 


7 


4 


7 


.39 


1 to 275 


1 


1.25 


2.5 


2 


2T5 


5 


7 


4.5 


10 


.69 


1 to 250 


1 


1.25 


3 


2.5 


350 


6 


7.5 


5 


10 


.85 


1 to 250 


1 


1.25 


3 


2.5 


375 


6.5 


8 


5 


12 


1.02 


1 to 250 


1 


1.25 


4 


4 


400 


7 


10 


6 


12 


1.46 


1 to 200 


1.25 


1.5 


4 


4 


450 


8 


12 


7 


12 


1.99 


1 to 200 


2 


2.5 


5 


5 


525 


9 


14 


8 


12 


2.61 


1 to 200 


2 


2.5 


5 


5 


600 


10 


16 


10 


16 


5.43 


1 to 200 


2.5 


3 


6 


6 





11 


18 


12 


24 


11.75 


1 to 180 


3.5 


4 


8 


6 





12 


20 


14 


24 


15.99 


1 to 180 


3.5 


4 


10 


8 


— 


13 


24 


16 


24 


20.79 


lto 150 


4 


4.5 


12 


10 


— 


14 


30 


18 


24 


26.43 


1 to 150 


5 


6 


14 


12 


- — 



Pumps arranged with other combinations of cylinders, for light service, wrecking, 
irrigation, etc. Vertical mining and Artesian well-pumps. 

Independent direct-acting steam -pumps have a special advantage in the sup- 
plying of boilers, as their speed can be adjusted to run continuously, and to main- 
tain the water at a uniform height. 

Valve motion being positive, under any conditions, these pumps can be worked at 
the slowest practicable speed. 

Special patterns for fire, tank, raining, brewer's, marine, tannery, oil, vacuum, 
drainage, and irrigating pumps, air - compressors, combined boilers and pumps, 
and compound condensing water-works pumping engines. Single or duplex. 

BOILEE. 

Its efficiency is determined by proportional quantity of heat of combus- 
tion of fuel used, which it applies to the conversion of water into steam, 
or it may be determined by weight of water evaporated per lb. of fuel. 

In following results and computations, water is held to be evaporated 
from standard temperature of 212°. 

Proportion of surplus air, in operation of a furnace in excess of that 
which is chemically required for combustion of the fuel, is diminished as 
rate of combustion is increased ; and this diminution is one of the causes 
wh} T the temperature in a furnace is increased with rapidity of combustion. 

When combustion is rapid, air should be introduced in a furnace above 
the grates, in order the better to consume the gases evolved. 

Natural Draught. 

Boiler (Land) set at an inclination downward of .5 inch in 10 feet. 
^ Grates (Coal) should have a surface area of 1 sq. foot for a combus- 
tion of 15 lbs. of coal to be consumed per hour, at a rapid rate of combus- 
tion, and set at an inclination toward bridge wall of 1 to 2 inches in every 
foot of length. When, however, rate of combustion is not high, in conse- 
quence of low velocity of draught of furnace, or fuel being insufficient, 
this proportion must be increased to 1 sq. foot for every 12 lbs. of fuel. 

Level of grate under a plain cylindrical boiler gives best effect with a fire 
5 ins. deep when but 7.5 ins. from lowest point. 

With Wood as fuel, their area should be 1.25 to 1.4 that for coal. 



592 STEAM-ENGINE. BOILERS. 



n from 
j steam 



Width of bars the least practicable, and spaces between them 
.5 to .75 of an inch, according to fuel used. 

Short grates are most economical in combustion, but generate 
less rapidly than long. 

Automatic (Vicar's). Its operation effects increased rapidity in firing 
and more effective evaporation. 

Ash-pit. — Transverse area of it, for a combustion of 15 lbs. of coal per 
hour, .25 area of grate surface for bituminous coal, and. 33 for anthracite. 

Furnace or Combustion Chamber {Coal). — Volume from 2.75 to 3 cub. feet 
for every sq. foot of its grate surface. {Wood.) Volume 4.6 to 5 cub. feet. 

The higher the rate of combustion the greater the volume of it, bitumi- 
nous coal requiring more than anthracite. 

Velocity of current of air entering an ash-pit may be estimated at 12 
feet per second. 

Volume of air and smoke for each cub. foot of water converted into steam is from 
coal 17S0 to 1950 cub. feet, and for wood 3900. 

Rate of Combustion. — In pounds of coal per sq. foot of grate per hour. 
Comiish Boilers, slowest, 4; ordinary, 10. Factory, 12 to 16. Marine, 
16 to 24. Quickest : complete combustion of dry coal, 20 to 23 ; of caking 
coal, 24 to 27 ; Blast or Fan and Locomotive, 40 "to 120. 

Combustion is the most complete with firings or charges at intervals 
of from 15 to 20 minutes. 

Admission of air above the grate increases evaporative effect, but di- 
minishes the rapidity of it. 

Air admitted at bridge-wall effects a better result than when admitted 
at door, and when in small volumes ; and in currents, it arrests or prevents 
smoke. It may be admitted by an area of 4 sq. ins. per sq. foot of grate. 

High wind increases evaporative effect of a furnace. 

Bridge-wall. — Cross section of it an area of 1.6 to 1.8 sq. inches for each 
lb. of coal consumed per hour, or from 24 to 27 sq. inches for each sq. foot 
of grate, for a combustion of 15 lbs. of coals per hour. 

Temperature of a furnace is assumed to range from 1500° to 2000°, and 
volume of air required for combustion of 1 lb. of bituminous coal, together 
with products of combustion, is 154.81 cub. feet, which, when exposed to 
above temperatures, makes volume of heated air at bridge-wall from 600 
to 750 cub. feet for each lb. of coal consumed upon grate. 

Hence, at a velocity of draught of about 36 feet per second, area at bridge- 
wall, required to admit of this volume being passed off in an hour, would 
be .7 to .8 of a sq. inch, but in practice it should be 1.6 to 1.8 inches. 

When 15 lbs. of coal per hour are consumed upon a square foot of grate, 15x1.6 
or 1.8=24 or 27 sq. inches are required, and in this proportion for other quantities. 

Calorimeter. — When area of flues is determined upon, and area over 
bridge-wall is required, it should be taken at from .7 to .8 area of lowei 
flues for a natural draught, and from .5 to .6 for a blast. 

When one half of tubes were closed in a fire tubular marine boiler, the 
evaporation per lb. of coal was reduced but 1.5 per cent. 

Heating Surfaces. 
Marine (SeaWater). — Grate and heating surfaces should loe Increased 
about .07 over that for fresh water. 

Relative Value of Heating Surfaces. 

Horizontal surface above the flame=l. j Horizontal beneath tlie hame =.1 

Vertical = .5 | Tubes and flues z=.56 

A scale one sixteenth of an inch in thickness will effect a loss of 14.7 per cent, of fuel. 

One square foot of fire surface is computed to be as effective as three of 
heating surface. 



STEAM-ENGINE. BOILERS. 593 

Boilers Avith. Internal Furnaces. 

For Coal, 13 lbs. per Hour per Square Foot of Grate. (Natural Draught.) 

Pressure of Steam 20 lbs. {Mercurial Gauge), and 20 Revolutions of the Engine 
per Minute. 

Fire and Flue Surface.* (Arches or Flues and Return Flues.) — For 
every cubic foot of steam to be expended in the steam cylinder, for a sin- 
gle stroke of the piston (computed only to the point of" cutting off), the 
length of the flues and steam chimney "not exceeding 45 or 50 feet, there 
should be from 48 to 54 square feet. 

(Arches or Flues, and Tubes, or Return Tubes.') Horizontal Return.-— 
The length of the tubes and steam chimne}- not exceeding 30 or 35 feet, 
there should be from 58 to 64 square feet. 

Vertical Water Tubes. — From 64 to 70 square feet. 

Grates. — For every cubic foot of steam as above, there should be from 
1.75 to 2.1 square feet. 

For Coal, 30 lbs. per Hour per Square Foot of Grate. (Blast or 
Exhaust.) 

Pressure of Steam 30 lbs. (Mercurial Gauge), and 20 Revolutions of the Engine 
per Minute. 

Fire and Flue Surface* (Arches or Flues and Return Flues.) — For 
every cubic foot of steam to be expended in the steam cylinder, for a 
single stroke of the piston (computed only to the point of cutting off), 
the length of the flues and steam chimney not exceeding 55 or 60 feet, 
there should be from 24 to 28 square feet. 

(Arches or Flues and Tubes.) Horizontal Return. — The length of the 
tubes and steam chimney not exceeding 30 or 35 feet, there should be 
from 29 to 32 square feet. 

Vertical Water Tubes. — From 32 to 35 square feet. 

Grates. — For every cubic foot of steam as above, there should be from 
1.15 to 1.35 square feet. 

Boilers with External Furnace and. Internal Flues. 
(Cylindrical Fine.) 

For Coal, 20 lbs. per Hour per Square Foot of Grate, or for Wood at 40 lbs. 
(Natural Draught.) 

Pressure of Steam 100 lbs. (Mercurial Gauge), and 20 Revolutions of the Engine 

per Minute. 

Fire and Flue Surface.^ — For every cubic foot of steam to be expended 
in the steam cylinder, for a single stroke Of the piston (computed only 
to the point of cutting off), the length of the flues and steam chimney not 
exceeding 55 to 60 feet, there should be from 100 to 108 square feet. 

Grates. — For every cubic foot of steam as above, there should be from 
3.8 to 4. square feet. 

Western Boilers. — In the boilers upon the Western lakes and rivers 
of the United States, where the coal consumed is of the very best quality, 
and the smoke pipes are carried to a great height, the combustion of coal 
per square foot of grate per hour readily reaches 40 lbs. 

1% cords of Western wood have been burned per hour upon 48 square 
feet of grate. 

In this case, the units above given may be reduced to 50 and 54 for 
heating surface, and the grate surface decreased to 1.85 and 2. 

* Estimated from nbove the grate bars, including steam chimney, and for sea water, 
f Estimated from above the grate bars, including steam chimney, where one exists, and for frusli 
water. 



594 STEAM-ENGINE. — BOILEES. 

Boilers -with. External Furnace and. Fine. (3?lain Cylin- 
drical.) 

For Coal, 20 lbs. per Hour per Square Foot of Grate, or for Wood at 
40 lbs. {Natural Draught.) 

Pressure of Steam 100 lbs. {Mercurial Gauge), and 20 Revolutions of the Engine 
per Minute. 

Fire and Flue Surface.* — For eveiy cubic foot of steam to be expended 
in the steam cylinder, for a single stroke of the piston (computed only to 
the point of cutting off), the length of the flues and steam chimney not 
exceeding 30 feet, there should be from 85 to 92 square feet. 

Grates. — For every cubic foot of steam as above, there should be from 
3.8 to 4. square feet. 

All of these units are based upon the volume of furnace, area of bridge- 
wall, or cross-section of flues or tubes, etc., as given in the preceding rules. 

The ranges given, of from 48 to 54, 24 to 4S, etc. , are for the purpose of meeting the 
ordinary differences of construction, thickness of metal, etc. 

When a heater is used, and the temperature of the feed- water is raised above that 
obtained in a condensing engine, the proportions of surfaces may be correspondingly 
reduced. 

Steam Room. — There should be from 2.5 to 3.5 times the volume of 
steam room that there are cubic feet of steam expended in the cylinder 
for each single stroke of the piston for 25 revolutions ; or the volume of 
it should be from 5 to 7 times the volume of the cjdinder, increasing in 
proportion with the number of revolutions. 

When there are two engines, or an increased number of revolutions, these propor- 
tions of steam room must be increased. 

Felt covering to a boiler and steam pipes effects a very material saving 
in fuel. 

Notes —Four copper boilers, with a natural draught and bituminous coal, flues 40 feet in length, 
Including steam chimney, with 14 square feet of fire and flue surface, and 6 of a square foot of grate 
surface for every cubic foot in the cylinders, furnished steam at 20 lbs pressure, cut off at V 2 of the 
stroke of the piston, for 18.5 revolutions. 

The mean of four cases, with iron boilers and anthracite coal, with a blast, flues 50 feet in length, 
gave, with 12.5 square 1'eet of fire and flue surface, and .5 of a square foot of grate surface for every 
cubic foot in the cylinders, steam at 35 lbs. pressure, cut off at V 2 of the stroke of the piston, for 22 
revolutions 

The space in the steam room of the boilers and chimney was about 5 times that of the cylinders in 
the preceding cases. 

To Compute tlie Heating and. G-rate Surface required for 
a given Evaporation, or Volume of* Cylinder and Revo 
lutions. 



; 

t- 

* 



Operation. — Reduce the evaporation to the required volume of cylin- 
der, number of revolutions of engine, pressure of steam, and point of cut- 
ting off; then reduce these results to the range of consumption of fuel 
per square foot of grate, pressure of steam, and number of revolutions 
given for the several cases at pp. 593 and 594, and multiply them b}' the 
units given for the surfaces required. 

Illustration. — There is required an evaporation of 492.24 cubic feet of salt 
water per hour, under a pressure of steam of 17.3 lbs. per square inch, stroke of en- 
gine 10 feet, cutting off at X stroke, revolutions 15 per minute, and consumption of 
fuel (coal) 13 lbs. per square foot of grate per hour, in a marine boiler having inter- 
nal furnaces and vertical tubes. 

Volume of steam at this pressure compared with water, S33. 
492.24 X 833-^60 = 6833.93 cubic feet of cylinder per minute. 
6833.93 -=- 15 X 2 = 227.79 cubic feet of cylinder at half stroke. 

* These proportions are for the evaporation of fresh water; if 6ea water is used, the surface 
must be increased .066. 



STEAM-ENGINE. 595 

227 79 X IT 3 197.04 X 15 
Then — = 197.04 cubic feet at 17.3 lbs. pressure, and — '—^. = 

147.78, which X 66, the unit for heating surface for a vertical tubular toiler at 20 
lbs. pressure and 20 revolutions = 9753.48 square feet. 
And 147.78 X 2 = the unit for grate under like condition = 295.56 square feet. 

Note. — The steamer Baltic has developed all the elements here given, and the sur- 
faces of her boilers and grates (for one engine) were 9742 and 293.9 square feet. 

To Compute tlie Consumption of Fuel in. the Furnace of 
a Boiler. 

The Dimensions of the Cylinder, the Pressure of the Steam, the Point of 
Cutting Off, the Revolutions, and the Evaporation of the Boilers per Pound 
of Fuel per Minute being given. 

Rule. — Ascertain the volume of water expended in steam, and multi- 
ply it by the weight of a cubic foot of the water used ; divide the product 
by the evaporating power of the fuel in the boiler under computation in 
pounds of water, and add thereto the loss per cent, by blowing off; 

BOILER PLATES AND BOLTS. 

Boiler Plates and Bolts. — Tensile strength of Wrought Iron plates 
and bolts ranges from 45 500 to 62 500 lbs. per sq. inch for plates, 59 000 
for English bolts, and 65000 for American, being increased when sub- 
jected to a moderate temperature. 

, The mean tensile strength of Steel plates and bolts ranges from 80 000 
to 96 000 lbs. Kirkaldy gives 85 966 as a mean. 

Bursting and. Collapsing Pressures. 

The computation for plates and bolts should be based, so far as may 
be practicable, upon their exact tensile strength. Whenever, then, the 
strength of plates is ascertained, there should be deducted therefrom one 
half for single riveting and three tenths for double riveting, and the re- 
mainder divided by a factor of safety of three. When the exact strength 
can not be ascertained, a factor of six should be used both for plates and 
bolts. 

The resistance to collapse of a flue or tube is much less than the re- 
sistance to bursting ; the ratio can not well be determined, as the resist- 
ance of a flue decreases with its length, or that of its courses. 

With an ordinary cylindrical boiler, 4 feet in diameter, single riveted, 
20 feet in length, with flues 153^ inches in diameter, shell % e thick, 
flues V in., the relative strengths are: Bursting, 350 lbs.; Collapsing, 
152 lbs. 

To Compute the Thickness, Maximum Working Press- 
ure, and. Diameter of a Wrought Iron Boiler or Flue. 

For Service in Salt Water. — Add one sixth to the thickness of the 
plate or diameter of the bolt. 

Thickness. Rule. — Multiply the diameter in inches by half the maxi- 
mum working pressure in lbs. per square inch, and divide the product by 
9 000 (one sixth of 54000) for single riveting, and 12 500 for double, and 
the result will give the thickness in decimals of an inch. 

Pressure. Rule.— Multiply the thickness by 9 000 or 12 500, as before 
given ; divide the product bv the diameter, and twice the quotient will 
^ive the maximum pressure in pounds. 



596 STEAM-ENGINE. — BOILERS. 

Diameter. Rule. — Multiply the thickness by 9 000 or 12 500, as before ; 
divide the product b}- half the maximum pressure, and the quotient will 
give the diameter in inches. 

Example. — The diameter of a single riveted wrought iron boiler is 42 inches, and 
the plates yi of an inch in thickness ; what is its maximum working pressure ? 

.25 X 9000 



and what it3 thickness at this pressure ? 



-X 2 = 110? 



42x110-^2 _ . 
9000 = ' 25 '™- 

To Compxxte the Thickness ofFlat Surfaces in a Wrought 
Iron Boiler. 

Rule. — Multiply the maximum working pressure by the square of the 
distance, or the area of the surface, between the centres of the stavs in 
inches; divide the product by 15 500, and the quotient will give the thick- 
ness in inches. 

Example. — Take the elements of the preceding case. 
110x62 



15 5U0 



= .255 in. 



To Compute the Diameter of* Stay Bolts. * 

Rule. — Multiply the distance between their centres in inches by the 
square root of the quotient of the maximum working pressure, divided by 
5500 for wrought iron, and 4100 for copper, and the result will give the 
diameter of the body of the bolt in inches. 

Example. — The maximum working pressure of a wrought iron boiler is 110 lb3., 
and the distance apart of the bolts is inches ; what should be their diameter? 

6 Xy/^ = 6 X V-02 = 6 X .1414 = .S5 in. 

To Compute the Distance Apart of Stay Bolts. 

Rule.— Multiply the square root of the quotient of 5500 for wrought 
iron, and of 4100 for copper, divided by the maximum working pressure, 
by the diameter of the bolt, and the product will give the distance in 
inches. 

Example. — The maximum working pressure of a wrought iron boiler is 110 lbs., 
and the diameter of the stay bolts is .85 inch ; what should be their distance apart ? 



4 



*^ X .85 = V50 X .S5 = 6 vis. 



Note. — Where stays are secured by keys, their ends should he IX times the di- 
ameter of the stay, the depth of the slot 1.0 diam. of stay, and the width .3. 

Stay Bolts. — Iron stay bolts, % ins. in diameter, screwed into a Copper 
plate % thick, drew at a strain of 18 260 lbs. 

A like stay bolt, screwed and riveted into an Iron plate, drew at a strain 
of 28 760 lbs. 

A like stay bolt of Copper, screwed and riveted into a Copper plat 
drew at a strain- of 16265 lbs. 

Hence, Stay bolts when screwed and riveted are % stronger than whei 
screwed alone. 



: 



STEAM-ENGINE. — BOILERS. 



597 



Flat Surfaces, 

The resistance of a flat surface decreases in a higher ratio than the 
space between the stays. 

Iron plates % in. thick, with stay bolts 5 ins. apart (from centres), gave 
way with a strain of 900 lbs., and with stay bolts 4 ins. apart at 1600 lbs. 

Thickness of* Boiler Iron, required and. Pressures allo\^ecl 
toy- the 1-jarws of* the U. S. 

Pressure equivalent to the Standard for a Boiler 42-in. in Diameter and X inch thick, 



Wire 


Thickness 
in 16ths. 








Diameter. 








Gauge. 


34 Inches. 


36 Inches. 


38 Inches. 


40 Inches. 


42 Inches. 


44 Inches. 


46 Inches. 


No. 




Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


Lbs. 


1 


5 


169.9 


160.4 


152. 


144.4 


137.5 


131.2 


125.5 


2 


4# 


158.5 


'149.7 


141.8 


134.7 


128.3 


122,5 


117.2 


3 


4& 


14T.2 


139.1 


131.8 


125.1 


119.2 


113.7 


108.8 


4 


4 


135.9 


128.3 


121.6 


115.5 


110. 


105. 


100.4 


5 


3% 


124.5 


117.6 


111.4 


105.9 


100.8 


96.2 


92.1 


o 


3X 


113.2 


106.9 


101.3 


96.2 


91.7 


87.5 


83.7 


7 


3 


101.9 


96.2 


91.2 


86.6 


82.5 


78.7 


75.3 



Riveted. Joints. 

Forms and Proportions of Riveted Joints. — (W. Fairbairn.) 



Thici- 


Diameter 

of 

Rivets. 


Multi- 
plier. 


Length 

of 
Rivets. 


Multi- 
plier. 


Centre 

to Centre 

of 

Rivets. 


Multi- 
plier. 


Lap 

in 
Single 
Joints. 


Multipliers. 


nets of 
Plate. 


Single 
Joints. 


Double 
Joints. 


Ins. 

% 

% 


Ins. 

% 
% 
% 
% 


2. 

2. 

2. 

2. 

1.5 

1.5 

1.5 


Ins. 

2.M 


4.5 

4.5 
4.5 
4.5 
4.5 
4.5 
4.5 


Ins. 

m 

2 

2^ 

3 


6.5 

6. 

5.2 

4.7 

4. 

4. 

4. 


Ins. 

IX 

1% 

2 

2^ 

2% 

3K 


6.8 

6. 

6. 

5.3 

4.5 

4.4 

4.3 


11.1 

10. 

10. 
8.8 
7.5 
7.3 
7.2 



The Length of a rivet, alike to a bolt, is measured from inside of its head. 

The Multipliers are for computing the Diameter, Length, and Distance 
between centres of the rivets ; also for the Laps for Single and Double 
Joints, by multiplying the thickness of the plate by the Multiplier for the 
element required. 

In Riveted Joints exposed to a tensile strain, the area of the rivets 
should be equal to the areas of the section of the plates through the line 
of the rivets, running a little in excess up to %, in., and somewhat less 
beyond that diameter of rivet. 

Relative Strength of Riveted Joints per Square Inch of Single Plate. 

Single Lapped. — Machine riveted. Rivets 3 diameters from centres, 
25 000 lbs. 

Hand riveted. Rivets 3 diameters from centres, 24000 lbs. 

Rivets set "staggered," and equidistant from centres, 30 500 lbs. 

Abut Joints. — Hand riveted. Rivets not "staggered," and equidistant 
from centres, single cover or strip, 30 000 lbs. 

Rivets set " square," single cover or strip, 42 000 lbs. ; double covers or 
strips, 55 000 lbs. 

Relative Mean Strength of Riveted Joints compared to that of the Plates, 
Allowances being made for Imperfections of Rivets, etc. 

Plates, 100; Double or " square" rivets, .7; " Staggered" rivets, ,65; 
Single rivets, .5. 

3E 



598 STEAM-ENGINE. LOCOMOTIVES. 

LOCOMOTIVE ENGINE. 

Proportion of Parts based upon Diameter of Cylinder. 



Crank Shaft diameter .4 

Connecting Rod, end " .16 

middle... " .21 
Piston Rod " .15 



Depth of Piston 2{ 

Blast Pipe, diameter 3 

Area Steam Port d 2 X .03 

" Exhaust Port d 2 X .28 






.0022 X — = w. 21.2 Vw 9 — H. .0022 — = g ; H and g representing 

areas of heating and grate surfaces in square feet, and w volume of water 
evaporated in cubic feet per hour. 

Heating Surface. — 70 to 85 times that of the grate surface. 

Grate Surface. — Area .011 of heating surface when coke is used, and 
.016 when bituminous coal is used. 

Area of Tube surface 10.5 times that of the furnace surface. 

Steam Room. — 6 to 7 cubic feet per square foot of grate surface. 

The Evaporation of water in a locomotive boiler is from 8.5 to 9.5 lbs. 
water from 212° per lb. of coke consumed. 

Memoranda. 

1192 square feet of heating surface and 74 feet of grate in a locomotive 
boiler (tubular direct), gave identical results with a plain cylindrical 
boiler having 1184 feet of heating surface and 96 feet of grate. 

The mean results of an experiment with a passenger train was 18 lbs. 
coke expended per mile run, and 26.3 lbs. per mile at a speed of 24.5 
miles. 

There is an increase of effect with large drums over small. 

.76 cubic feet of water evaporated per hour will produce 1 horse power. 

In the Soemmering locomotive, a power of 4.2 horses has been attained 
per ton of engine and tender; and 1 lb. of wood has raised 13 600 lbs. 
1 foot in height per minute = 14.6 lbs. wood per horse power; and in a 
Freight locomotive, Northwestern Railwa}-, No. 227 (Eng.), a power of 
3.55 horses has been attained per ton of engine and tender. 

With two locomotives of R. Stephenson & Co., cylinders 14 ins. in diam. 
and 22 ins. stroke, driving wheels 3.5 feet, locomotives secured together 
and operated as one, weight 50 tons, 150 tons were elevated a grade of 1 
in 36 at 15 miles per hour. 

Speed of Trains. 

60. -f- minutes in running 1 mile = miles per hour. 
3600 -;- seconds " 1 mile = " " 

Railway Train. — At a speed of 33 miles per hour, a distance of 57 vards 
is required within which a train can be arrested; and at a speed "of 63 
miles per hour, a distance of 273 yards is required. 

Brakes. — The resistance of brakes, as determined by experiments, is 
about 129 lbs. per ton of train. 

GOVERNOR. 

187.5 /187.5\ 2 



> /187.5\ 2 _ . 

nT~ n ' \ n ) " re P re senting vertical height from plane of 

n volution to point of suspension in inches, and n number of revolutions per 
minute. 



STEAM-ENGINE. — COST, WEIGHTS, ETC. 



599 



Dxxty- of* Steam Engines. 

The conventional duty of an engine is the number of pounds raised by 
it 1 foot in height by a bushel of bituminous coal (112 lbs.). 

Cornish Engine. — Average dutv 70 000 000 lbs. ; the highest duty rang- 
ing from 47 000 000 to 101 900 000"lbs. 

An actual horse-power per hour, in a condensing marine engine, work- 
ing with steam at 15 lbs. (mercurial gauge), cut off at % stroke, will re- 
quire 2.07 lbs. bituminous coal. 

Evaporation 10.5 lbs. water per lb. of Welsh coal consumed. 

Portable Engine. — Cylinder 6J£ ins. in diam. by 4 foot stroke of piston, 
revolutions 115 per minute, consumption of anthracite coal 17.28 lbs. per 
hour, steam cut off at % stroke. 

Pumping Engine (Condensing). — The engine of the Brooklyn Water* 
works, N. Y., elevated 611 114 lbs. water 1 foot in height per minute, with 
a consumption of 1 lb. anthracite coal. Friction of engine between cylin- 
der and pumps 7.4 per cent. ; loss of action in pumps 1.69 per cent. 

This operation is in excess of all previous essays. 

The Leeghwater engine, in the Harlaem Meer, elevated 11 926 642 lbs. 
1 foot in height per minute. 

lielative Cost ofvarious Engines for Equal Effects. 
In Pounds of Coal per Horse-power per Hour. 

Lbs 

A theoretically perfect steam-engine , 66 

A Cornish condensing steam-engine 2 .38 

Ericsson's air engine 3.86 

A marine condensing steam-engine 2 to 6 . 



WEIGHTS OF STEAM-ENGINES AND WHEELS OR PROPELLERS. 

Side Wlieels. — American Marine {Condensing). 



Engine. 


Frame. 


Water 
Wheels. 


No. of 
Cylinders. 


Volume of 
Cylinders. 


Weight per 
Cubic Feet. 


Service. 










Cub. feet. 


Lbs. 




Vertical beam . . 


Wood.* 


Wood. 


1 


63. 


1100 


River. 


■do. 


Wood.* 


Wood. 


2 


21G. 


1040t 


Coast. 


do. 


Wood.* 


Iron. 


1 


530. 


1500 


Sea. 


do. 


Wood.* 


Iron. 


1 


725. 


losat 


Sea. 


Steeple 


Iron. 


Iron. 


1 


12.8 


3800 


River and Coast. 


Oscillating 


Iron. 


Iron. 


2 


540. 


850 


Sea. 


do 


Iron. 


Iron. 


2 


1090. 


760* 


Sea. 


Overhead direct 


Iron. 


Iron. 


2 


261.3 


1400 


Sea (Gorgon). 


Inclined direct. . 


Iron. 


Iron. 


2 


534.5 


1100 


U. S. Navy. 


do. 


Iron. 


Iron. 


2 


353. 


1316 


U. S. Navy. 


Side Lever 


Iron. 


Iron. 


2 


534.5 


1390 


U. S. Navy. 


Horizontal 


Wood. 


Wood. 


2. 


— 


— 


River. 



English, per Nominal Horse-power. 



Side Lever. 



Lbs. 

1546. 



Engines and) 

Wheels j" 

Boilers (tubular) 798 . 

Coal Bunkers 77. 3 

Water in boilers 515. 



Oscillating. 

Lbs. 

Engines 560 

Wheels 246 

Boilers 4S1 

Coal Bunkers 67 

Water in Boilers 2.M 



* Without frame. 



f With frame 1109. 



J Single frame. 



600 



STEAM-EXGIXE. — SPACE, WEIGHTS, ETC. 
Screw P»ropellers. —Marine. (Condensing.) 



Engine. 


No. of 
Cylinders. 


Volume of 
Cylinder. 


Weight per 
Cubic Feet. 


Service. 




2 
1 

1 
2 
4 
2 
2 


Cubic Feet. 

12.5 

69. 

33. 
215. 
506. 
193. 

6S. 


Lbs. 
5600 
4660 
3650 
2200 
3010 
2080 
4260 


Sea. 


do 


Sea. 


do 


Coa^t. 


Vertical compound single 

do. double 

Trunk « 


Sea. 
Sea. 
Sea. 


Horizontal back-action 


Sea. 



Kon-condensing. 



No. of 
Cylinders. 



Volume of 
Cylinder. 



Weight per 
Cubic Feet. 



Horizontal direct . 

Oscillating 

Vertical direct 

Inclined direct. . . . 



Cubic Feet. 
3.9 
7.6 
9.3 
4. 



Lbs. 
4990 
5500 
7800 
5200 



Lan-d. Engines. 



Engine. 



Engine. 



(Xon-condensing. ) 

I Spur Wheel | a ._.„ ' _ ., 

and Sugar Mill ; Boilers, 

Connections.' Complete. Grates, etc. 



River. 

Sea. 

Coast. 

Coast. 



Engine per 
Cubic Feet 
of Cylinder. 





Lbs. 


Lbs. 


Lbs. 


Vertical beam, IS ins. X 4 feet.. 


67 200 


37800 


89 600 


do. 30 ins. X 5 feet.. 


105 000 


137179 


265 S79 


Horizontal, 14 ins. X 2 feet 


10914 


— 


— 


do. 22 ins. X 4 feet 


56000 


— 


— 



Lbs. 
26 8S0 
75(K'0 

S200 
30140 



Lbs. 
9600 

98S4 
5100 
5600 



Jrtelative Space occa.ipi.ed. and "Weight of different Forms 
of jVTariixe Engines, omitting Water Wheels, Shafts, 
and Cranks, being common to all.— (Horatio Allen.) 





Space. 


Weight. 


Engine.— Equal Volume of Cylinder of 315 
Cubic Feet. 


Total. 


Per Cubic 

Foot of 
Cylinder. 


Total. 


Per Cubie 

Foot of 
Cylinder. 


Vertical beam (overhead), 1 cylinder. . . . 

Side lever 1 u .... 

Oscillating 1 " 

Double cylinder, inclined at right angles, ) 
volume of high pressure or full stroke V 
cylinders 150 cub. ft. — (a. G. Stimkes.) ) 


Cub. Feet. 

14 000 

12 000 

8 000 

10000 


Cub. Feet. 
44.4 
38. 
25.4 

17. 


Lbs. 
2S0 000 
295 000 

220 000 

448 000 


Cub. Feet. 
8S5 
935 
6J7 

760 



The spaces include all passages about the engine; but with the vertical beam, 
fchat portion of the frame and engine which is above the spar deck is not included. 

Proportionate Weights, Space occupied, and Cost of English Side-wheel 
Engines, Boilers, Wheels, or Screw Propeller, etc., per 'Nominal Horse- 
power. — (Admiralty.) (Mean of 18 cases.) 

"WEIGHTS. 



Engines 5S8 

B dton and Appendages 336 

Water in do 224 

Wheels or Propeller 112 



Lbs. 

Extra pieces 84 

Coal Bunker for 1GS0 lbs. per horse-) R , 

power j 

Total 1428 



Space occupied by Engines alone, exclusive of boilers, coal bunkers, and passage- 
iv.i\ -, may be taken at % of a square foot per horse-power, and for Boilers alone at 
1 square foot. 



STEAM-ENGINE. — WEIGHTS OF BOILERS, ETC. 



601. 



cost (1856), 

Engines $120 

Boilers 60 

Coal Bunkers 10 



Wheels $12 .50 

Extra pieces .. 12. 50 

Total $215. 



WEIGHTS OF BOILERS. 

Weights of Iron Boilers (including Doors and Plates, and exclusive of Smoke 
Pipes and Grates) per Square Foot of Heating Surface. 

Measured from Grates to Top of Steam Chimney or Base of Smoke Pipe. 

Description of Boiler. oi- Weight. 

in Lbs. 



Do. 


do. 


Drop 


do. 


Do. 


do. 


Do. 


do. 


Single 


do. 


Do. 


do. 



Double return, Flue* water bottom . . 

Single do. do.f do 

do _ 

do.t water bottom. . 

do., and over furnace do 

do. do. .... — 

do. , Multifluet water bottom . . 

do. do — 

Horizontal return, Tubular* water bottom. . 

Do. do. do.t — 

Do. do. do.f — 

Vertical do. do. J water bottom. . 

Do. do.,t back of furnace do. 

Horizontal direct, Tubulart water bottom. . 

Do. do. do.t — 

Cylindrical, external furnace,§ 36 ins. in diam., % in. thick 
Do. Flue do. §36 to 42 do. X do. 

Horizontal direct, Tubular Locomotive . . . 

Vertical Cylinder direct, Tubular _ 



S. 
L. 
L. 
L. 

S. 



31.5 to 33.3 

25.6 to 32,9 

24. to 30. 

30.4 to 40.8 
29. to 38. 
2T. to 36. 
27. to 45. 

25. to 43* 

22.5 to 35. 
21. to 33. 
17.T to 26.T 
18.5 to 26.5 
25. to 31.2 
19.S to 23.8 
17. to 21. 
23.5 to 24. 
18.1 to 18.6 
16.3 to 17.3 
24. to 26. 



Note. — The range in the units of weight here given arises from peculiarities of 
construction, consequent upon the proportionate number of furnaces, thicknesses of 
metal, volume of boilers compared with heating surface, character of staying, etc. 

2. The boiler of the British Admiralty is a Horizontal return Tubular, with water 
bottom, and its weight varies from 28 to 33 lbs. per square foot as above. 

Consumption of Fuel in Pounds per Hour per Square Foot of Grate 
for several Marine and Land Boilers. 

Natural Draught. 

Western steamboats, bituminous 40 

, locomotive, marine 16 

, do., land 13 



"Costa Eicn," anthracite 11 

(compound), bituminous 12 



11 Arctic, 1 ' bituminous 13 



Blast. 



' Daniel Drew," anthracite 36 | • 



-, locomotive, land.. 



.90 



SATURATION IN MARINE BOILERS. 

Sea water contains 3.03 parts of its weight in saline matter, and is sat- 
urated when it contains 36.37 parts. 

Blowing OfF. 

To Compute the Loss of Heat by the Blowing Off of Saturated Water from a 
Steam-boiler. 



S— TxE+< 
t 



--proportion of heat lost, S — T xE = degrees of heat re* 



* Section of furnace square. Shell, top arched, bottom square. 

+ Rection of furnace square. Shell cylindrical. J Section of furnace and shell square. 

§ Wrought iron heads, 3gths thick, flues J^ in., and surface computed to half diameter of shell. 

3E* 



602 STEAM-ENGINE. — SATUKATION. 



. 



quired from the fuel for the water evaporated, and - — - — - = loss of 

O 1 X-ti ~T~ t 

heat per cent.. S representing sum of sensible and latent heats of water evap' 
orated, T temperature of feed water, t difference in temperature of water 
bloiced off and that supplied to the boiler, E volume of water evaporated, 
proportionate to that blowed off, the latter being a constant quantity, and 
represented by 1. 

Values of E at the following degrees of saturation, viz. : 
1.25 1.5_ r/75_ %_ 2^ hl--i* 

32 ~' 25; "32 _ - 5; 32 - '' 5; 32 _ 1; 32 ~ 1M '> 32 ~ 1>u 5 

2.75 , „ c 3 ft 3.25 ft OK 3.5 o B 

^=1.V5 3T2 = 2; ^--2.2o.,-^- = 2.5, etc. 

2 
Thus, when the water in a boiler is maintained at a density of — , 1 vol 

ume of it is evaporated, and an equal volume, or 1, is blowed off. Hence 
1 + 1—1 =1 z=l ratio of volume evaporated to the volume bloiced off; and 

1 25 
when it is maintained at * , .25 volumes of it are evaporated, and 1 

blowed off. Hence 1 -^ .25 = 4 volumes blowed off. 

2 
Illustrations. — The point of blowing off is — , the pressure of the steam 15.3 

lbs. mercurial gauge, and the density of the feed water — . 

S = 1202°, T = 100°, * = temp. of 15.3 +14.7 = 251.6° - 100 = 151.6°, E = l. 

_. 1202 — 100x1 + 151.6 QO _ _. ,_' . , 

Then — ■ ■ = 8.21 = proportion of heat lost 

151 6° 
1202 — 100x1 + 151. 6°= 1253.6° = total heat required from the fuel; and ' 

o 12oo to 

= 12.093 per cent loss by blowing off at — at the temperatures given. 

Oil 

Note If the temperature of the feed water in this case had been 150°, the I033 

would have been but 8.S1 per cent. 

To Compute the Degree of Saturation to Contain x Parts of Saline Matter. 

The quantity of saline matter entering and the quantity blowed off in the 

same time, will be equal when 3,03 (s + b) =xb . hence — '■ = b, and 

3.03 5 ono tf-3.03 

— — — x — 3.03. 
o 

Illustration.— The volume of water used for steam in an engine is 1, and the 
volume blowed out 4; what is the degree of saturation ? 

Note.— As Saline Hydrometers are graduated to 3 parts of saline matter in 100, 
or 1 in 33, it is preferable to use 3 instead of 3.03 as above. 

Here x degrees of saturation = 1.25. 

Then i J* *., — o= z in = 4 '^ aQ d ^P = .75, or(1.25x3-3)=*. 
1.25 X o — 6 to 4 

To Compute the Volume of Water Blowed Off to that Evaporated. The Degree 
of Saturation being Given. 

Rule. — Divide 1 by the proportionate volume of water evaporated to 
that blowed off, or the value of E as above, for the degree of saturation 
given, and the quotient will give the number of volumes blowed off to that 
evaporated. 



STEAM-ENGINE. — HORSE POWER. 603 

2.25 

Illustration.— The degree of saturation in a marine boiler is -— - - ; what is the 

volume of water bio wed off? 

Value of E 1.25. Then — — = .8 blowed off. 
1.20 

To Compute the Economy attained by the Use of Fresh, Water in a Marine Boile? 
compared with the Use of Sea Water. 

966.6 -r- T — t = r, or ratio of increase of temperature to which feed water must be 
subjected. 

F — E=V, or volume to be blowed off. 

Y-±- r = v , or volume of water which could be converted into steam by the temper* 
ature lost by blowing off. 

Hence v X the pounds of fuel necessaiy to evaporate a cubic foot of water will give 
the amount of fuel economized. 

T and t representing temperature of fresh water due to the pressure of the steam, 
and the mean temperature of feed water, and F and E the volumes of feed-water and 
that evaporated. 

Note.— This economy is exclusive of the loss of heat consequent upon the incrust- 
ation upon the heating surfaces of a boiler when sea water is used. 

HORSE POWER. 

As this is the universal term used to express the capability of first 
movers, of magnitude, it is essential that the estimate of it should be 
uniform. 

Its estimate is the elevation of 33 000 pounds avoirdupois one foot 
in height in one minute, and it is designated as being Nominal, Indi- 
cated, or Actual. 

The first designation being adopted and referred to by Manufac- 
turers of steam-engines in order to express the capacity of an engine, 
the elements thereof being confined to the dimensions of the steam 
cylinder, and a conventional pressure of steam and sp'eed of piston ; 
the second to designate the full capacity of an engine, as developed in 
operation, without any deduction for friction ; and the last referring 
to its actual power as developed by its operation, involving the ele- 
ments of the mean pressure upon the piston, its velocity, and a just 
deduction for the friction of the operation of the machine. 

In reviewing the various modes for the computation as submitted by 
Engineers and Manufacturers, there is no proper formula that presents the 
essential element of being in conformity with an} T other, and as conformity 
in a rule for this purpose, if based upon an assimilation to the capacity 
of an engine, is all that is requisite, it would have been preferable to have 
adopted an existing formula to the introduction of a new one, had it been 
practicable to have done so. It occurs, further, that there is not only a 
want of conformity in the various rules essayed by authors, but they have 
neither reached the cases of both condensing and non-condensing engines, 
nor have thej'- properly approached to the actual power of an engine ; and 
as the practice of operating engines since the adoption of existing formulas 
has materially altered, both m an increase of pressure and velocity of 
piston, the following rules are submitted. 

Nominal Horse's Power. {Condensing Engine.) 

d?v 

^— as horse s power : d representing diam. of cylinder in inches, and v 

the velocity of the piston in feet per minute. 
This is alike to the rule of the British Admiralty, substituting 3000 foi 



604 



STEAM-ENGINE. — HORSE POWER. 



6000, and it is based upon a uniform steam pressure of 10 lbs. per square 
inch (steam gauge, or above the pressure of the atmosphere), cut off ai 
one half the stroke, deducting one fifth* for friction and losses, with a 
mean velocity of piston of 250 feet per minute for an engine of long 
stroke, and of 200 feet for one of short stroke. 

The rule of the British Admiralty is based upon a uniform and effective 
pressure of 7 lbs. per square inch at full stroke, and a mean velocity of 
piston of 205 feet per minute ; viz., 170 feet for a stroke of 2.5 feet, and 240 
feet for a stroke of 8 feet. 



d 2 v 
1000 



N on -condensing Engine. 



= horses' power. 



This is based upon a uniform steam pressure of GO lbs. per square inch 
(steam gauge), cut off at one half the stroke, deducting one sixth for 
friction and losses, with a mean velocity of piston of 250 feet per minute. 

jN"ominal Horse Power of several Non-condensing 
Engines. 



Computed from formula .j— 7r ft : 



HP. 



Horses' 
Power 


Diameter 
and Stroke 
of Cylinder 


Revo- 
lutions 


Horses' 
Power 


Diameter 
ani Stroke 
of Cylinder 


Revo- 
lutions. 


Horses' 
Power 


Diameter 
and Stroke 
of Cylinder 


Revo 

lutions. 


No. 


Ins 


Feet 


Min 


No 


Ins. Feet. 


Min. 


No. 


Ins Feet 


Mid 


9. 


6X1. 


125 


461 


12X4.5 


32 


159.7 


22X5.5 


30 


9.2 


6 


1.5 


85 


55.3 


14 3. 


47 


160.7 


22 6. 


2S 


12.2 


7 


1. 


125 


56.3 


14 3.5 


41 


163.6 


22 6.5 


26 


12.5 


7 


1.5 


S5 


58. 


14 4. 


37 


169.4 


22 7. 


25 


10. 3 


8 


1.5 


85 


60. 


14 45 


34 


183.7 


24 5.5 


29 


10.9 


8 


1.75 


75 


60. S 


14 5. 


30 


193.5 


24 6. 


2S 


21.1 


9 


1.5 


87 


64.8 


15 3. 


48 


194.7 


24 6.5 


26 


21 3 


9 


1.75* 


75 


66.1 


15 3.5 


42 


193.5 


24 7. 


24 


21.4 


9 


2. 


66 


66.6 


15 4. 


37 


198.7 


24 7.5 


23 


21.5 


9 


2.5 


53 


66. S 


15 4.5 


33 


227.1 


26 6. 


28 


26.1 


10 


1.5 


87 


67.5 


15 5. 


30 


22S.5 


26 6.5 


26 


26.6 


10 


175 


76 


77.1 


16 3.5 


43 


227.1 


26 7. 


24 


27.2 


10 


2. 


68 


77.8 


16 4. 


3S 


233.2 


26 7.5 


23 


27.5 


10 


2.5 


55 


78.3 


16 4.5 


34 


237.9 


26 8. 


22 


2S.2 


10 


3. 


47 


79.4 


16 5. 


31 


266. 


28 6.5 


26 


2S7 


10 


3.5 


41 


81.7 


16 5.5 


29 


274.4 


2S 7. 


£5 


28.8 


10 


4. 


36 


82.9 


16 6. 


27 


'270.5 


2S 7.5 


23 


33.9 


11 


2. 


70 


99.1 


18 4.5 


34 


275.8 


28 8. 


22 


33.3 


11 


25 


55 


103.7 


18 5. 


32 


279.9 


28 8.5 


21 


33.4 


11 


3. 


46 


103.4 


18 5.5 


29 


304.2 


30 6 5 


26 


33.9 


11 


3.5 


40 


105. 


IS 6. 


27 


315. 


30 7. 


25 


34.9 


11 


4. 


36 


12S. 


20 5. 


32 


324. 


30 7.5 


24 


39.2 


12 


2. 


68 


127.6 


20 5.5 


29 


331 2 


30 8. 


23 


39.6 


12 


2.5 


55 


129.6 


20 6. 


27 


336 6 


30 8.5 


22 


40.0 


12 


3. 


47 


130. 


20 6 5 


25 


340.2 


30 9. 


21 


41.3 


12 


3.5 


41 


134.4 


20 7. 


24 


359.1 


30 9 5 


21 


41.5 


12 


4. 


36 


154.9 


22 5. 


32 


860. 


30 10. 


20 



Indicated Horse Power. 

This is the gross power exerted by an engine, without an}- deduction 
for friction, the mean pressure upon the piston being determined by an 
Indicator, or by a computation based upon the actual initial pressure in 
the cylinder. 

* The frirtion nn>l losses in a marine engine may be taken at 1.5 to 2 lbs per square inch for work 
ir.£ the engine, and 5 to 1\u per cent upon the remainder for the friction of the load. 



STEAM-ENGINE. — HORSE POWER. 605 

Actual or Effective Power.- Condensing Engine. 



A X P* f\ X 2 s r 

qqTw) ' = Worse's power. A representing area of cylinder in 

square inches, P mean effective pressure upon cylinder piston in lbs. per 
square inch, inclusive of the atmosphere, f the friction of the engine in all its 
parts, added to the friction of the load in lbs. per square inch, s stroke of 
piston in feet, and r number of revolutions per minute. 

The Power required to work the air-pump of an engine varies from .7 to 
.9 lbs. per square inch upon the cylinder piston. 

Illustration. — The diameter of cylinder of a marine steam-engine is 60 ins., the 
stroke of its piston 10 feet, its revolutions 15 per minute, and the pressure of the 
steam per gauge, cut off at one fourth the stroke, is 20 lhs. per square inch. 

A = 2S27.4 sq. ins. P (per Ex., p. 580) 20.S55* lbs. /=1.5+ 20.855- 1.5 X .05 

= 2.46T Ws. Then »»T.4x^-^TX 8 X IPX tt = M8jg4 ^ ^ 

From which is to he deducted in Marine Engines the power necessary to discharge 
the water of condensation at the level of the load-line, which is determined by the 
pressure due to the elevation of the water, the area of the air-pump piston, and the 
velocity of its discharge in feet per second. 

Non-condensing Engine. 



AxP-(J^+14.7)x2sr , 

33W)0 = h ° rseS P ° Wer ' 

The sum of these resistances is from 12^ to 20 per cent, according- to 
the pressure of the steam, being least with the highest pressure. 

.Illustration. — The diameter of cylinder of a non-condensing engine is 10 ins., 
the stroke of the piston 4 feet, its revolutions 45 per minute, and the mean pressure 
of the steam in the cylinder .(per steam gauge) is 60 lbs. per square inch. 

A = 78. 54 sq. ins. P 60 -f 14. 7 = 74. 7 lbs. f—% 5 -f (60 + 14.7 — 2.5) X. 075 = 

n no /,. mu 79.54X (60+ 14.7 -7.92+14.7)X2x4x45 .... 

7 . 92 lbs. Then qTooo ~ horses. 

Note. — The power of a non-condensing engine is sensibly affected by the charac- 
ter of its exhaust, as to whether it is into a heater, or through a contracted pipe, to 
afford a blast to combustion. 

Note 2. If an Indicator is not used to determine the pressure of the steam in a 
cylinder, a safe estimate of it, when acting expansively, is .9 of the full pressure, 
and when at full stroke from .75 to .8. 

To Compute tlie Horses 1 3Po"we:r of* an Engine necessary 
to raise Water to any G-iven Height. 

Rule. — Multiply the weight of the column of water by its velocity in 
feet per minute, and divide the product by 33 000. 

Example.— It is required to raise a column of fresh water, 16 inches in diameter 
by 06 feet in height, with a velocity of 128 feet per minute ; what power is necessary ? 

* This value is best obtained by an Indicator : when one is not used, refer to rule and table, 
page 579. In estimating th$ value of P, add 14.7 lbs., for the atmospheric pressure, to that indi- 
cated by the steam gauge or safety valve. When, however, an Indicator is not used, a safe esti- 
mate is .85 that of the boiler pressure. 

f This value may be safely estimated in engines of magnitude at 1.5 to 2 lbs, per square inch, 
for the friction of the engine in all its parts, and the friction of the load may be taken at 5 to 7^ 
per cent, of the remaining pressure. 

The sum of these resistances in ordinary marine engines is from 10 to 20 per cent , according to 
the pressure of the steam, exclusive of the power required to deliver the water of condensation at 
the level of the load-line. For the pressure representing the friction for different designs and ca- 
pacities of engines as estimated by English authority, see page 347. 

t Clearance of piston at each end of cylinder is included in this estimate. 

o This value may be safely estimated at 2.5 lbs. per square inch for the friction of the engine in 
all its parts, and the friction of the load may be taken at 7 %, per cent, of the remaining pressure. 



606 STEAM-ENGINE. HOESE POWER. 

The height of a column of fresh water equal to a pressure of 1 lb. per sq. in. = 2.31. 
Then SO -H 2.31 = 37.23 lbs. 

Area of 1(3 ins. =201.06 ins., which X 37.23 = 7455.46, and 74S5.46 X 123^-33 000 
= 20.03 horses' 1 power. 
To which should be added an allowance of fully .2 for friction, leakages, waste, etc. 

To Compute tlie "Velocity necessary to -Di.sch.arge a GJ-iven 
Volume ofWater iix any Griven Time. 

Role. — Multiply the number of cubic feet by 144, and divide the prod- 
uct by the area of the pipe or opening in inches. 

Example. — The diameter of a pipe is 16 inches, and the volume of water 179 cubic 
feet per minute ; what is its velocity ? 
179 X l-'4 

Area of pipe 201.06 ins. * = 12S.2 feet. 

To Compute tlie Area of a reciuiired 3?ipe, tlie Velocity ancl 
Volume of tlie Water being given. 

Rule. — Proceed as above, and divide the product by the velocity. 

To Compute tlie Volume of Water required, to "be 
Evaporated in a Steam-engine. 

Rule. — Multiply the volume of steam expended in the cylinder and 
steam chests by twice the number of revolutions, and multiply the product 
by the density of the steam at the pressure given. 

Example.— What quantity of water will an engine require to he evaporated per 
revolution, the diameter of the cylinder being 70 ins., the stroke of the piston 10 feet, 
and the pressure of steam 32 lbs. per square inch, including the atmosphere, cut off 
at one half of the stroke ? 

Area of cylinder=3S4S.4ms. 10 X 12-^2=60 ins., then 60 XSS4S. 4=230904 cub. ins. 

Add, for clearance at one end and volume of nozzle, steam chest, etc., 1731S cub. ins. 

Then 230904-}- 1731S-Hl72Sx2 = 2S7. 29 cub feet, which X .0012, the density of 
steam at 32 lbs. pressure (p. 574) =.3447 cubic feet. 

Note.— This refers to the expenditure of steam alone; in practice, however, a 
large quantity of water (differing in different cases) is carried into the cylinder in 
mechanical combination with the steam. 

To Compute the Area of* an Injection JPipe. 

Rcel. — Ascertain the volume of water required by the rule, p. 576. in 
cubic inches per second, multiply it by the number" of volumes of water 
required for condensation, by rule, p. o77, and divide it by the velocity 
due to the flow in feet per second, and again, b} r 12, and the quotient will 
give the area in square inches. 

Example. — An engine has the following elements at its maximum operation; 
what should be the area of its injection pipe? 

Cylinder, 70 ins. diameter and 10 feet stroke of piston ; revolutions, 15 per minute ; 
steam, 17. 8 lbs., mercurial gauge, cut off one half. 

Volume of cylinder 267.25 cubic feet, cut off at }£ = 133.625. 

Density of steam at 32 lbs. (17.3-}- 14.7) = .0012. Velocity of flow of injected 
water (computed from vacuum and elevation of condensing water) 33 feet per second. 

Then 133.625 X l- r > X 2 x 1723 -=-60 = 115452 cubic inches steam per second, and 
115 452 x. 0012 = 138.54 cubic inches water per second. 

Tlie maximum volume of water required to condense steam is about 70 times the 
volume of that evaporated, which only occurs in the Gulf of Mexico ; the ordinary 
requirement is not one half of it. 

Hence 138.54 + 10.39 ins. f=T# per cent, for leakage of valves, etc.) = 148.93, 
which X TO, a* above, =10 425.1 cubic ins., which -r- 33 = 315.91, and again by 12 
tins, in a foot) =26.32 square ms. area, which ^-.7, the coefficient for velocity of 
flow of water in a pipe under like conditions, = 37.6 square ins. 

Note. — This is the required capacity for one pipe for navigation in the Gulf of 
>. It i.-i proper and customary that there should be two pipes % to meet the 
Ingeney of the operation of one being arrested. 



BLOWING ENGINES. 607 

To Compute tlie Volume of* Discharge tlirougli a^L 
Injection. 3?ipe. 

Rule. — Multiply the square root of the product of 64.333 and the depth 
of the centre of the opening into the condenser, from the surface of the 
external water in feet, added to the height in feet of a column of water due 
to the vacuum in the condenser, by the area of the opening in square 
inches ; and .7 the product divided by 2.4 (144 -^ 60) will give the volume 
in cubic feet per minute. 

Example. — The diameter of an injection pipe is 5% ine., the height of the external 
water above the condenser 6.13 feet, and the vacuum 24.45 ins., mercurial gauge; 
what is the volume of the flow per minute ? 

24 45 in"* 

Area of 5% ins. = 22.69 ins. Vacuum ' "' = 12 lbs., and 12 lbs. X 2.231) 

feet (sea water) =26. 87 feet, and 26.8T + 6.13 = UfeeL 

_. V64.333X 33X22 . 69X. 7 T 31.82 on OQ ,. . 
Then — = — — — = 304.93 cubic feet. 

To Compute tl\e Area of* a Feed Pump. For Sea Water. 

Rulk. — Ascertain the volume of water required in cubic inches per min- 
ute ; divide it by the number of single strokes of the piston of the pump 
per minute, and divide the quotient by the stroke of the piston in inches ; 
multipl} T this quotient by 6 (for waste, leaks, ** running up," etc.), and 
the product will give the area of the pump in square inches. 

Example. — Take the elements of the preceding cases. 

13S.54 cubic inches per seconds 8312.4 per minute, stroke of pump o% feet. 

Then ??^i — 554.16, which -7-3.5x12 = 13.19, and 13.19x6 = 79.14 square ins 
15 

Note In fresh water, this proportion may be reduced two thirds. 



BLOWING ENGINES. 

For Smelting. 

The volume of oxygen in air is different at different temperatures. Thus, 
dry air at 85° contains 10 per cent, less oxygen than when it is at the 
temperature of 32° ; and when it is saturated with vapor, it contains 12 per 
cent. less. If an average supply of 1500 cubic feet per minute is required 
in winter, 1650 feet will be required in summer. 

Manufacture of Pig Iron. 

Coke or Anthracite Coal. — 18 to 20 tons of air are required for each ton. 

Charcoal. — 17 to 18 tons of air are required for each ton. 

(1 ton of air at 34°= 29 751, and at 60°= 31 366 cubic feet.) 

Pressure. — The pressure ordinarily required for smelting purposes is 
equal to a column of mercury from 3 to 7 inches. 

Reservoir. — The capacit}' of it, if dn% should be 15 times that of the 
cylinder if single acting, and 10 times if double acting. 

Pipes. — Their area, leading to the reservoir, should be .2 that of the 
blast cylinder, and the velocit} r of the air should not exceed 35 feet per 
second. 

A smith's forge requires 150 cubic feet of air per minute. Pressure of 
blast % to 2 lbs. per square inch. A ton of iron melted per hour in a cu- 
pola requires 3500 cubic feet of air per minute. A finery forge requires 
100 000 cubic feet of air for each ton of iron refined. A blast furnace re- 
quires 20 cubic feet per minute for each cubic yard capacity of furnace. 






608 BLOWING ENGINES. 

To Comptite the Power of a Blowing Engine. 

Pa vf= power in lbs. to be raised 1 foot per minute, and „' =horses' 

power required. P representing pressure of blast in lbs. per square inch ; 
a area of cylinder in square inches; v velocity of piston in feet per minute; 
f friction of piston and from curvatures, etc., estimated at 1.25 per square 
inch of piston. 
Note.— If the cylinder is single acting, divide the result by 2. 

To Compute tlie Dimensions of a Driving Engine 

Rule. — Divide the power in pounds by the product of the mean effective 
pressure upon the piston of the steam cylinder in pounds per square'inch, 
and the velocity of the piston in feet per minute, and the quotient will give 
the area of the cylinder in square inches. 

2. — Divide the velocity of the piston by twice the number of revolutions, 
and the quotient will give the stroke of the piston in feet. 

The quantity of air at atmospheric density delivered into the reservoir, in conse- 
quence of escape through the valves, and the partial vacuum necessary to produce 
a current, will be about .2 less than the capacity of the cylinder. 

See p. 620 for dimensions of Furnace, Engines, etc. 

To Compute the Volume of* .A^ir transmitted, "by an En- 
gine, When the Pressure, Temperature, etc., are given. 

34. 5 \J h I ,' — J C = v. h representing pressure of blast in inches 

of mercury ; t temperature of blast ; H height of barometer in inches ; and 
v velocity in feet per second. 

Then a v x 60 = V in cubic feet per minute. 

Illustration. — A furnace having 2 tuyeres of 5 inches diameter, the pressure and 
temperature of the blast 3 inches and 350°, and barometer 30 inches ; what is the 
volume of air transmitted per minute? 

Coefficient for a conical opening .94. 

34 ' 5 ^ 3 ( 1± iT§^ X - M = 845 \/ 8 (w) = 34 -5X.46Tx.94^15.14/, f f 

velocity per second. 

' Then, area 5 ins.= 19.635, which X 2 = 39.27 ins., and 39.27x15.14x60 -=-144 = 
247.73 cubic feet. 

FAN BLOWERS. 

Proportions of Parts.— Blades. Their width and length should be at 
least equal to £ or ^ the radius of the fan. 

Openings. The inlet should be equal to the radius of the fan ; and the 
outlet, or discharge, should be in depth not less than % the diameter, its 
width being equal to the width of the fan. 

An increase in the number of blades renders the operation of the fan smoother, 
but does not increase its capacity. 

"When the pressure of a blast exceeds .7 inch of mercury per square inch, .2 
will be a better proportion for the width and length of the fan than that above given. 

The pressure or density of a blast is usually measured in inches of mer- 
cury, a pressure of 1 lb. per square inch at 60°=2.0376 inches. 

When water is used as the element of measure, a pressure of 1 lb.= 27.671 inches. 

The eccentricity of a fan should be .1 of its diameter. 

By the experiments of Mr. Buckle, he deduced 

1. That the velocity of the periphery of the blades should be .9 that of 
their theoretical velocity ; that is, the velocity a body would acquire in 



BLOWING ENGINES. 609 

falling the height of a homogeneous column of air equivalent to the re- 
quired density. 

2. That a diminution of the inlet from the proportions here given in- 
volved a greater expenditure of power to produce the same density. 

3„ That the greater the depth of the blade, the greater the density of 
air produced with the same number of revolutions. 

The operation of a blower requires about 2.5 per cent, of the power of the attached 
boiler. 

To Compute tlie Density of a Blast. 

( o~ao )' **■ 939.454 = d in inches of mercury, v representing velocity of 
periphery of fan in feet per second. 
Illustration. — The velocity of a blast is 123 feet per second. 
123 ~ S.02 2 -f- 939.454 =a . 25 inch. 

To Compute tlie "Velocity of* a Blast. 



V 939.454 x 64.333 = v in feet per second. 
Illustration. — The required density of the air is 1 inch. 

V939.454X 64.333 = 245 . 8 feet. 

Hence the velocity of the periphery of the fans should be .9x245.8 = 221.2 feet. 

To Compute tlie "Volnxne of^ir dlsdiarged per Minute. 
axvx60 

160" 
Illustration. — The area of the discharge is 40 inches, and the velocity 123 feet 

40X123X60 



=V in cubic feet, a representing area of discharge in square ins. 
per second, 



^1S45 cubic feet. 

To Compute tlie Horses' Power. 
_ - = horses' power, or a P .000014 v 7 . P representing the pressure of 

blast in lbs. per square inch. 

Illustration. — The velocity of air discharged is 123 feet per second, the area of 
the opening 40 square inches, and the density .25 inch of mercury. 

=3.07 horses, independent of the friction of the blast in the pas- 
sages and tuyeres. 

A Ton of pig iron requires for its reduction from the ore 310 000 cubic 
feet of air, or 5.3 cubic feet of air for each pound of carbon consumed. 
Pressure, .7 lb. per square inch. 

An ordinary Eccentric Fan, 4 feet in diameter, with 5 blades 10 inches 
wide and 14 inches in length, set 1% ins. eccentric, with an inlet opening 
of 17.5 inches in diameter, and an outlet of 12 inches square, making 870 
revolutions per minute, will supply air to 40 tuyeres, each of 1% inches 
in diameter, and at a pressure per square inch of, .5 inch of mercury. 

An ordinary eccentric fan blower, 50 inches in diameter, running at 1000 revolu- 
tions per minute, will give a pressure of 15 inches of water, and require for its oper- 
ation a power of 12 horses. Area of tuyere discharge 500 square inches. 

A non-condensing engine, diameter of cylinder 8 ins., stroke of piston 1 foot, press- 
ure of steam 18 lbs. (mercurial gauge), and making 100 revolutions per minute, will 
drive a fan, 4 feet by 2, opening 2 feet by 2, 500 revolutions per minute. 

Such a blower was applied as an exhausting draught to the smoke-pipe of the 
Steamer Keystone State, cylinder SO ins. by S feet, and the evaporation was doubled 
over that of when the wind was calm. 

3F 



610 



BELTS. 
BELTS. 



, 



The resistance of belts to slipping is independent of their breadth, con- 
sequently there is no advantage derived in increasing this dimension be- 
yond that which is necessary to enable the belt to resist the strain it is 
subjected to. 

The ratio of friction to pressure for belts over wood drums, is for leather 
belts, when worn, .47; when new, .5; and when over turned cast-iron 
pulleys, .24 and .27. 

A leather belt will safely and continuously resist a strain of 350 lbs. per 
square inch of section, and a section of .2 of a square inch will transmit 
the equivalent of a horse's power at a velocity of 1000 feet per minute 
over a wooden drum, and .4 of a square inch over a turned cast-iron pulley. 

A vulcanized India-rubber belt will sustain a greater stress than leather, 
added to which its resistance to slipping is from 50 to 85 per cent, greater. 

In high speed belting, the tension, or the breadth of the belt should be 
increased, in order to prevent the belt from slipping. Long belts are more 
effective than short ones. 

To Compute tlie Stress a Belt or Cord, is capa"ble oftrans- 
mitting.— Aide Memoire. 

Rule. — Multiply the value of C from the following table by the stress in pounds. 



Proportion of Arc 




Value of Coefficient C. 




embraced to the Cir 








cumference of the 


Leather Belts. 


Cords on Wooden Sheaves. 


Driving Pulley. 


On Wood Drums, 


On Iron Pulleys. 


Rough. 


Polished. 


.2 


1.8 


1.4 


1.9 


1.5 


.3 


2.4 


l.T 


2.6 


1.9 


.4 


3.3 


2. 


3.5 


2.3 


.5 


4.4 


2.4 


.4.8 


2.8 


.6 


5.9 


2.9 


6.6 


3.5 


.T 


7.9 


3.4 


9. 


4.2 



C = the ratio of the resistance of a drum or pulley to slipping a belt or cord when 
the resistance of the belt or cord upon the under or slack side is known. 

Example. — What is the stress a belt is capable of transmitting when the arc em- 
braced upon the surface of the driving and wooden drum is .4 of its circumference, 
and the power or tension of the belt is 200 lbs. ? 

3.3 X 200 = 660 lbs. 

To Compute tlie Stress wliicli is transmitted to a Belt 

or Cord.. 

Rule.— Divide the power in pounds transmitted to the periphery of the pulley by 
the velocity of the surface of the drum. 

Example.— A cast-iron pulley, 4 feet in diameter, driven by a power of 4 horses, 
makes 160 revolutions per minute ; what is tke stress upon the belt ? 

33 000 X 4 = 132 000 lbs. 1 foot per minute. 

4 X 3.1416 X 100= 1256.64 feet velocity. 

Then „, = 105 lbs. = difference of the stress upon the belt and the resistance 



1256.64 
of the under side of it, 



S 



== S, and S-j-s = P. P representing the stress trans- 
it — 1 

mitted by a belt, s the resistance of its under side, and P the sum ofS-\-s } or the 
stress and resistance. 

Illustration-.— What should be the resistance of the under side of a leather belt 
running over the semi-circumference of a cast-iron pulley, 1 foot in diameter, driven 
by a power of 200 lbs. ? 



CONSTRUCTION OF VESSELS. 611 

To Compute tlie "Width of* a Leather Belt. 

Illustration.— An engine of 4 horses' power is to be transmitted through a leath- 
er belt over a cast-iron pulley, embracing .4 its circumference, 4 feet in diameter, 
and making 100 revolutions per minute ; what should be the width of the belt ? 

Power, as per preceding example, 132 000 lbs. 

Velocity " M 1256.64 " 

g « « 105 " andC = 2. 

Then -5-=^^- = 105, and S + s = P = 105 + 105 = 210 lbs. 
C — 1 2 — 1 

The resistance or tensile strength of a leather belt is from 2T0 to 330 lbs. per square 
inch ; and, assuming the thickness of it to be .15 of an inch, then 300 X .15 = 45 lbs. 
Hence 210 -f- 45 = 4.67 inches. 

Illustration.— A belt, 11 inches in width and .22 inch thick, over a drum 4 feet 
in diameter, C = .5, making 60 revolutions per minute, is sufficient to transmit the 
power from an engine working at 990 000 lbs. per minute. 

Then 890 000 = g^ = 1313 .3 l b s., and ^ = 386.17, which X 2 = 
4X3.1416X60 753.98 ' 4.4 — 1 1 

772 35 
772. 35 lbs. Hence, 300 X . 22 = 66, and — ^— = 11.7 ins. 

India !Rn."b"ber Belting.— (Vulcanized.) 

Results of Experiments upon the Adhesion of India Rubber and Leather Belting. 
—(J. H. ChebvekJ 

Rubber.* Leather. 

Lbs. Lbs. 

Rubber belt slipped on iron pulley at 90 Leather belt slipped on iron pulley at 48 

« i* leather " 128 u " leather " 64 

" " rubber " 1S3 « « rubber " 128 

Hence it appears that a Rubber Belt for equal resistances with a Leather Belt may 

be reduced, under the circumstances here given, respectively 46, 50, and 30 per cent. 

from the results to be obtained by the foregoing Rule. 

Memoranda. 

Two leather belts, 15 ins. in width, over a driver 6 feet in diameter, running with 
a velocity of 2128 feet per minute, transmit the power from the water-wheel at Rocky 
Glen Factory, the dimensions of which are given on page 441. 

A belt, 11 ins. in width, over a driver 4 feet in diameter, running from 1200 to 
2100 feet per minute, will transmit the power from two steam cylinders, 6 ins. in 
diameter and 11 ins. stroke, averaging 125 revolutions per minute, with a pressure 
of 60 lbs. per square inch. 

The computations here given are based upon the actual horses 1 power. 



. , 

CONSTRUCTION OF VESSELS. 

^Results of Experiments upon tlie Form of "Vessels. 
(Wm. Bland.) 

Cubical Models. 
Head resistance. — Increases directly with the area of its surface. 
Resistance to Weight. — Increases directly as the weight. 

Vessel's Models. 

Lateral Resistance. — About one twelfth of the length of the body im- 
mersed, varying with the speed. 
The centre of lateral resistance moved forward as the model progressed. 
The centre of gravity had no influence upon the centre of lateral resistance. 

* Manufacture of the New York Belting and Packing Co , Park Row, N. Y. 



612 



CONSTRUCTION OF VESSELS. 



Length. — Increased length gave increased speed or less resistance. 

Amidship Section. — Curved sections gave higher speed than angled. 

Sides. — Curved sides with one fourth more beam gave equal speeds with 
straight sides of less beam. Keels. — Length of keel has greater effect than 
depth. Stem. — Parallel-sided after body gave greater speed than a ta- 
pered sided. 



Relative Speeds, 
Form of Bow. 



Order of Speed. 



Isosceles triangle, sides slightly convexed 

u u u right lines 

a " " slightly concave at entrance and running 
out convex 



Spherical equilateral triangle, compared to equilateral triangle, speed was as 11 to 
12. Equilateral triangle, with its isosceles sides bevelled off at an angle of 45°, 
compared to bow with vertical sides, was as 5 to 4. When bow had an angle of 14° 
with plane of keel, compared with one of 7°, its speed was greater. 

Relative Resistance to Lee Way. 



Form of Amidship Section. 


Order of 
Resistance. 


Form of Amidship Section. 


Order of 
Resistance. 


Rectangular section. . . . 
Semicircular u 


1 

2 


Elliptic section 

Triangular section 


3 

4 



No. 1 resisted best with a depth of keel of % an inch ; No. 2, best with varying 
depths of keel and without any ; No. 3 resisted best without any keel, and No. 4 had 
the least resistance without any keel. 

Bodies Inclined Upward from Admidslup Section. 

1. Model, with bow inclined from g$, had less resistance than model without any 
inclination. 

2. Model with a stern inclined from g$, had less resistance than model without 
any inclination. 

Model 1 had less resistance than model 2. Model with both bow and stern inclined 
from g$, had less resistance than either 1 or 2. 



IMMERSED SURFACE OF VESSELS. 

To Compute tlie External, or Bottom and Side Surf&oe 
of tlie Hull of a "Vessel. 

Bottom and Side. Rule. — Multiply the length of the curve of the mid- 
ship section, taken from the top of the tonnage or main deck beams upon 
one side to the same point upon the other (omitting the width of the keel) 
by the mean of the lengths of the keel and between the perpendiculars in 
feet, and multiply the product by .85 or .9 (according to the capacity of 
the vessel), and the product will give the surface required in square feet. 

Example. — The lengths of a steamer are as follows : length of keel 201 feet, and 
between the perpendiculars 210 feet, length of the curved surface of the midship sec* 
tion 76 feet; what is the surface ? 

Coefficient = .S7. 

210 + 201-4-2 = 205.5, and 7Gx205.5x.S7 = 135ST square feet. 

Note. — The exact surface is 13 650 square feet. 

Bottom Surface. Rule. — Multiply the length of hull at the load line 
by its breadth, and this product by the depth of the immersion (omitting 
the depth of tlie keel) in feet ; and this product multiplied by from .07 
to .08 (according to the capacity of the vessel) will give the surface re- 
quired in square feet. 



CONSTRUCTION OF VESSELS. 613 

Example The length upon the load line of a vessel is 310 feet, the beam 40 feet, 

the depth of the keel 1 foot, and the draught of water 20 feet; what is the bottom 
or wet surface ? 

Coefficient = .073, 



810X40X20 — IX. 073 = 17 199 square feet. 

DISPLACEMENT OF VESSELS. 

To Compute the Displacement of a "Vessel. 

Rule.— -Multiplply the length of the vessel at the load line hy the 
breadth, and the product by the depth (from the load line to the underside 
of the. garboard strake) in feet, and this product by the unit of volume or 
coefficient of the mass, and divide by 35 for salt water and 36 for fresh 
water ; the quotient will give the displacement in tons. 

Note. — This rule is deduced by Mr. S. M. Pook, Naval Constructor, U. S. N., and 
he gives the ranges of the units of volume as follows : 

Amidship sections range from .7 to-.9 of their circumscribing square and the mean 
of the horizontal lines from .55 to .75 of their respective parallelograms. Hence 
the ranges for vessels of the least capacity to the greatest are .7x.55 = .3S5, and 
.9X.75 = .675. 

Coefficients or Decimals of Proportionate Volume of 
"Vessels. 



Merchant Ship, very full 6 to .7 

" " medium..... .58 to .62 

River Steamer, stern wheel.. . .6 to .65 

Ship of the Line .5 to .6 

Naval Steamer, 1st class 5 to .6 

52 to .58 

Merchant Steamer, sharp 54 to .5S 

Half Clipper 52 to .56 

Brigs, Barks, etc 52 to .56 

River Steamer, tug boat, m'd'm .52 to .56 



Merchant Steamer, medium . . .52 to .54 

Clipper ........ 5 to .54 

Schooner, medium 48 to .52 

River Steamer, tugboat, sharp .45 to .5 

River Steamer, medium 45 to .5 

" " sharp 42 to .45 

Schooner, sharp 46 to .5 

Yachts, sharp 4 to .45 

" very sharp 36 to .4 

River Steamers, very sharp . . .36 to .42 



To Compute Power required, in a Steam Vessel, tlie 






Capacity of another "being given. 



In Vessels of similar Models.* — =V: ="V"; — =C; and — =R. 

J a ' $3 ' r ' ^ 2 r 

v representing product of volume of given cylinder and revolutions in cubic feet; 
a and A areas of immersed section of given and required vessel at like revolu- 
tions, and speed of given vessel in cubic feet; s and S speeds of aiven and 
required vessel; V volume of cylinder of required vessel at revolutions of 
given vessel in cubic feet ; r and r' revolutions of given and required vessel; 
and C product of volume of combined cylinders and revolutions for required 



Illustration.— A steam vessel having an area of amidship section of 675 square 
feet, has 2 cylinders of a combined capacity of 533.33 cubic feet, and runs 10^ knots 
per hour, with 15 revolutions of her engines. Required the volume of steam cylin- 
ders, with a stroke of 10 feet, for a section of 700 feet, and a speed of 13 knots at 14^ 
revolutions. 

t> = 533.33 X 15 = 8000 cubic feet. 80Q0 _ X 7 °° ' = 8296.3 cubic feet. 133 * 82963 

675 10. o« 

15 X 15 745 2 
z= cubic feet, and — — — = 10288.1 cubic feet. 

* As a means of comparison, if required, and of models, the elements here given are based upon 
the performances of the Steamers Arctic and Powhatan, for the capacities of which, see pp. 638 and 
639, and the area of cylinders deduced are such as the Arctic would have required at the iiuuier. 
eion given and speed required. 

3F* 



614 



CONSTRUCTION OF VESSELS. 



16 288 1 
Then _ , J =1123.3, which -r- 2 for the number of cylinders 



14.5 



561 G5, which 
56.165, which -f- 12 for the inches in a foot, and 



again divided by 10 for the stroke 

X 1728 for the inches in a cubic foot = 8087.76 square inches = area of the cylinder, 

or a diameter of cylinder of 101.5 — inches. 

Results of Experiments upon the Resistance of Screw Propellers, at high Velocities 
and immersed at varying Depths of Water. 

Immersion of 
Screw. 



Immersion of 1 Woe .; ofoTW ,„ 1 
Screw. J Resistance. | 


Immersion of 
Screw. 


Resistance. 


Surface. 1. 
1 foot. 1 5. 


2 feet. 

3 " 


7.5 



4 feet. 
5 " 



7.8 
8. 



Thrust of a Screw Propeller. 

Indicated horse power 1000. Resistance of hull or thrust of propeller, 
20 000 lbs. 

; 
Approximate Rules to Compute Speed and Indicated 
Horses' Power of* Steam-boats and Steamers. 



VP TH P V 3 A 

— -r — = V. p 3= IHP. C representing coefficient of vessel, A 

area of immersed amidship section, and V velocity of vessel in knots per hour. 
Note. — When there exists rig, or any element that affects the unit or coefficient 
for the class of vessel given, a corresponding addition to, or decrease of the follow- 
ing units is to be made. 

The Range of Coefficients, as deduced from Observation, are as follows : 



Side Wheel. 


Section. 


Speed. 


C. 


Screw Propeller. 


Section. 


Speed. 


C. 


River Steam-boat, 


Sq. Feet. 


Miles. 




River Steam-boat, 


Sq. F«et. 


Miles. 




Medium lines 

41 It 

Fine " 


43 
150 

200 


12 
15 
20 


470 
560 
600 


Medium lines 

Fine " 

Very fine lines . . . 
t< f« it 


45 

150 
27 
27 


12 
15 
15.3 
20 


500 
530 
600 
500 


Sea Steamer, 
Medium full lines* 


675 


Knots. 
10 


650 


Sea Steamer, 
Medium full lines* 
" fine " * 


550 

400 


Knots. 
9 
12 


570 

500 


u it u t 


880 


15 


650 


Fine lines 


1200 


15 


625 



* Ship-rigged. 



t Barque-rigged. 



Coefficients as determined by several Steamers of H. B. M. Service. 
(C. Mackrow, M. I. N. A.) 



Length. 


Length 

-=r- Beam. 


Area of 
Section at 


Displace- 
ment. 


IHP. 


Speed. 


IHP 


Feet. 




Sq. Feet. 


Tons. 




Knots. 




185 


6.53 


23tv 


775 


782 


10.34 


333 : 


212 


5.89 


377 


1554 


1070 


10.89 


456 


360 


7.33 


814 


5898 


2084 


11.5 


598 


270 


6.43 


632 


3057 


2046 


12.3 


574 


380 


6.52 


1308 


9487 


3205 


12.05 


714 


400 


6.73 


1198 


9152 


5971 


13.8S 


536 


362 


7.33 


778 


5600 


3945 


14.06 


548 


400 


6.73 


1185 


9071 


6867 


15.43 


634 



CONSTEUCTION OF VESSELS. 615 

Helative Capacities of Screw Propellers and. Side 

"Wheels. 
H. B. M. Screw-propeller Frigate Arrogant and the U. S. Mail Side- wheel Steamer 
Pacific had very similar dimensions, like draughts of water, and area of immersed 
amidship sections. 
The Pacific was 75 feet longer, and consequently had easier lines. 

Results of Performances. 

Arrogant. Pacific. 

Indicated Horses' power. . 729 1964 

Mean speed in knots per hour 8. 35 12 

KM 

S.353 = 5S2, and 123 - 1T2S. Therefore, 1964 : 172S : : 729 : 663, and ~ = .877. 

Hence, the application of power in the Pacific exceeded that in the Arrogant as 
1 to .S77. 

Peninsular and Oriental Screw-propeller Steamer Himalaya and Side- wheel Steam- 
er Atrato had similar dimensions of immersed section, displacement, and draught 
of water, and ran at like speeds. 

The Himalaya was 22 feet longer, and consequently had easier lines. 

Results of Performances. 

Himalaya. Atrato. 

Indicated Horses' Power 2050 3070 

Mean speed in knots per hour 13.78 13.97 

13.783 = 2016, and 13.973 = 2726. 
2616 
3070: 2726: :2050: 1820, and 7^ = 1.437. Hence, the application of power in 

the Himalaya exceeded that in the Atrato as 1.437 to 1. 

speed in knots 3 XlMS „ . ■ . . _.. 

Again : TJ = coefficient of immersed amidship section =± 716, 

inr 

494, and — - = 1.44, as before obtained. 
494 
Slip of Propeller, 15 per cent. ; of Side-wheel feathering blades, and taking axes 
of blades as the centre of pressure, 23 per cent. 

Resistance of Wind to Steam Vessels. — The resistance of air to a square 
foot of surface at right angles to the course of a vessel is about .33 lbs., 
and when the surface is inclined to the direction of the wind, the pressure 
varies as the sine of the angle of incidence. 

The mean of the angles of the surface of a steamer exposed to the wind 
may be taken at 45° ; hence their resistance is about .25 lbs. per square 
foot when the wind has a velocity of 10 knots per hour. 

Assuming the sectional area of a steamer's hull above water to be 750 
square feet, the resistance to the air at a speed of 10 knots in a dead calm 
would be 750 x .25 = 187.5 lbs., and the resistance to the smoke-pipe, spars, 
and rigging (brig rigged) would be 201 lbs. 

Location of Masts. - \ ulinmtli I \ '. '. '.'. 'A of t f he ^ n ^ of * he f load 
I Mizenmast 9 ) water llne from the stem « 

Balance of Sails. — The effect of the jib is equal to that of all the sails 
upon the mainmast, and the sails upon the mizenmast balance those of 
the foremast. 

Lee-way. — A full modeled vessel, with an immersed section of 1 to 6 of 
her longitudinal section, and with an area of 36 square feet of sails to 1 
of immersed section, will drift to leeward 1 mile in 6. A medium mod- 
eled vessel, with an immersed section of 1 to 8, and with like areas of sail 
and section, will drift 1 in 9. 

Motive Power. — A sailing vessel having a length 6 times that of her 
breadth, requires for a speed of 10 knots per hour, an impelling force of 
48 lbs. per square foot of immersed section. 



616 



CONSTRUCTION OP VESSELS. 



Proportion of Power "Utilized, in. a Steam "Vessel. 
P — z 
Side Wheel. . 000002 59 x ^3 x r 2 = a P representing gross Indicated Horses' 

Power, z loss of effect by slip and oblique action of wheel (or propeller), d diameter 
of wheels at centre of effect, r revolutions per minute, and C coefficient for the 
vessel. 

Illustration. — The indicated horses' power of the engines of a side-wheel steamer 
is 11'20 ; the slip of the wheels and loss by oblique action, 33.37 per cent. ; the diame- 
ter of the centre of effect of the wheels is 29.5 feet, and the number of revolutions 13.5 
per minute ; what is the coefficient, and what the power applied to propel the vessel ? 

Note. — The slip of the wheels from their centre of effect in this case is 15.3T per 
cent., and the loss by oblique action IS per cent. Hence, representing the total 
power by 100, 100 _ (IS + 15.37) = 66.63 per cent, of the power applied to the wheels. 

As the assumed power that operates upon the wheels in this case is taken at 86.12 
per cent, of the power exerted by the engines, 86.12 X 33.37 = 2S. 74 per cent, for the 
sum of loss bv the wheels. 

1120 — (1120x28. 74 -=-100) 7rs.ll _" co _ . . 

— ' = = 65.63 coefficient. 

.0000025. X 29.53X13.52 12.16 M 

The speed of the vessel beimr 10 knots per hour=17.05 feet per second, the power 
applied to propel the vessel at this speed = 65.63 X 17.052— 19 076.13, and the horses' 

, 19 076. 13 X 17.05 X 60 Kni oc _ 
power exerted = ^^ = 591.36 horses' power. 



33000 

Fric tion of engines 1.5 lbs. upon 384S sq. ins. X 13.5 revolutions 

Xl0x2-r- 33 000X2 = 
Friction of load 6 per cent, upon pressure of steam, less the 2 lbs. 

for friction of engine, as above — 
Oblique action of wheels = 
Slip of wheels = 
Absorbed by propulsion of vessel 



Horses' 

Power. 

94.45| 

60.45) 
201.6 
172.14 
591.36 



Per Cent, 
of Power. 

13.88 

18. 

15.37 

52.8 






1120. 



100. 



Screw Propeller. 



Friction of engines == 
" of load = 

" of screw surface and resistance of 
edges of blades = 
Slip of screw = 
Absorbed by propulsion of vessel 



Auxiliary Propeller. 



Horses' 
Power. 

96.06 
81.48 

53.44 
205.55 
375.92 



Per Cent, 
of Power. 



} 



782.45 
Area of Sails. 



18.83 

6.83 

26.27 
48.04 

100. 



17.9 



12. 

13.7 

56.4 

100. 






Sails. 


3 Yards upon 
each Mast. 


4 Yards upon 
each Mast. 


Sails. 


3 Yards upon 
each Must. 


4 Yards upon 
each Mast. 


Jib 


.08 

.295 

.417 


.08 

.295 

.417 


Mizeniuast. . . . 
Spanker 


.127 
.0S1 


.14 


Foremast 

Mainmast 


.068 



Proportional Area of Sails upon each Mast under above Divisions. 



Sail. 


Fore. 


Main. 


Mizen. 


Proportion to 1. 


Course 


.115 
.105 
.075 

.03 


.097 
.09 
.003 
.045 

.08 


.102 
.149 
.106 


.13S 
.121 

.089 
.063 


.075 
.052 

.081 


.063 
.045 
.032 
.068 


.3S9 
.358 
.253 


.33 


Topsail 

Topgallant sail 

Royal 

Spanker (Driver) 

Jib 


.303 
.215 
.152 




.295 


.2!; 5 


.417 


.417 


.288 


.288 


1. 


1. 






CONSTRUCTION OF VESSELS. 61? 

The areas of the sails upon the masts of a ship should be in the follow 
ing proportion : 

Fore 1.414. Main 2. Mizen 1. 

When, therefore, the main yard has a breadth of sail of 100 feet, the 
fore yard should have 70.71 feet, and the mizen 60 feet ; the topgallant 
and royal yards and sails being in the same proportion. 

Assuming the centre of lateral resistance to be in the middle of the 
length of a vessel, which is the case when she is upon an even keel, the 
fore and after moments of sails, the masts and sails being set and pro- 
portioned in accordance with the preceding rules, will be alike. 

Resistance of Bottoms of Hulls at a Speed of. One Knot per Hour. 

Smooth wood or paint 01 lb. I Copper 007 lb. 

Iron bottom, painted 014 " | Grass and small barnacles. .06 "- 

To Compute tlie Resistance to tlie "Wet Surface of* tlie 
Hull of a "Vessel. 

C a v 2 = R. C representing a coefficient of resistance, a area of wet sur~ 
face in square feet, and v velocity of hull in feet per second. 
Values (.007, clean copper. I .014, iron plate, 
of C, (.01, smooth paint. | . 019, iron plate, moderately foul. 
The power required to propel 1 square foot of immersed amidship section at gj is 
.073 that of smooth wet surface. 

Estimate of Volumes and Weights for a Cargo and Passenger Steamer 
for a Route of 1500 Miles. 

Volume. Weight. 

Cubic feet. Tons. 

Cargo 25 000 500 

Passengers, 1st Cabin, 25 6 250 2 .5 

,U ." '.M " '20 3000 2 

Engines, boilers, and water / 500 150 

Fuel and engine-stores 10000 200 

Hull and fittings 17 500 350 

Provisions and water 2 500 50 

Officers, engineers, and crew 7 500 10 

Equipment and sea-stores 7 500 150 

Spare space and weight 3 250 85.5 

90 000 1500 

Costs of Vessels per Ton— (English, 1865.) 
Hull, Joiner Work, Equipment, and Fittings. 

Wood. 

Merchantman, 650 tons. 

Wood in hull, masts, and spars. . $41. 



Iron. 
Merchantman, 500 tons $88. 

Passenger ship or steamer, 800 tons. 115. 

Passenger steamer, 1500 tons 147. 

u " 1800 " 122. 

Hull (materials) $29.50 

Labor 14 50 

Plant* and labor 14.50 

Wood work 12.25 

Fittings and launching 14.25 

Equipment 17. 

Cabins and passenger fit- 
tings 20. 

" $122. 



Yellow metal, iron bolts, and labor 10.30 

Joiner work and labor 5.15 

Labor on hull 20. 

Boats, etc., outfit 12.30 

Rope and sails 8. 

Anchors, chains, and tanks 4.25 

Yellow metal sheathing 4. 

$105. 
Steamer. 
Cabins and fittings, 1st class passn'r $25. 
. " " 2d " 12.5 

Engines, boilers, and machinery com- 
plete, $225 to $275 per nom'l horse-power. 



Rent, Machinery Tools, etc. 



618 CONSTRUCTION OF VESSELS. 

Sh.ip-fcyu.ild.ing. 

Weights. — A man requires in a vessel a displacement of 488 lbs. pel 
month for baggage, stores, water, fuel, etc., in addition to his own weight, 
which is estimated at 175 lbs. 

A man and his baggage alone averages 225 lbs. 

A sailing ship, 150 feet in length, 32 feet beam, and 22 feet 10 ins. in 
depth, or 664 tons, C. H. (old measurement), has stowed 2540 square and 
484 round bales of cotton. Total weight of cargo 1254448 lbs., equal to 
4.57 bales, weighing 1889 lbs., per ton of vessel. And a full built ship of 
1625 tons, N. M., can carry 1800 tons' weight of cargo, or stow 4500 bales 
of cotton, New Orleans pressed. 

Hull of the iron steam-boat John Stevens — length 245 feet, beam 31 feet, and hold 
11 feet: weight of iron 239 440 lbs. And of one— length 175 feet, beam 24 feet, and 
8 feet deep : weight of iron 159 190 lbs. 

The average weight of a cubic foot of bottom of the ship Great Republic at &, in- 
dependent of decks and keelsons, was 180 lbs. 

The weight of a square foot of bottom of the iron steamer Great Britain at }g$, in- 
dependent of deck and keelsons, was 42 lbs. = 1T0 cubic inches of metal. 

The weight of hull of a vessel with an iron frame and oak planking, compared with 
a hull entirely of wood, is as 8 to 15. An iron hull weighs about 45 per cent, less 
than a wooden hull. 

The weight of hull of an 80-gun ship-of-the-line, 198.5 feet in length by 55 feet 
beam, is 1995 tons ; ballast, 70 tons ; crew, equipment, and stores fcfr four months, 
including armament (447 tons), 1612 tons. Total displacement at load line, 3677 tons. 

Sails. 

That a vessel's sail may have the greatest effect to propel her forward, 
it should be set between the plane of the wind and that of the ship's course, 
that the tangent of the angle it makes with the wind may be twice the 
tangent of the angle it makes with the ship's course. 

Thus, with the wind abeam, a vessel's sails should be braced at an angle 
of 54° 45' from the wind, as the tangent of this angle is twice that of 
35° 15'. or 90°-54° 45'. 



PASSAGES OF STEAMEES AND SAILING VESSELS. 618* 

PASSAGES OF STEAMERS AND SAILING VESSELS. 

Distances in Geographical Miles or Knots. 

STEAMERS — SIDE- WHEELS. 

1807, PhceniXy of Hoboken, N. J. (John Stevens), New York, N. Y., to Philadelphia, 
Penn. First passage of a steam vessel at sea. 

1814, Morning Star, of Eng., River Clyde to London, Eng. First passage of an 
English steamer at sea. 

181T, Caledonia, of Eng., Margate, Eng., to Cassel, Germ., 180 miles, in 24 hours. 

1819, Savannah, of N. Y., about 340 tons O. M., Tybee Light, Savannah River, Ga., 
to Rock Light, Liverpool, Eng., 3640 miles, in 25 days 14 hours; 6 days 21 hours 
of which were under steam. 

1825, Enterprise, of Eng., 500 tons, Falmouth, Eng., to Table Bay, Africa, in 51 
days; and to Calcutta, India, in 113 days. First passage of a steamer to India. . 

1S30, Hugh Lindsay, 411 tons, 80 horse-power, Bombay, India, to Suez, Egypt, 
3103 miles, in 31 days running time. 

1839, Great Western, of Eng., Liverpool to New York, N. Y., 301T miles, in 12 
days IS hours. 

1852, Arctic, of N. Y., New York to Liverpool, Eng., 301T miles, in 9 days 17 hours 
15 min. 

1856, Persia, of Eng., New York to Liverpool, Eng., 3017 miles, in 9 days 1 hour 
45 min. 

1856, Ocean Bird, of N.Y., New York to Havana, Cuba, 1167 miles, in 4 days 4 
hours. 

1S5S, Pacific, of Eng., from St. John's, N. F., to Galway, Ireland, 1665 miles, in 
6 days 1 hour. 

1859, Baltic, of N. Y, Aspinwall to New York, N. Y., 2000 miles, in G days 21 
hours. 1853, Liverpool to New York, 3017 miles, in 9 days 16 hours 33 min. 

1859, Vanderbilt, of N. Y, from the Needles, Eng., to New York, N. Y, 3053 miles, 
in 9 days 9 hours 26 min. 1857, New York to the Needles, in 9 days S hours; equal 
to 9 days 11 hours to Liverpool. 

1861, Adriatic, of N. Y., ran the measured mile at Stokes's Bay, Eng., at an aver- 
age speed of 15.9 knots per hour. 1860, New York to Liverpool, Eng., 3017 miles, 
in 9 days 13 hours 30 min. ; Galway to Quarantine, N. Y., touching at St. John's, 
N. B., 2S65 miles, in 8 days 12 hours 20 min. For 1 day ran 365 miles. 

1862, Columbia, of N. Y., Washington Navy Yard, D. C, to the Battery, N. Y, 425 
miles, in 30 hours. 

1S65, Henry Chauncey, of N. Y., Aspinwall to Canal Street, N. Y., 2000 miles, 
in 6 days 3 hours 40 min. 

1S65, Colorado, of N. Y., New York to San Francisco, Cal., 14 549 miles, in 61 days 
21 hours 4 min. 

1866, Santiago de Cuba, of N. Y., Grey town to New York, N. Y., 2090 miles, in 6 
days 19 hours. 

1866, Guiding Star, of N. Y, New Orleans to New York, N. Y., 1699 miles, fully 
laden, in 5 days 11 hours 30 min. 

1SG6, Morro Castle, of N. Y, Havana, Cuba, to New York, N. Y., 1167 miles, in 3 
days 15 hours. 

1866, Mahroussa, of Egypt, Southampton, Eng., to Malta, Mediterranean, 2130 
miles, in 6 days 13 hours— 13. 6 knots per hour. 

1867. Nevada, of N, Y, New York to Panama, Isthmus of Darien, 11370 miles, 
in 43 days 5 hours, running time. 

1870, Scotia, of Eng., Queen stown, Ireland, to Sandy Hook, N.Y., 2780 miles, in 
S days 7 hours 31 min. 1866, New York to Queenstown, 2798 miles, in 8 days 2 
hours 48 min. ; thence to Liverpool, Eng., 270 miles, in 14 hours 59 min. ; total, 8 
days 17 hours 47 min. 



619 PASSAGES OF STEAMERS AND SAILING VESSELS. 



STEAMERS — SCREW. 

1S67, Hammonia, of Bremen, Southampton, EDg., to Sandy Hook, N. Y., 3103 
miles, in 9 days 9 hours 25 min. 

186S, E. B. Souder, of N. Y., Sandy Hook, N.Y., to Charleston Bar, S. C, 615 miles, 
in 49 hours. 

1868, Pereire, of Havre, New York, N. Y., to Brest, France, 2 933 miles, in 8 days 

10 hours SO min, ; 1S70, Brest, France, to New York, in 8 days 15 hours 38 min. 

1869, City of Boston, of Eng., Halifax, N. S., to Queenstown, Ireland, 2226 miles, 
in 6 days 22 hours. 

1S69, Wm. Lawrence, of Boston, Norfolk, Va., to Boston, Mass., 585 miles, in 2 day? 
2 min. 

1872, City of Houston, New York to Key West, Fla., 1 145 miles, in 4 days 7 hours 
25 min. ; thence to Galveston, Texas, 855 miles, in 3 days 1 hour 10 min. 

1574, India Government Boat, Steel, length 87 feet, beam 12 feet, draught of water 
3.75 feet, mean speed for one mile 20.77 miles per hour, and maintained a speed of 
IS. 92 miles in 1 hour. 

1576, Oceanic, of Eng., Yokohama, Japan, to San. Francisco, CaL, 4750 miles, in 
14 days 15 hours 20 min. 

1874, Schiller and Goethe, of Hamburg, Germany, from Hamburg to New York, 
N.Y., 3 577 miles, each in 10 days 20 hours. 

1874, City of Waco, Galveston, Texas, to New York, N. Y., anchorage to dock, via 
Key West, 2 000 miles, in 6 days 18 hours 40 min. 

1575, Hudson and Knickerbocker, of N.Y., wharf at New Orleans to wharf at New 
York, 1 820 miles, in 5 days 6 hours 15 min. 

1874, Knickerbocker, of N. Y., wharf at New York to S.W. Pass, La., 1 676 miles, 
in 5 days 10 hours 20 min. 

1577, City of Peking, of N. Y., San Francisco, Cal., to Yokohama, Japan, 4 750 
miles, in 15 days 9 hours. 

1576, Britannic, of Eng., Sandy Hook, N. Y., to Queenstown, Ireland, 2 780 miles, 
in 7 days 12 hours 46 min., and Queenstown to Sandy Hook, N. Y. (2S54 miles by 
observations), in 7 days 13 hours 11 min. ; thence to Liverpool, 270 miles, including 
all detention, in 8 days 5 hours 15 min. ; 1877, Queenstown to Sandy Hook, N. Y., in 
7 days 10 hours 53 min. 

1679, Arizona, of Eng., Sandy Hook, N. Y., to Queenstown, Ireland, 2 780 miles, 
in 7 days 23 hours 45 min. 

18T9, Saratoga, of N. Y., Sandy Hook, N. Y., to Morro Castle, 1 165 miles, in 3 days 
14 hours 50 min. 

1S79, City of Washington, of N. Y., Morro Castle to Sandy Hook, N. Y., 1 165 
miles, in 3 days 3 hours 21 min. 

1S76, Illinois, of Phila., Cape Henlopen, Del., to Queenstown, Ireland, 2908 miles, 
in 8 days 18 hours. 

1865, Henry Chauncey, of N. Y., Aspinwall to New York, 2 005 miles, pier to pier, 
in 6 days 3 hours 40 min. 

1879, Torpedo Launch, of Yarrow & Co., Eng., Steel, length 86 feet over all, I 

11 feet, measured miles (knots) 21.9 per hour. 

SAILING VESSELS. 

1851, Chrysolite (clipper snip), of Eng., Liverpool, Eng., to Anjer, Java, 13000 
miles, in 88 days. The Oriental, of N. Y., ran the same course in S9 days. 

1851, Flying Cloud (clipper ship), of Boston, Mass., from New York to San Fran- 
cisco, Cal., 13 610 miles, in 89 days and 18 hours, and sailed 374 miles in 1 day. 

1852-3, Flying Dutchman (clipper ship), of N. Y., New York to San Francisco, 
Cal., and return, discharged and loaded, wharf to wharf, 27 220 miles, in 6 months 
21 days; 1853, San Francisco to the Equator, 2380 miles, 11 days 9 hours, and 
rounded Cape Horn, 6 380 miles, in 35 days. 

1853, Trade Wind (clipper ship), of N. Y., San Francisco, Cal., to New York, 
N. Y., 13 610 miles, in 75 days. 

1S54, Lightning (clipper ship), of Boston, Boston, Mass., to Liverpool, Eng., 282T 
miles, in 13 days — hours ; and Melbourne, Australia, to Liverpool, Eng., 12 190 
miles, in 64 days. 



PASSAGES OF SAILING VESSELS. 619* 

1854, Comet (clipper ship), of N. Y., Liverpool, Eng., to Hong Kong, China, 13040 
miles, in 84 days. 

1854, Sierra Nevada (schooner), of N. H. , Hong Kong, China, to San Francisco, Cal., 
6 000 miles, in 34 days. 

1854, Red Jacket (clipper ship), of N. Y., Sandy Hook, N. Y., to Liverpool bar, 
Eng., 3 000 miles, in 13 days 11 hours 25 min. ; and New York to Melbourne, Aus- 
tralia, 12,720 miles, in 69 days 11 hours 1 min. 

1855, Euterpe (half-clipper ship), of Rockland, Me., New York to Calcutta, India, 
12 500 miles, in 78 days. 

1S55, Mary Whitridge (clipper ship), of Baltimore, Baltimore, Md., to Liverpool, 
Eng., from Cape Henry, Va., 3 400 miles, in 13 daijs 7 hours. 

, Richard Busteed (half-clipper ship), of Boston, Sidney, N. S. W., to Calcutta, 

India, 5 800 miles, in 42 days. 

1SG0, Dawn (clipper bark), of N. Y., Buenos Ayres, Brazil, to New York, N. Y. 6 010 
miles, in 36 days. 

1S60, Andrew Jackson (clipper ship), of Boston, New York, N. Y., to San Francisco, 
Cal., 13610 miles, in 80 days 4 hours. 

1S65, Dreadnought (clipper ship), of Boston, Honolulu, Sandwich Islands, to New 
Bedford, Mass., 13 470 miles, in 82 days; I860, Sandy Hook, N; Y., to off Queenstown, 
Ireland, 2 760 miles, in 9 days 17 hours; and 1859, Sandy Hook, N. Y., to Rock 
Light, Liverpool, Eng., 3 000 miles, in 13 days 8 hours. 

, Ocean Telegraph (half-clipper ship), of Boston, Callao, Peru, to Boston, Mass., 

9 970 miles, in 58 days. 

, Sovereign of the Seas (medium ship), of Boston, Mass., in 22 days sailed 5 391 

miles=:245 miles per day. For 4 days sailed 341.78 miles per day, and for 1 day, 
375 miles. 

, Northern Light (half-clipper ship), of Boston, San Francisco, Cal., to Boston, 

Mass.., 13 550 miles, in 76 days 8 hours. 

, North Wind (medium clipper ship), of N.Y.,the Downs, Eng., to Port Philip 

Head, Australia, 12500 miles, in 67 days. 

1866, Henrietta (schooner yacht), of N. Y., Sandy Hook, N. Y., to the Needles, 
Eng., 3 053 miles, in 13 days 21 hours 55 min. 16 sec. 

1866, Ariel and Serica (clipper ships), of England, Foo-chou-foo Bar, China, to 
the Downs, Eng., 13 500 miles, in 98 days. 

1867, Thornton (full ship), of N. Y., Sandy Hook, N. Y., to Rock Light, Liverpool, 
Eng , 3 000 miles, in 13 days 9 hours. 

1867, John J.Ward (schooner), of N. Y., Alexandria, Va., to Jersey City, N.J., 425 
miles, in 4S hours. 

1867, Frank Peren (schooner), of Chicago, Chicago, 111., to Buffalo, N. Y., 945 miles 
(1 100 statute miles), in 3 days 5 hours 30 min. 

1868, Mercury (ship), of N. Y., New York to Havre, France, 3 068 miles, in 12 days 
— hours. 

1569, Sappho (schooner yacht), of N. Y., Light-ship off Sandy Hook, N. Y., to 
Kinsale Head, 2754 miles, in 12 days 7 hours 51 mm., and to Queenstown, Ireland, 
2 857 miles run, in 12 days 9 hours 34 min. 

1869, Minnie A. Smith (half brig), of Millbridge, Me., from New York, N. Y., to 
Salerno, Italy, 4 210 miles, in 27 days. 

1869, Dauntless (schooner yacht), of N. Y., Light-ship off Sandy Hook, N. Y., to 
Queenstown, Ireland, 2 770 miles run, in 12 days 17 hours 6 min. 12 sec. 

18 — , James Barnes, of Boston, Boston, Mass., to Liverpool, Eng., 2827 miles, in 12 
days 6 hours. 

1869, Ocean Spray (bark), Galveston, Texas, to River Mersey, Eng., 4850 miles, in 
26 days. 

1873, Young America (clipper ship), of N.Y., Liverpool, Eng., to San Francisco, 
Cal., 13,800 miles, in 96 days. 

1870, Telegraph (bark), of Quebec, New York, N. Y., to Cronstadt, Russia, 45S8 
miles, in 25 days. 

1570, Nunquam Dormio (medium ship), of New York, N. Y., Mobile, Ala., to Havre, 
France, 4 700 miles, in 25 days. 

3G 



620 PASSAGES OF STEAM-BOATS. 

NORTHERN AND EASTERN. 

Distances in Statute Miles. 

1807, Clermont, of N.Y., New York to Albany, 145 miles, in 32 hours = 4.53 miles 
per hour, neglecting effect of the tide. 

1849, Alida, of N.Y., Caldwell's, N. Y., to Pier 1, North River, 43^ miles, in 1 hour 
42 min., ebb tide = 2.75 miles per hour. Speed = 22.19 miles per hour. 1860, 30th 
Street, N. Y., to Cozzens's Pier, West Point, 49^ miles, in 2 hours 4 min., and to 
Poughkeepsie, 18% miles, in 3 hours 27 min., making 5 landings. Flood tide. And 
1853, Robinson Street to Kingston Light, 90% miles, in 4 hours, making 6 landings. 
Flood tide. 

1S52, Reindeer, of N.Y., New York to Hudson, 116^ miles, in 4 hours 57 min., 
making 5 landings. Flood tide. 

1S64, Daniel Dreiv, of N. Y., Jay Street, N. Y., to Albany, 148 miles, in 6 hours 51 
win., making 9 landings. Flood tide. Speed of boat = 22.6 miles per hour. 

1864, Chauncey Vibbard, of N.Y., Desbrosses Street, N.Y., to Rhinebeck, 91 miles, 
in 4 hours 42 min., making 4 landings. From Rhinebeck to Catskill, 22 miles, 
in 1 hour 6 min. Left New York at time of high-water. 

1S67, Mary Poivell, of N.Y., Desbrosses Street, N.Y., to Newburg, 60>£ miles, in 2 
hours 50 min., making 3 landings ; and to Rondout, 91% miles, in 4 hours 23 mm., 
making 7 landings (from Poughkeepsie to Rondout Light, 15% miles, in 39 min.). 
Flood tide. 1873, Milton to Poughkeepsie, light draft and flood tide, 4 miles, in 9 
min.; and 1874, Desbrosses Street to Piermont, 24 miles, in 1 hour; to Caldwell's, 
43^ miles, in 1 hour 50 min. ; to Cozzens's Pier, 50)4 miles, in 2 hours 9 min. ; 
and to Poughkeepsie, making 6 landings, in 3 hours 39 min. Speeds 22.77 to 23 
miles per hour. 

Runs from New York to Albany, 146 miles, by different Boats. 
1826. Sun, 12 hours 16 min. 1851. New World,\\ 7 hours 43 min. 



1826. North America* 10 " 20 

1540. Albany, 8 " 27 
1841. TrovJ 8 "10 

1541. South A mericij. 7 " 28 
1849. Alidad 7 " 45 



1852. Fr. Skiddy,\\ 7 " 24 

1S52. Reindeer^ 7 " 27 

1S60. Armenia^ 7 " 42 

1S64, Daniel Drew,t 6 c ' 51 

1864, Chrfcy Vibbard,t 6 '« 42 
1S74, Sylvan Dell, 7 hours 43 min. 



* 7 landings. f 4 landings. % 9 landings. § 12 landings. | 6 landings. ^[ 11 landings. 

Timing Distance.— From 14th St., Hudson River, N. Y., to College at Mount St. Vincent. 13 miles. 

Note. — Where landings have been made, and the river crossed, the distance between the points 
given is correspondingly increased. 



SOUTHERN AND WESTERN. 

Distances in Statute Miles. 

1811, New Orleans, of Pittsburg, Penn. (non-condensing and stem-wheel), Pittsburg 
to Louisville, Ky., 650 miles, in 2 days 22 hours. 

1844, J. M. White, of St. Louis, Mo. (non-condensing), St. Louis to New Orleans, 
La., 1200 miles (at that time), 600 tons 1 freight, 4.5 to 5.5 miles per hour adverse 
current, in 3 days 16 hours; and returned to St. Louis, making all regular landings, 
and losing 2 hours 30 min. by wooding, etc., in 3 days 23 hours 9 min. 

1850, Buck Eye State, of Pittsburg, Penn. (non-condensing), Cincinnati to Pittsburg, 
500 miles (200 passengers), making 53 landings, in 1 day 19 hours; 4 miles per hour 
adverse current. Speed = 15.63 miles and i. k 23 landings per hour. Average depth 
of water in channel 7 feet. 



PASSAGES OF STEAM-BOATS. — ICE-BOATS. 620* 

1853, Shotwell, of Louisville, Ky. (non-condensing), New Orleans to Louisville, 
1 450 miles, making 8 landings, in 4 days 9 hours ; 4.5 to 5.5 miles per hour adverse 
current. Speed = 18.81 miles per hour. 

Note.— In 1817-18 the average duration of a passage from New Orleans to Louisville was 27 days 
12 hours ; the shortest, 25 days. 

1855, Princess, of New Orleans (non-condensing), New Orleans, La., to Natchez, 
Miss., 310 miles, in IT hours 30 min. ; 3.5 to 4 miles per hour adverse current. 
Speed = 20.98 miles per hour. 

1861, Atlanta, of St. Louis, Mo. (non-condensing), New Orleans to Hard Times 
Landing, La., making 1 landing, 365 miles, in 24 hours; 4.5 to 5.5 miles per hour 
adverse current. Speed = 20.29 miles per hour. 

, Telegraph, No. 3 (non-condensing), of Cincinnati, Ohio, Louisville to Cin- 
cinnati, 150 miles, in 9 hours 55 min. ; 5.5 miles per hour adverse current. Speed 
= 20.63 miles per hour. 

1S70, Natchez, of Cincinnati (non-condensing), New Orleans, La., to St. Louis, Mo., 
1 180 miles, making 14 landings, 4 to 5 miles per hour adverse current, and lost 
1 hour 35 min., in 3 days 21 hours 58 min. ; and from New Orleans to Natchez, 
Miss., 295 miles, in 16 hours 51 min. 30 sec. 

1870, R. E. Lee, of St. Louis (non-condensing), New Orleans to St. Louis, Mo., 
1180 miles (without passengers or freight), 4 to 5 miles per hour adverse current: to 
Natchez, in 17 hours 11 min. ; Vicksburg, 1 day 38 min. ; Memphis, 2 days 6 hours 
9 min. ; Cairo, 3 days 1 hour ; and to St. Louis, 3 days 18 hours 14 min., inclusive of 
all stoppages. {Natchez, of Cincinnati, Ohio, starting in company with the R. E. Lee, 
with passengers, making 2 landings for passengers alone, in 4 days 1 hour 8 min., 
including all stoppages for fuel, repairs, and passengers, which were computed at 
7 hours 28 min.) And from New Orleans to Baton Konge, 120 miles, in 7 hours 
40 min. 42 sec. ; to Natchez, 295 miles, in 16 hours 36 min. 47 sec. 

18 — , Chrysopolis, Sacramento to San Francisco, Cal., 125 miles, in 5 hours 18 min. 
Runs from New Orleans to Natchez, 295 miles, by different Boats. 



1814, Orleans, 6 days 6 hours 40 min. 
1S28, Tecumseh, 3 days 1 hour 20 min. 
1840, Edward Shippen, 1 day 8 hours. 



1844, Old Sultana, 19 hours 45 min. 
1S56, New Princess, 17 hours 30 min. 
1S70, R. E. Lee, 16 hours 36 min. 47 sec. 



ICE-BOATS. 

Distances in Statute Miles. 

1866, Una, of Poughkeepsie, N. Y., Newburg to New Hamburg, 6>£ miles, in 7 
min. 

1870, Ella, of Poughkeepsie, N. Y., Hudson Kiver, 8 miles South and back, esti- 
mated by courses at 24 miles, in 23 min. 10 sec. 

1872, Haze, of Poughkeepsie, N. Y., Poughkeepsie to buoy off Milton, 4 miles, in 4 
min. 

1872, Whiz, of Poughkeepsie, N. Y., Poughkeepsie to New Hamburg, 8% miles, in 
8 min. 



620** KIFLE AXD PIGEON SHOOTLNG, ETC., ETC. 



RIFLE-SHOOTING. 

1864, Col. H. Berdan, Utica, N. Y., 1200 yards, muzzle rest, 4 consecutive bull'i 
eyes ; and 1S4S, at Chicago, 111. , 40 rods, 10 shots, muzzle rest, 1% inches. 

1869, Hamilton. Montreal vs. Hamilton, Can., 6 men on each side, 500, 600, 700, 
S00, and 1000 yards ; 7 shots each range ; scored 4S6 ins. 

1S74, Major Henry Fulton, Creedmoor, L. I., N. Y., 171 points out of ISO, at 800, 
900, and 1000 yards; bull's eyes counting 4. 

1875, American Team, Dellymount, Ireland, 90S points out of 1080, at 800, 900, 
and 1000 yards. 

1ST7, L. C. Bruce, Creedmoor, L. L, N. Y., 219 points out of 225; viz., 74 at 800, 
72 at 900, and 73 at 1000 yards. 

1S77, Dudley Selph, New Orleans, La. (practice score), 21 consecutive bull's eyes, 
14 at 900, and 7 at 1000 yards. 

1878, Joseph Partello, Washington, D. C. (practice score), 27 consecutive bull's 
eyes, 15 at 800, and 12 at 900 yards ; also, 224 points out of 225, at 800, 900, and 
1000 yards. 

1S79, W. M. Farrow, Creedmoor, L. I., 50 points out of 50, 10 of 200 yards, off-hand. 

1879, J. F. Brown, Boston, Mass., 150 points out of 150, 15 at 500", and 15 at 900 
yards. 



PIGEON-SHOOTING. 

1S65, A . B. Holabird, Cincinnati, Ohio, double birds, 21 yards' rise, 100 yards' 
bounds, \y z oz. shot, killed 90 out of 100, and 2 fell out of bounds. 

1S6S, — Root and — Brieth, Newport, Ky., single birds, 21 yards, SO yards' bounds, 
IX oz. shot, killed 23 out of 24. 

1S6S, Major Whittingstall, London, Eng., single birds,' 30 yards' rise, IX oz. shot, 
killed 14 out of 17. 

1869, A. H. Bogardus, Chicago, 111., single tame birds, 21 yards' rise, SO yards' 
bounds, single barrel, heavy gun, shot unlimited, plunge trap, killed 100 in succes- 
sion; 1S71, Union Course, L. I., single birds, 21 yards' rise, 80 yards' bounds, 1% oz. 
shot, ground trap, killed 46 out of 50, and loaded and killed, from spring and plunge 
traps, 73 birds in 6 mm. 31% sec. ; 1875, Prospect Park, L. I., single birds, 5 traps, 
30 yards' rise, 80 yards' bounds, IX oz. shot, 5 drams powder, No. 12 gun, No. 8 and 
7 shot (Hurlingame rules), killed 84 out of 100, and 50 out of the last 52 (5 dead out 
of bounds). 

1869, Ira Payne, Secaucus, N. J., single birds, 21 yards' rise, 100 yards' bounds, \}£ 
oz. shot, killed 44 out of 50 from ground trap, 39 out of 40 and 92 out of 100 from 
spring trap. 

1871, John Taylor, Greenville, N. J., single birds, 21 yards' rise, 80 yards' bounds, 
ground trap, IX oz. shot, killed 45 out of 50 ; and 1SC5, double birds, IS yards' rise, 
100 yards' bounds, \% oz. shot, killed 94 put of 100. 

1871. M. Johnson,Union Course, L. I., double birds, 18 yards' rise, 100 yards' bounds, 
IX oz. shot, ground traps, killed 28 out of 30. 



TRAP-SHOOTING. 

1S77, A. H. Bogardus, New York, 1000 glass balls out of 1028, 2 traps, 18 yards, 
in 1 hour 17 min, 40 sec, and 1878, 1000, 2 traps, 15 yards, in 1 hour 1 min. 54 sec. 



BIRD-SHOOTING. 
18—, M. Campbell, Scotland, 220 brace of grouse in 1 day. 

1856, Seth Green, Charlotte, N.Y., 2 double guns, No. 9, No. 8 shot, 990 pigeons, in 
flocks, in 3 hours. 



RAT-KILLING. 

1R62, Jacko, London, Eng., 13 lbs., 00 rats in 2 min. 43' sec, 100 in 5 min. 28 sec, 
200 in 14 min. 37 sec, and 1000 in less than 100 min. ; 1S01, 25 in 1 min. 28 sec. 
1861, , Philadelphia, Penn., 25 rats in 1 min. 2S sec. 



CEMENTS. 621 

CEMENTS. 
!Ru.st Joint.— (Quick Setting) 

1 lb. Sal-ammoniac in powder, 2 lbs. Flower of sulphur, 80 lbs. Iron borings. 
Made to a paste with water. 

(Slow Setting.)— -2 lbs. Sal-ammoniac, 1 lb. of Sulphur, 200 lbs. Iron borings. 
The latter cement is best if the joint is not required for immediate use. 

For Steam-boilers, Steam-pipes, etc. 

Soft. — Red or white lead in oil, 4 parts ; Iron borings, 2 to 3 parts. 
Hard. — Iron borings and salt water, and a small quantity of Sal-ammoniac with 
fresh water. 

Mialtna, or GJ-reek HVXastic. 

Lime and Sand mixed in the manner of mortar, and made into a proper consist- 
ency with milk or size without water. 

For China. 

Curd of milk, dried and powdered, 10 oz. ; Quick-lime, 1 oz. ; Camphor, 2 drachms. 
Mix, and keep in closely-stopped bottles. When used, a portion is to be mixed with a little water 
into a paste. 

For Earthen and. Grlass Ware. 

Heat the article to be mended a little above 212°, then apply a thin coating of 
gum Shellac upon both surfaces of the broken vessel. 

Or, dissolve gum Shellac in alcohol, apply the solution, and bind the parts firmly 
together until the cement is dry. 

For Holes in. Castings. 

Sulphur in powder, 1 part ; Sal-ammoniac, 2 parts ; powdered Iron turnings, 80 
parts. Make into a thick paste. 

The ingredients composing this cement should be kept separate, and not mixed until required ibr use. 

For Marble. 

Plaster of Paris, in a saturated solution of alum, baked in an oven, and reduced 
to powder. Mixed with water. It may be mixed with various colors. 

For ]VEarble Workers and. Coppersmiths. 

White of egg, or mixed Avith finely-sifted Quick-lime, will unite objects which are 
not submitted to moisture. 

Transparent— for Grlass. 

India-rubber, 1 part in 64 of chloroform ; add gum Mastic in powder, 1G to 24 parts. 
Digest for two days with frequent shaking. 

To Mend Iron Ware. 
Sulphur 2 parts, fine Black Lead 1 part. Put the sulphur in an iron pan, over a 
fire, until it melts, then add the lead ; stir well ; then pour out. When cool, break 
into small pieces. A sufficient quantity of this compound being placed upon the 
crack of the ware to be mended, can be soldered by an iron. 

For Cisterns and. Water-casks. 
Melted glue, 8 parts ; Linseed -oil, 4 parts ; boiled into a varnish with litharge. 

This cement hardens in about 48 hours, and renders the joints of wooden cisterns and casks air and 
Water tight. 

Hydraulic Cement Faint. 
Hydraulic cement mixed with oil forms an incombustible and water-proof paint for 
roofs of buildings, out-houses, walls, etc. 

Entomologists' Cement. 
Thick Mastic Varnish and Isinglass size, equal parts. 

I* G* 



622 BROWNING, LACKERS, GLUE, ETC. 

BROWNING, OR BRONZING LIQUID. 

Sulphate of copper, 1 oz. ,' Sweet spirit of nitre, 1 oz. ; Water, 1 pint. 
Mix. In a few days it will be fit for use. 

Browning for Gun Barrels. 

Tinct. of Mur. of Iron, 1 oz. ; Nitric Ether, 1 oz. ; Sulph. of Copper, 4 scruple , 
rain water, 1 pint. If the process is to be hurried, add 2 or 3 grains of Oxyinuriate 
of Mercury. 

When the barrel is finished, let it remain a short time in lime-water, to neutralize any acid which 
may have penetrated; then rub it well with an iron wire scratch-brush. 

Bronzing Fluid for Guns, etc. 

Nitric acid, sp. gr. 1.2 ; Nitric ether, Alcohol, and Muriate of iron, each 1 part. 
Mix, then add Sulphate of copper 2 parts, dissolved in water 10 parts. 

LACKERS. 
For Small ^rms, or Water-proof Paper, 

Beeswax, 13 lb3. ; Spirits turpentine, 13 gallons ; Boiled linseed-oil, 1 gallon. 

All the ingredients should be pure and of the best quality. Heat them together in a copper or 
earthen vessel over a gentle fire, in a water-bath, until they are well mixed. 

For Briglit Iron. Work. 

Linseed-oil, boiled 80. 5 I White lead, in oil 11 . 25 

Litharge 5 . 5 j Resin, pulverized 2 . 75 

Add the litharge to the oil ; let it simmer over a slow fire 3 hours ; strain it, and add the resin and 
white lead ; keep it gently warmed, and stir it until the resin is dissolved. 

INKS. 
Ind.eli.fole, for jVXarlring Linen, etc. 

1. Juice of Sloes, 1 pint ; Gum, % an ounce. 

This requires no " preparation" or mordant, and is very durable. 

2. Nitrate of silver, 1 part ; Water, 6 parts ; Gum, 1 part. Dissolve. 
Marking.— Lunar caustic, 2 parts ; Sap green and Gum arabic, each 1 part ; dis- 
solve with distilled water. 

The " Preparation." — Soda, 1 ounce; Water, 1 pint. Sap green, % drachm. Dis- 
solve, and wet the article to be marked, then diy and apply the ink. 

Perpetual, for Tomb-stones, Marble, etc. — Pitch, 11 parts; Lamp-black, 1 part; 
Turpentine sufficient. Warm and mix. 

Copying Ink. — Add 1 oz. of Sugar to a pint of Ordinary Ink. 

GLUES. 

For Parchment. 

Parchment shavings 1 lb. : Water. 6 quarts. 

Boil until dissolved, then strain and evaporate slowly to the proper consistence. 

Rice Glue, or Japanese Cement. 
Rice flour ; Water, sufficient quantity. 
Mix together cold, then boil, stirring it all the time. 

LiqnicL. 

Glue, Water, and Vinegar, each 2 parts. Dissolve in a water-bath, then add Al. 
cohol, 1 part. 

Or, Cologne or strong glue, 2/2 Lbs, ; Water, 1 quart ; dissolved over a gentle heat; 
add Nitric acid 36°, 7 oz., in small quantities. 
.Remove from the fire and cool. 

Or, White glue, 1G oz. ; White lead, dry, 4 oz. : Rain water, 2 pinte. Add Alco< 
hoi, 4 oz., and continue the heat for a few minutes. 



VARNISHES. 623 

IMiarine. 

Dissolve India-rubber, 4 parts, in 34 parts of Coal-tar naphtha; add powdered 
Shellac, 64 parts. 

While the mixture is hot it is poured upon metal plates in sheets. When required 
for use, it is heated, and then applied with a brush. 

Or, 1 part India-rubber, 12 parts of Coal tar ; heat gently, mix, and add 20 parts 
of powdered Shellac. Pour out to cool. When used, heat to about 250°. 
* Or, Glue, 12 parts ; Water, sufficient to dissolve ; add yellow Resin, 3 parts ; and, 
when melted, add Turpentine, 4 parts. 

Mix thoroughly together. 

Strong Glue.— Add Powdered chalk to common glue. 

Gum Mucilage.— A little oil of cloves poured into a bottle containing gum mucilage* 
prevents it from becoming sour. 

Grlne to resist ZMoistnre. 

5 parts Glue, 4 parts Resin, 2 parts Red ochre, mixed with the least practicable 
quantity of water. 

Or, 4 parts of Glue, 1 part of Boiled oil by weight, 1 part Oxide of iron. 
Or, 1 lb. of glue melted in 2 quarts of skimmed milk. 

VARNISHES. 

Waterproof. 

Flower of sulphur, 1 lb. ; Linseed-oil, 1 gall. ; boil them until they are thoroughly 
combined. 

This forms a good varnish for waterproof textile fabrics. 

Another is made of Oxyde of lead, 4 lbs. ; Lamp-black, 2 lbs. ; Sulphur, 5 oz. ; 
and India-rubber dissolved in turpentine, 10 lbs. 

Boil together until they are thoroughly combined. 

To Adhere Engravings or Lithographs tipon Wood. 

Sandarach, 250 parts; Mastic in tears, 64; Resin, 125; Venice turpentine, 250; 
and Alcohol, 1000 parts by measure. 

For Harness. 

India-rubber, X lb. ; Spirits of turpentine, 1 gall. ; dissolve into a jelly ; then takt 
hot Linseed-oil, equal parts with the mass, and incorporate them well over a slow fire. 

For Fastening Leather on Top Rollers. 

Gum Arabic, 2% oz., dissolved in water, and a like volume of Isinglass dissolved 
in water. 

To Preserve Grlass from the Rays of* the Snn. 

Reduce a quantity of Gum Tragacanth to fine powder, and let it dissolve for 24 
hours in white of eggs well beat up. 

For "Water-color Drawings. 

Canada balsam, 1 part ; Oil of turpentine, 2 parts, mixed. 
Size the drawing before applying the varnish. 

For O "ejects of UNTatxiral History, for Shells, Fish, etc. 

Mucilage of Gum Tragacanth and Mucilage of Gum Arabic, each 1 oz. 
Mix, and add spirit with Corrosive sublimate, so as to precipitate the more stringy 
part of the gum. 

For Articles of Iron and. Steel. 

Clear grains of Mastic, 10 parts; Camphor, 5 parts; Sandarach, 15 parts; and 
Elemi, 5 parts. Dissolve in a sufficient quantity of alcohol, and apply without heat. 

This varnish will retain its transparency, and the metallic brilliancy of the articles will not be 
obscured. 






624 MISCELLANEOUS. 

3Tor G-u.li Barrels, after Browning. 
Shellac, 1 oz. ; Dragon's-blood, % oz. ; rectified Spirit, 1 quart. Dissolve and filter. 

Black- 
Heat to boiling, 10 parts of Linseed-oil varnish with burnt. Umber, 2 parts, and 
powdered Asphaltum, 1 part. 
When cooled, dilute with Spirits of turpentine as required. 

Balloon. 

Melt India-rubber in small pieces with its weight of boiled Linseed-oil. 
Thin with Oil of turpentine. 

Transfer. 

Alcohol, 5 oz. ; pure Venice turpentine, 4 oz. ; Mastic, 1 oz. 

To Clean Varnish. 
Mix a ley of Potash, or Soda, with a little powdered Chalk. 

Composition for rendering Canvas Waterproof and. 

Bliaole. 
Yellow soap, 1 lb., boiled in 6 pints of water, add, while hot, to 112 lbs. of Paint. 

STAINING. 
Staining Wood and Ivory. 

Yellow. Dilute Nitric acid will produce it on wood. 

Red. An infusion of Brazil wood in stale urine, in the proportion of 1 lb. to a 
gallon for wood, to be laid on when boiling hot, and should be laid over with alum 
water before it dries. Or, a solution of Dragon's-blood in Spirits of wine. 

Black. Strong solution of Nitric acid. 

Mahogany. Brazil, Madder, and LogAvood, dissolved in water and put on hot. 

Blue. For Ivory : soak it in a solution of Verdigris in Nitric acid, which will turn 
it green ; then dip it into a solution of Pearlash boiling hot. 

Purple. Soak ivory in a solution of Sal-ammoniac into four times its weight of 
Nitrous acid. 

MISCELLANEOUS. 

To Clean Marble. 

Chalk, powdered, and Pumice-stone, each 1 part; Soda, 2 parts. Mix with water. 
Wash the spots, then clean and wasli off with soap and water. 

To Extract GJ-rease from Stone or jVLarble. 

Soft soap, 1 part ; Fuller's earth, 2 parts ; Potash, 1 part. Mix with boiling water. 
Lay it upon the spots, and let it remain for a few hours. 

Paint for Window Grlass. 

Chrome green, % oz. ; Sugar of lead, 1 lb. ; ground fine, in sufficient Linseed-oil 
to moisten it. Mix to the consistency of cream, and apply with a soft brush. 

The glass should be well cleansed before the paint is applied. The above quantity is sufficient for 
about 200 feet of glass. 

DnraTDle Paste. 

Make common flour paste rather thick (by mixing some flour with a little cold 
water until it is of uniform consistency, and then stir it well while boiling water is 
being added to it) ; add a little brown Sugar and Corrosive sublimate, which will 
prevent fermentation, and a few drops of Oil of lavender, which will prevent it be- 
coming moldy. When this paste dries, it may be used again by dissolving it in water. 

It will keep for two or three years in a covered vessel. 
Dn"bniiig. 

Resin, 2 lbs. ; Tallow, 1 lb. ; Train-oil, 1 gallon. 



MISCELLANEOUS. 625 

Slacking for Harness. 

Bees' -wax, % lb. ; Ivory black, 2 oz. ; Spirits of turpentine, 1 oz. ; Prussian blue 
ground in oil, 1 oz. ; Copal Varnish, ^ oz. 

Melt the wax and stir it into the other ingredients before the mixture is quite cold ; 
make it into balls. Rub a little upon a brush, and apply it upon the harness, then 
polish lightly with silk. 

To Prevent Iron from Resting. 

Warm it ; then rub with White wax ; put it again to the fire until the wax has per- 
vaded the entire surface. 
Or, immerse tools or bright work in boiled Linseed-oil and allow it to dry upon them. 

[Paper for Draughtsmen, etc. 

Powdered Tragacanth, 1 part ; Water, 10 parts. 

Dissolve, and strain through clean gauze, then lay it smoothly upon the paper, 
previously stretched upon a board. 

This paper will take either oil or water-colors. 

To Remove old. Ironmold. 

Remoisten the part stained with ink, remove this by the use of Muriatic acid di- 
luted by 5 or 6 times its weight of water, when the old and new stain will be removed. 

Pastiles for Fumigating. 

Gum Arabic, % oz. ; Charcoal powder, 5 oz. ; Cascarilla bark, powdered, % oz. % 
Saltpetre, ^ dram. Mix together with water, and make into shape. 

ITor "Writing upon Zinc La"bels— Horticultural. 

Dissolve 100 gr. of Chloride of Platinum in a pint of water ; add a little Mucilage 
and Lamp-black. 
Or, Sal-ammoniac, 1 dr. ; Verdigris, 1 dr. ; Lamp-black, ^ dr. ; water, 10 dr. Mix. 

Booth's Grrease for Railway Exiles. 

Water 1 gall. Palm-oil 6 lbs. I Or, Tallow 8 lbs. 

Clean tallow 3 lbs. Common soda. .. y 2 lb. | Palm-oil... 10 " 

To be heated to about 212°, and to be well stirred until it cools to T0°. 

^nti-friction Grrease. 

100 lbs. Tallow, TO lbs. Palm-oil. Boiled together, and when cooled to S0°, strain 
through a sieve, and mix with 2S lbs. of Soda and \% gallons of water. 
For Winter, take 25 lbs. more oil in place of the tallow. 
Or, Black Lead, 1 part ; Lard, 4 parts. 

Hiiard.. 

50 parts of finest Rape-oil and 1 part of Caoutchouc, cut small. Apply heat until 
it is nearly all dissolved. 

Stains. 

To Remove. — Stains of Iodine are removed by rectified spirit. Ink stains by ox- 
alic or superoxalate of potash. Ironmolds by the same ; but if obstinate, moisten 
them with ink, then remove them in the usual way. 

Red spots upon black cloth from acids are removed by Spirits of Hartshorn, or 
other solutions of Ammonia. 

Stains of Marking-ink, or Nitrate of Silver. — Wet the stain with fresh solution 
of Chloride of lime, and after 10 or 15 minutes, if the marks have become white, dip 
the part in solution of Ammonia or of Hyposulphite of soda. In a few minutes wash 
with clean water. 

Or, stretch the stained linen over a basin of hot water, and wet the mark with 
tincture of Iodine. 

Preservative Paste for Objects of Natural History. 

White Arsenic, 1 lb. ; Powdered Hellebore, 2 lbs. 



626 MISCELLANEOUS. 






To Preserve Eggs. 
Cover the shell with a solution of gum-arabic, glue, or any viscous substance that 
will exclude the air ; and pack them diy. 

To Extract Grease from Paper. 
Warm the spotted part and press blotting-paper upon it until all the grease is ex- 
tracted that is practicable by this method ; then with a brush apply very hot essen- 
tial oil of turpentine, and wash off with alcohol. 

Water-proof Mixture for Boots. 
Tallow, 1 pint ; oil of turpentine, X P ia * '•> yellow wax and Burgundy pitch, each 
X of a pound. Melt and mix. 

Brilliant "Whitewash. 

Lime, X bushel, slaked with warm water, strained through a sieve ; add a strong 
solution made with a peck of salt and warm water, 3 lbs. of boiled ground rice- 
paste, % lb. of powdered Spanish whiting, 1 lb. of dissolved glue, and 5 gallons of 
hot water ; all intimately mixed, and retained three days before use. 

"Varnish for Wood-Patterns. 
Gum-shellac, 3 oz., and resin, 1% oz., dissolved in a pint of wood naphtha. 

Asphalt Eloor. 

Pitch and resin, each 18 parts, sand 60, gravel 30, and slaked lime 6. To be laid 
2 inches in thickness. 

Eire and. Water Proof Cement. 

Unoxidized iron filings, finely sifted, 2 parts; loam, dry and fine, 1 part; mixed 
with strong vinegar, aud worked to a homogeneous mass. To be made only as re 
quired to be used. 

Rice Cement. 

Mix rice flour with cold water, and simmer it over a fire. 

When made of the consistency of plaster, molds, busts, etc., may be made of it. 

Impervious Cement. 

Sand 300 lbs., melted resin 120 lbs., to be well mixed. Before hardening, fine 
sand should be sprinkled over it to prevent abrasion. 

To Repair Cement. 

Linseed-oil and resin, each 2 parts, and ground pumice-stone 1 part; boil the oil, 
mix the resin with it, and add the pumice-stone. 

To Preserve Polished. Steel and. Iron Surfaces from 

Rust. 
Cover with a mixture of lime and oil. 

Painting of Brick-work. 

A square yard of new brick wall requires for the first coat of paint in oil % lb., 
for the second .3, and for the third .4. 

Gun-barrels. 
An ointment of corrosive sublimate and lard will preserve them from the corro- 
sive effects of sea air. 



ALLOYS AND COMPOSITIONS. 627 

Paste fbr Cleaning ]VIetals. 
Oxalic acid, 1 part ; Rottenstone, 6 parts. Mix with equal parts of Train-oil and 

Spirits of turpentine. 

Watchmaker's Oil, v^liich never Corrodes or Thickens. 
Place coils of thin sheet lead in a bottle with Olive-oil. Expose it to the sun for a 
few weeks, and pour off the clear oil. 

Blacking, withoxxt IPolishing. 

Molasses, 4 oz. ; Lamp-black, % oz. ; Yeast, a table-spoonful; Eggs, 2; Olive-oil, 
a teaspoonful ; Turpentine, a teaspoonful. Mix well 
To be applied with a sponge, without brushing. 

To Preserve Sails. 
Slacked lime, 2 bushels. Draw off the lime-water, and mix it with 120 gallons 
water, and with Blue Vitriol, % lb. 

"Whitewash . 

For outside exposure, slack Lime, % a bushel, in a barrel; add Common Salt, 
1 lb. ; Sulphate of Zinc, % lb. ; and Sweet Milk, 1 gallon. 

To Preserve Woodwork. 

Boiled oil and finely-powdered Charcoal, each 1 part; mix to the consistence of 
paint. Lay on two or three coats with it. 

This composition is well adapted for casks, water-spouts, etc. 

To IPolish "Wood. 



Rub surface with Pumice-stone and water until the rising of the grain is removed, 
Then, with powdered Tripoli and boiled Linseed-oil, polish to a bright surface. 

Files. 
Lay dull file3 in diluted Sulphuric acid until they are bitten deep enough. 

To Clean Brass Ornaments. 

Brass ornaments that have not been gilt or lackered may be cleaned, and a very 
brilliant color given to them, by washing them in alum boiled in strong ley, in the 
proportion of an ounce to a pint, and afterward rubbing them with strong Tripoli. 

Adhesive Cement fbr Fractures of* all Kinds. 
White Lead ground with Linseed-oil Varnish, and kept out of contact with the air. 
It requires a few weeks to harden. 

When stone or iron are to be cemented together, use a compound of equal parts of 
Sulphur and Pitch. 

| : 

ALLOYS AND COMPOSITIONS. 

Alloy is the proportion of a baser metal mixed with a finer or purer, 
as when copper is mixed with gold, etc. 

Amalgam is a compound of Mercury and a metal — a soft alloy. 

All Compositions of copper contract in the admixture, and all Amalgams 
expand. 

In the manufacture of alloys and compositions, the more infusible met- 
als should be melted first. 

In compositions of Brass, as the proportion of Zinc is increased, so is 
the malleability decreased. 

The tenacity of Brass is impaired by the addition of Lead or Tin. 

Steel alloyed with gJo tn P art 0I * platinum, or silver, is rendered harder, 
more malleable, and better adapted for cutting instruments. 



628 



ALLOYS AXD COMPOSITIONS. 



ALLOTS AND COMPOSITIONS. 



Argentan 55. 

Argentiferous 50. 

Babbitt's metal* 3. T 

Brass, common ...... 84.3 

» " T5. 

« " hard . 79.3 

44 Mathematical 

instruments. . 92.2 

" pinchbeck SO. 

41 red tombac 88. 8 

" rolled 74.3 

44 tutenag 50. 

11 very tenacious. 88.9 
t: wheels, valves . 90. 

" white 10. 

11 wire 67. 

11 yellow, fine 66. 

Britannia metal — 

When fused, add ... 

Bronze, red 87. 

11 red 86. 

11 vellow 67.2 

" Cymbals SO. 

44 gun metal, large 90. 
11 " small 93. 

44 Medals 93. 

44 Statuary 91.4 

Chinese silver 65.1 

Chinese Avhite copper. 40.4 

Church bells 80. 

44 69. 

Clock bells 72. 

Cocks, Musical bells . . 87.5 

German silver 33.3 

" 40.4 

44 " fine... 49.5 

Gongs S1.6 

House bells 77. 

Lathe bushes 80. 

Machinery bearings . . 87.5 

44 hard 77.4 

Metal that expands in 

cooling 

Muntz metal 60. 

Pewter, best 



Printing characters . . 

Sheathing metal 

Speculum " 

44 " 

Telescopic mirrors 

Tempert 

Type and stereotype 

pbites 

White metal 

44 " hard . 

Oreide 



24. 
2.5 

572 

25. 
6.4 



20. 
11.2 
22.3 
31. 



13. 

11.1 
31.2 



5.5 
19.3 
25.4 

5.6 



33.4 
25.4 
24 



56. 

66. 

50. 

66.6 

33.4 



7.4 
69. S 

73. 



1 See page 628 for directions. 



- 21. 
2.5 40. 



10.5 
14.3 

7.S 



3.4 

8?3 
10. 
10. 



25. 



2.9 

1.6 
20. 
10. 

7. 

7. 

1.4 

2.6 
10.1 
31. 
26.5 
12.5 



7.4 
25. S 

12.3 



1S.4 

23. 

20. 

12.5 

15.6 



19. 



13. 
31.6 



33.3 
31.6 
24. 



2.5 



22. 
29. 
33.4 
66.6 



2S.4 
4.4 
I Mngnesia . 



75. 



7.3 



25. 



•25. 



16.7 

1-1. 

20. 



15 5 
56. S 



8.3 



15.5 



2.4S 



12. 



2.5 



1.5 



2.6 
2.5 



12. 



- 4.4 Cream of tartar 6.5 
(Sal-ammoniac 2.5 Quick-lime 1.3 

f For adding small quantities of copper. 



ALLOYS AND COMPOSITIONS. 



629 









SolcLers. 








Copper. 


Tin. 


Lead. 


Zinc. 


Silver. 


Bis- 
muth. 


Gold. 


Calci- 
mine. 


Antimony. 


Tin 


— 


25 
58 


75 

16 


— 


— 


16 


- 


— 


_ 


ii 


10 


" coarse, 




melts at 500°.. 


— 


33 


67 


— 


— 








— 


— 


Tin, ordinary, 




















melts at 360°.. 





67 


33 


— 


— ■ 





— 


— 


— 


Spelter, soft 


50 


— 


— 


50 


— 





— • 


— 


■ — 


" hard .. . 


6T 





— 


33 











— 


— 


Lead 


13 
50 


33 


67 


~5 

50 


82 


- 


- 


- 










Brass or Copper. . 


— 


I ine brass 


47 


— 


— 


47 


6 





— 


— 


— 


Pewterers' or Soft 





33 


45 


— 





22 


— 


— 


— 


U It 





50 


25 


— 





25 


— 


— 


' — 


Gold 


4 

66 


— 


— 


34 


7 


— 


89 


z 





" hard 





" soft 


— 


66 


34 


— 


__ 


— 


— 


— 


— 


Silver, hard 


20 








— 


80 





— 


— 


— 


" soft 


12 





— 


— 


67 


— 





21 


- — 


Pewter 


66 
53 


40 
47 


20 


33 


' 


40 


- 


- 





Iron 


1 


Copper 





A Plastic Metallic Alloy.— See Journal of Franklin Institute, vol. xxxix., page 55, 
for its composition and manufacture. 

Composition for Welding Cast Steel. 

Borax, 10 parts ; Sal-ammoniac, 1 part. Grind or pound them roughly together ; 
fuse them in a -metal pot over a clear fire, continuing the heat until all spume has 
disappeared from the surface. When the liquid is clear, pour the composition out 
to cool and concrete, and grind to a fine powder ; then it is ready for use. 

To use this composition, the steel to be welded should he raised to a bright yel- 
low heat ; then dip it in the welding powder, and again raise it to a like heat as be- 
fore ; it is then ready to be submitted to the hammer. 



iF'u.si'ble Compounds. 



Compounds. 



Zinc. 


Tin. 


Lead. 





25 


25 


33.3 


. 


33.3 





19 


31 


— 


12 


25 



Rose's, fusing at 200° 

Fusing at less than 200° , 

Newton's, fusing at less than 212 c 
Fusing at 150° to 160° 



50 
33.4 
50 
50 



13 



Soldering Fluid for use with soft Solder. 

To 2 fluid oz. of Muriatic acid add small pieces of Zinc until bubbles cease to rise. 
Add % a teaspoonful of Sal-ammoniac and 2 fluid oz. of Water. 

By the application of this to Iron or Steel, they may be soldered without their sur- 
faces being previously tinned. 

Fluxes for Soldering or Welding. 

Iron Borax. Zinc Chloride of zinc. 

Tinned iron Resin. Lead Tallow or resin. 

Copper and Brass ... Sal-ammoniac. Lead and tin pipes. Resin and sweet oil. 

Steel— Sal-ammoniac, 1 part; Borax, 10 parts. Pound together, and fuse until 
clear, and, when cool, reduce to powder. 

Babbitt's Anti- attrition Metal. 

Melt 4 lbs. Copper; add, by degrees, 12 lbs. best Banca tin, 8 lbs. Regulus of anti- 
mony, and 12 lbs. more of Tin. After 4 or 5 lbs. Tin have been added, reduce the 
heat to a dull red, then add the remaiuder of the metal as above. 

This composition is termed hardening ; for lining, take 1 lb. of this hardening, 
melt with it 2 lbs. Banca tin, which produces the lining metal for use. Hence, the 
proportions for lining metal are 4 lbs. of copper, 8 of regulus of antimony, and 96 of tin. 

3 11 



630 U. S. MEASURES. MISCELLANEOUS ILLUSTRATIONS. 



EQUIVALENTS OF OLD AND NEW U. S. MEASURES. 



Length.. 

Meters. 

llnch = .02540005 
lFoot = .3048006 
lYard = .9144018 
1 Chain = 20.1168396 
1 Furlong = 201.168396 
IMile =1609.347168 



"Volume. 



1 Fluid Drachm 
1 Fluid Ounce 
1 Fluid Pound 
1 Gill 

1 Wine Pint 
1 Dry Quart 
1 Wine Quart 
1 Wine Gallon 



Liters.* 

= .0036967 
= .0295739 
=s .35488656 
= .1182955 
= .4731821 
=1.1012344 
= .9463642 
=3.7854579 



Surface. 

Square Meters. 

1 Inch = .000645161 
lFoot = .092903184 
1 Yard = .836128656 
IRod = 25.292891844 
1 Rood =1011.71567376 
1 Acre =4046.86269504 

"Weight. 

Grams. 

1 Grain = .0648004 

1 Scruple = 1.296008 

lDwt. = 1.5552096 

1 Drachm = 3.888024 
1 Ounce (Troy) = 31.104192 

1 Ouncef = 28.350175 

1 Pound =453.6028 

lTon =1016070.272 



Note. — A square Meter is 1549.9969 square inches, hut by Act of Congress it is 
declared to he 1550 square inches; hence the Liter (cubic decimeter)= 61.023 377 953 
cubic inches. In the Act of Congress, a Liter is declared to be 61.022 cubic inches, 
which is erroneous, as here shown, by the .001 -f- of an inch. 



MISCELLANEOUS ILLUSTRATIONS. 

1. The turret of an Iron-clad steamer has an internal diameter of 21 
feet, and is composed of 10 courses of plates, 1 inch in thickness, to which 
an allowance of half a sixteenth (^d part) of an inch must be made for 
space between each course. What is the diameter of the inner surface of 
the outer course in feet and the decimal of a foot ? 

21 feet-f-(10— 1X1+1) ins. +(10— 1x^X2) sixteenths=22 feet G ins. 9 sixteenths. 
Hence, to reduce this to feet and the decimal of a foot, 
Feet. Inches. Sixteenths. 

22 6 9 

12 



270 
16 
1629 
270 
4329 



and 16)4329 



12)270.5625 



22.546875 /e*rf. 

2. How man}- changes may be rung with 4 bells out of 8 ? 
Operation.— S X 7 X 6 X 5 = 1680 changes. 

3. How many changes are there in the throws of 5 dice ? 
Operation.— 6x6x6x0x6 = 7776 changes. 

4. It is required to lay out a tract of land in form of a square, to be in- 
closed with a post and rail fence, 5 rails high, and each rod of fence to 
contain 10 rails. What must be the side of this square to contain just as 
many acres as there are rails in the fence ? 

Operation.—], mile = 320 rods. Then 320x320 -f- 160, the square rods in an acre 
= 640 acres ; and 320 X 4 sidesX 10 rails = 12 800 rails per mile. 

Then, as 640 acres : 12 S00 rails : : 12 S00 acres : 256 000 rails, which will inclose 
256000 acres, and V 256 000X69.5701, the number of yards in the side of a square 
acre and -4- 1760, yards in a mile = 20 miles. 



* 61.02* cubic inches. 



f AvoirdnpoiB. 



MISCELLANEOUS ILLUSTRATIONS. 631 

5. A invested in a Company $150 for 14 months, B a certain sum for 
one year, and C $100 for a certain time ; some time afterward they ascer- 
tained that their stock and profits were equal to $475, of which sum A was 
credited $195, B $153, and C $127 ; how much did B invest, and for how 
long a time was C's stock invested ? 

Operation.— A was credited $195, and he invested $150; hence 195 — 150 = 45, 
his profit in 14 months. 

Then, as 14 months : 45 : : 12 months : 38.5T, profit on $150 in 12 months. 
And as 188.5T (150 + 38.57) : 150 : : 153 (B's stock) : 121.7, B's investment, 
C was credited $127. Then (127 — 100) = 27, C's profit. 
And as 150 : 45 : : 100 : 30, C's ratio of profit for 14 months. 
Therefore, as 30 : 14 : : 27 : 12.6 months, the time C's stock was invested. 

6. How many fifteens can be counted with four fives ? 

4X3X2X1 24 . 
Operation. ____ = _ = 4 

7. Assume there are 4 companies, in each of which there are 9 men ; 
it is required to ascertain how many ways 4 men may be chosen, one out 
of each company. 

Operation. 9x9x9x9 = 6561. 

8. What are the chances in favor of throwing one point with three dice ? 
Operation. — Assume a bet to be upon the ace. Then there will be 6x6x6 = 216 

different ways which the dice may present themselves, that is, with and without an ace. 

Then, if the ace side of the die is excluded, there will be 5 sides left, and 5x5x5 
= 125 ways without the ace. 

Therefore there will remain only 216 — 125 = 91 ways in which there could be an 
ace. The chance, then, in favor of the ace is as 91 to 125 ; that is, out of 216 throws, 
the probability is that it will come up 91 times, and lose 125 times. 

9. If a hodj should move through the length of 1 barley-corn in a sec- 
ond of time, one inch in the second second, three inches in the third, and 
so continue increasing its motion in triple geometrical proportion, how 
man} r yards would it advance in half a minute ? 

Operation. 012345 6 7 8 9 

1 3 9 27 81 243 729 2187 6561 19683 

Then 9 + 9 + 9 + 3 = 30, the number of seconds or terms. 

And 19 683 X 19 6S3 X 19 6S3 X 27 = 205 S91 132 094 649 = 30th power, -which -4- 3^1, 
the ratio less 1 = 102 945 566 047 324 barley-corns, and again by 3, 12, and 3, to re- 
duce it to yards = 953 199 685623 yards lfoot 1 inch and 1 barley-corn. 

10. A person expended $100 for 100 head of live-stock, consisting of 
cows, sheep, and pigs ; for the cows he paid $10 per head, for the sheep 
$1, and for the pigs 50 cents. How man}' did he purchase of each kind ? 

Operation.— The average cost of the stock was $1. Hence, the cost of a pig was 
50 cents below the average, and that of a cow $9 above. 

Therefore, if he had purchased cows and pigs only ; in the inverse ratio of their 
cost, he would have had 5 cows and 90 pigs = 95 animals costing $95. Then, by 
adding 5 sheep, at $1 per head, he had 

5 cows, at $10. = $50 

90 pigs at .50= 45 

5 sheep, at 1. = 5 

100 $100 

11. The hour and minute hand of a clock are exactly together at 12; 
when are they next together ? 

Operation — As the minute hand runs 11 times faster than the hour hand, then, 
11 : 60 : : 1 : 5 min. 27^- sec. The time, then, is 5 mm. 27f T sec. past 1 o'clock. 

12. The time of the day is between 4 and 5, and the hour and minute 
hands are exactly together ; what is the time ? 

Operation.— The difference of the speed of the hands is as 1 to 12 = 11. 

4 hours X 60 = 240, which -h 11= 21 min. 49.09 sec, which is to be added to 4 hours. 



632 MISCELLANEOUS ILLUSTRATIONS. 

13. Assume a cubic inch of glass to weigh 1.49 ounces troj T , the same of 
sea- water .59, and of brandy .53. A gallon of this liquor in a glass bot- 
tle, which weighs 3.84 lbs., is thrown into sea-water. It is proposed to 
determine if it will sink, and, if so, how much force will just buo} T it up ? 

Operation. — 3. 84x12 -=-1.49 = 30.92 cubic inches of glass in the bottle. 

231 cubic inches in a gallon X- 53 = 122. 43 ounces of brandy. 

Then, bottle and brandy weigh 3.84x12 -(-122.43 ounces = 168.51 ounces, and 
contain 261.92 cubic inches, whichx.59 = 154.63 ounces, the weight of an equal bulk 
of sea-water. 

And, 168.51 — 154.53 z= 13.98 ounces, the weight necessary to support it in the 
water. 

14. Three men, viz., A, B, and C, drink a quantity of wine ; A can drink 
it b} T himself in 12 daj-s, B in 10, and C in 15, when the days are 12 hours 
long. In what time" can they drink out the whole, drinking together, 
when the days are 10 hours long, and what will be each one's share? 

Operation.— -If A can drink it in 12 days, he can drink ^ °f it; in 1 day ; and, for 
like reason, B can drink ^ of it in 1 day, and C ^ of it in 1 day. Therefore, 
i2~^io ~^"i5~iro ~i' Me quantity they unitedly will drink in 1 day. Conse- 
quently, if they drink }£ of it in 1 day, they will drink the whole of it in 4 days of 
12 hours each, and 4xl2-M0 = 4.S days o/lO hours. 

Again : their ability to drink being represented by £j, ^b" , and ^, their share 
of drinking will be Jg- . J§ . Jf = .833 for A, 1. for B, and .666 for C. 

15. A fountain has 4 supply cocks, A, B, C, and D, and under it is a 
cistern, which can be filled b) T the cock A in 6 hours, by B in 8 hours, by 
C in 10, and by D in 12 hours ; now, the cistern has 4 holes, designated 
E, F, G, and lit, and it can be emptied through E in 6 hours, F in 5 hours, 
G in 4 hours, and H in 3 hours. Suppose the cistern to be full of water, 
and that all the cocks and holes were opened together, In what time 
would the cistern be emptied ? 

Operation.— Assume the cistern to hold 120 gallons. 



ho. 


gall. 


ho. 


gall. 


ho. 


gall. 


ho. 


gall. 


If 6 


120 : 


: 1 


20 at A. 


6 


120 : 


: 1 


: 20 at E. 


8 


120 : 


: 1 


: 15 at B. 


5 


120 : 


: 1 


: 24 at F. 


10 


120 : 


: 1 


: 12 at C. 


4 


120 : 


: 1 


30 at G. 


12 


120 : 


: 1 


10 at D. 


3 


120 : 


: 1 


40 at H. 



Run in in 1 hour, 57 gallons. Run out in 1 hour, 114 galloiis. 

5T 

Run out in one hour more than run in, 57 gallons. 
Then, as 57 gallons : 1 hour : : 120 gallons : 2.15S -f- hours. 

16. A cistern, containing 60 gallons of water, has 3 unequal cocks for 
discharging it ; one will empty it in 1 hour, a second in 2 hours, and a 
third in 3 hours ; in what time will it be emptied if they are all opened 
together ? 

Operation. — First, X would run out in 1 hour by the second cock, and % by the 
third; consequently, by the 3 was the reservoir supplied one hour. >£-|->£-{-l = 
& -f-* -f- J, being reduced to a common denominator, the sum of these 3 —^ ; whence 
the proportion, 11 : 60 : : 6 : 32^- minutes. 

17. A reservoir-has two cocks, through which it is supplied ; by one of 
them it will fill in 40 minutes, and by the other in 50 minutes ; it has also 
a discharging cock, by which, when full, it may be emptied in 25 minutes. 
If the three cocks are left open, in what time would the cistern be filled, 
assuming the velocity of the water to be uniform ? 

Operation. — The least common multiple of 40, 50, and 25, i3 200. 

Then, the 1st cock will fill it 5 times in 200 minutes, and the 2d, 4 times in 200 
minutes, or both, 9 times in 200 minutes; and, as the discharge cock will empty it 
8 times in 200 minutes, hence 9 — - S =: 1, or once in 200 minutes =3.2 hours. 



MISCELLANEOUS ILLUSTRATIONS. 633 

18. Out of a pipe of wine containing 84 gallons, 10 were drawn off, and 
the vessel refilled with water, after which 10 gallons of the mixture were 
drawn off, and then 10 more of water were poured in, and so on for a third 
and fourth time. It is required to compute how much pure wine remained 
in the vessel, supposing the two fluids to have been thoroughly mixed. 

Operation.— 84 — 10 = 74, the quantity after the 1st draught. 

Then, 84 : 10 : : 74 : 8.8095, and 74 — 8.8095 = 65.1905, the quantity after the 2d 
draught. 

84 : 10 : : 65.1905 : 7.7608, and 65.1905 — 7.760S = 57.4297, the quantity after the 
3d draught. 

84 : 10 :: 57.4297 : 6.8367, and 57.4297 — 6.8367 = 50.593, the quantity after the 
Uh draught, which is the result required. 

19. A reservoir having a capacity of 10 000 cubic feet, has an influx 
of 750 and a discharge of 1000 cubic feet per day. In what time will it 
be emptied ? 

10000 
Operation. _____ = 40 da^. 

Contrariwise : The discharge being 1000 and the influx 1250 cubic feet per hour. 
In what time will it be filled ? 

Operation. — -r — — — — 40 hours _ 1 day 16 hours. 

20. A son asked his father how old he was. His father answered him 
thus : If you take away 5 from my years, and divide the remainder bj*- 8, 
the quotient will be % of your age ; but if } r ou add 2 to j r our age, and 
multiply the whole by 3, and then subtract 7 from the product, you will 
have the number of years of my age. What were the ages of father and son ? 

Operation. — Assume the father's age 37. 

Then 37 — 5 = 32, and 32 -f- 8 = 4, and 4 X 3 = 12, son's age. Again : 12 + 2 = 14, 
and 14X 3 = 42, and 42 — 7 _ 35. Therefore 37 — 35 = 2, error too little. 

Again : assume the father's age 45 ; then 45 — 5 = 40, and 40 -r- 8 = 5. Therefore 
5x3 = 15, son's age. Again : 15 + 2 = 17, and 17x3 = 51, and 51 — 7—44. There- 
fore 45 — 44=1, error too little. 

Hence (45 sup. X 2 error)— (37 sup. X 1 error)=90 — 37 = 53, and 2 — 1 =1. 

Consequently, 53 is the father's age. Then 53 — 5 = 4S, and 4S-^8 = 6 = % of 
the son's age, and 6X3 = 18 years, the son's age. 

21. Two companions have a parcel of guineas. Said A to B, if }^ou will 
give me one of your guineas I shall have as many as you have left. B 
replied, if you will give me one of your guineas I shall have twice as many 
as you will have left. How many guineas had each of them ? 

Operation. — Assume B had 6. 

Then A would have had 4, for 6 — 1 = 4 -f- 1 = 5. Again : 4 (A's parcel) — 1 = 3, 
and 6 + 1 = 7, and 3x2 = 6. Therefore 7 — 6 = 1, error too little. 

Again : assume B had 8. 

Then A would have 6, for 8 — 1 = 6 + 1 = 7. Again : 6 (A's parcel) — 1 = 5, and 
S + 1 = 9, and 5x 2 = 10. Therefore 10 — 9 = 1, error too great. 

Hence 8x1 = 8, and 6x1 = 6. Then 8 + 6=14, and 1 + 1=2. Whence, di- 
viding the products by sum of the errors, 14 -r- 2 = 7 = B's parcel, and 7 — 1=5 +1 
= 6 for A when he had received 1 of B ; also 5 — 1X2 = 7 + 1 = 8 = B's parcel when 
he had received 1 of A. 

22. If a traveller leaves New York at 8 o'clock in the morning, and 
walks toward New London at the rate of 3 miles per hour, without inter- 
mission ; and another traveller starts from New London at 4 o'clock the 
same evening, and walks toward New York at the rate of 4 miles per hour 
continuously ; assuming the distance between the two cities to be 130 
miles, whereabouts upon the road will they meet? 

Operation. — From 8 to 4 o'clock in the morning is 8 hours; therefore, 8x3 = 24 
miles, performed by A before B set out from New London ; and, consequently, 130 — 
24 = 106 are the miles to be travelled between them after that. 

Hence, as (3 + 4) 7 : 3 : : 106 : 2i$ — 45^ more mi i es travelled by A at the meet- 
ing ; consequently, 24+35^ — 693, miles from New York is the place of their meeting. 

3 11* 



634 



MISCELLANEOUS ILLUSTRATIONS- 



23. There are two casks of equal capacity, the one % full of wine, the 
other % full of water; now, assume the cask containing the wine to be 
first filled from the water-cask, and then the water-cask to be filled from 
the wine-cask, and so on, alternately filling the one from the other. As- 
suming that the fluids mix uniformly at each time, how much wine and 
how much water will the wine-cask contain, and how much water and how 
much wine will the water-cask contain, after each have been filled 5 times ? 

Operation. — First, the wine-cask being filled from the water, it will contain J of 
wine and i of water. 

Second, the water-cask is filled from the wine-cask by drawing from it J ; that is, 
f °/f of wine = ^ of wine, and j o/J of water — g- of water; the water-cask con- 
tains, therefore, g of water and 4 of wine, and there remains in the wine^cask J 
of wine and ^ of water. 

Again : the wine-cask is filled, and contains 3^ of wine and i3. of water, and 
there will remain in the water-cask -^ of water and -^ of wine. 

The Denominators of the fractions expressing the proportion of wine and water 
are the successive powers of 3, and the Numerators are ascertained by dividing the 
denominators into two parts, differing from each other by a unit. 
Thus, after the 1st filling, the wine-cask has been filled once, and the quantity 
i(3H-l) 2 &( 3i-l) 1 
— = -, and the water is — = ^ 



of wine is - 



in it is 



After the 2d filling, the water-cask has been filled once, and the quantity of wine 
a (B2_l) 4 £(32 + 1) 5 
= -, and the water is 3 — - = -. 

After the 3d filling, the wine -cask has been filled twice, and the quantity of 

£(33+D I 4 , . *(3 3 — 1) 13 
wine is - — — ■ = — , and the water is - — — =s*i etc., etc. 

O -I O Jit 



Thus, 



Water-cask 
filled 1st time. 



2d time. 



3d time. 



36 5 
729 



4th time, 3.28.1 

6501 



JL&8JL 

19 83 



5th time, fffff 



Wine. 



40 
81 



JUL 
243 



364 

729 



3280 
19084 



m 



^524 
9 49 



Wine-cask 
filled 1st time. 



Jf 2d time. 



14 
81 



3d time. 



123 

729 



jUlff 4th time. 



1094 
G561 



24. If 9 men or 15 women eat 17 apples in 5 hours, and 15 men and 9 
women can eat 47 apples of like size in 12 hours, the apples growing uni- 
formly, how maivy boys can eat 3G0 apples in 60 hours, assuming that 120 
boys can eat as man}- as 18 men and 26 women ? 



MISCELLANEOUS ILLUSTRATIONS. 635 

Operation— If 9 men = 15 women, 1 man = 3^5 or £ women, and 15 men = 15xf 
= 25 women. Therefore 15 men and 9 women = 25 -j- 9 = 34 women. 

If 15 women eat IT apples in 5 hours, 34 women will eat Jf of 17 = W = 38^ 
apples in the same time ; and if they eat them in 5 hours, they will eat in 12 hours 
^ of 3S T 8 ^ = 92|J apples, provided the quantity is uniform. But as they are grow- 
ing whilst being eaten, 92 JJ — 47 = 45JJ. apples, the growth of them in 12 — 5 = I 
hours. 

Now, if the growth of 47 apples in 7 hours = 45j§, the growth of 360 in the 
same tirae = 3fLG of 45|§=81||^ = 34S^- apples, and in 55 hours (60—5), the 
growth of 360 will be & of 348 $fe = *%%^ = 2737$& apples. 

Hence, 360 + 2737j3 8 2 6 g = 3097g 3 ^ apples to be eaten in 60 hours. 

Again : if 15 women eat 17 apples in 5 hours, 1 woman will eat Jg of 17 apples in 
the same time, and in 60 hours she will eat 12 times as many = 12 X J- J = e 5 8 = 13§ 
apples ; and if 1 woman eat 13| apples in 60 hours, 3097^ -4- % 8 - = 3097^ X ^ 
— 5fi|4||5 _ 227111^ women to eat the apples. 

If 9 men = 15 women, 18 men and 26 woinen = ig 8 xl5 = 30 women, and 30-f-26 
= 56 women. 

Finally, if 56 women are equivalent to 120 hoys, 227 J||f \ women = 227 |||^J 

25. There is a fish, the head of which is 9 inches long, the tail as long 
as the head and half the body, and the body as long as both the head and 
tail. Required the length of the fish. 

Operation— Assume the body to be 24 inches in length. Then 24-=- 2-f 9 = 21, 
the length of the tail. 

Hence 2i 4- 9 = 30, the length of the body, which is 6 inches too great. 

Again : assume the body to be 26 inches in length. Then 26-^-2 -|- 9=22, the length 
of the tail. Hence 22 -f-9 — 31, the length of the body, which is 5 inches too great. 

Therefore, by Double Position, divide difference of products (see rule) by differ- 
ence of errors (the errors being alike), 26x6 — 24x5 = 36=^ ifference of products, 
and 6 — 5 = 1 = difference of errors. 

Consequently, 36 -4- 1 = 36, the length of the body, and 36 -4- 2 -f- 9 = 27, the length 
of the tail, and 36 -4- 27 -|- 9 = 72 inches, the length required. 

26. A hare, 50 leaps before a greyhound, takes 4 leaps to the grey- 
hound's 3, but 2 leaps of the hound are equal to 3 of the hare's. How 
many leaps must the greyhound take before he can catch the hare ? 

Operation As 2 leaps of the greyhound equal 3 of the hare, it follows that 6 of 

the greyhound equals 9 of the hare. 

Whilst the greyhound takes 6 leaps, the hare takes 8 ; therefore, while the hare 
takes 8, the greyhound gains upon her 1. 

Hence, to gain 50 leaps, she must take 50xS = 400 leaps; but, whilst the hare 
takes 400 leaps, the greyhound would take 300, since the number of leaps taken by 
them are as 4 to 3. 

27. If from a cask of wine a tenth part is drawn out and then it is filled 
with water; after which a tenth part of the mixture is drawn out; again 
is filled, and again a tenth part of the mixture is drawn out : now, assume 
the fluids to mix uniformly at each time the cask is replenished, what 
fractional part of wine will remain after the process of drawing out and 
replenishing has been repeated ten times ? 

Operation. — Since .1 of the wine is drawn out at the first drawing, there must re- 
main .9. After the cask is filled with water, .1 of the whole being drawn out, there 
will remain .9 of the mixture ; but .9 of the mixture is wine; therefore, after the sec- 

9a 
ond drawing, there will remain .9 o/.9 of wine, or — — ; and after the third drawing, 

93 10 2 

there will remain .9 of .9 of .9 of wine, or — — . 

Hence, the part of wine remaining is expressed by the ratio .9, raised to a power 
the exponent of which is the number of times the cask has been drawn from. 

910 
Therefore, the fractional part of wine is -— = .34867S440J 



(336 MISCELLANEOUS ILLUSTRATIONS. 

28. If a basket and 1000 eggs were laid in a right line 6 feet apart, and 
10 men (designated alphabetically from A to J) were to start from the 
basket and to run alternately, collect the eggs singly, and place them in 
the basket as collected, and each man to collect but 10 eggs in his turn, 
how many yards would each man have to run over, and what would be 
the entire" distance run over? 

Operation. — A'a course would be Ox^feet (first term) + 10x6x2 feet (last term) 
= lo2 == sum of first and last terms of the progression. 

Then 132 ?j- 2 x 10 = 660 feet = number of times X half the sum of the extremes =. 
the sum of all the terms, or the distance run by A in his first turn. 

B'3 course would be 11x6x2 = 132 feet (first term) + 20x0x2 = 240 feet (last 
term) = 372 = sum of first and last terms. 

Then 372-7-2x10 = lS6 = sMm of all the times, or B' a first turn. 

A's last course would be 90LX6X2 = 10 812 feet for the first term, aud 910X6x2 
= 10 920 feet for the last term of his last turn. 

Then 10 812 + 10 920 -t- 2 x10 = 1^8 660 = sum of the terms, or distance run. 

B's last course would 'be 911x6x2 = 10 932 feet for the first term, and K20Xux2 
= 11 040 feet for the last term of his last turn. 

Then 10932 + 11040 -7-2x10 = 109 860 = sum of the terms or distance run. 

Therefore, if A's first and last runs = 660 and 10S 660 fee t, and the number of 
terms 10, then, by Progression, the sum of all the terms = 546 600/e^. 

And if B's first and last runs = 1S60 and 100 S60 feet, and the number of terms 
10, then the sum of all the terms =558 $00 feet. 

Consequently, 55S600 — 546 600 = 12 000 = the common difference of the runs, 
which, being added to each man's run = the sum of all the runs, or the entire dis- 
tance run over. 

A's run, 546 600= 182 200 yds. Fs run, 606 600 = 202 200 yds. 

B's " 558 600 = 186 200 " G's " 618 600 = 206 200 " 

C's " 570 600 = 190 200 " H's " 630 600 = 210 200 " 
I's " 642 600 = 214 200 " 
J's " 654 600 = 21S 200 " 



D\s " 582 600=194 200 
E's " 594 600 = 198 200 



6 006 000 feet, which-7-52S0 = 1137.5 miles. 

29. If, in a pair of scales, a body weighs 90 lbs. in one scale, and but 
40 lbs. in the other, what is the true weight? 

V (40X90) =60 lbs. 

30. If a steam-boat, running uniformly at the rate of 15 miles per hour 
through the water, were to run for 1 hour with a current of 5 miles per 
hour, then to return against that current, what length of time would she 
require to reach the place from whence she started ? 

Operation.— 15 + 5= 20 miles, the distance run during the hour. 

Then 15—5 = 10 miles is her effective velocity per hour when returning, and 
20-=-10 = 2 hours, the time of returning, and 2 + 1 = 3 hours, or the whole time oc- 
cupied. 

Or, let d represent the distance in one direction, t and V the greater and less times 
of running (in hours), and c the current or tide. 

j±£ 

o (J vXt' 

Then, = velocity of boat through the water, and - — - = c. 

' txt' J t' 

31. The flood-tide wave in a given river runs 20 miles per hour, the cur- 
rent of it is 3 miles per hour. Assume the air to be quiescent, and a float- 
ing body set free at the commencement of the flow of the tide ; how long 
will it drift in one direction, the tide flowing for 6 hours from each point 
of the river? 

Operation. — Let x be the time required; 20# = distance the tide has run up, to- 
gi ther with the distance which the floating body has moved; 3a: = whole distance 
which the body has floated. 

Then 20a? — 3a: = 6x20, or the length in miles of a tide. 
20 

x = X 6 = 7 hours, 3 minutes, 31.7C47 seconds. 



MISCELLANEOUS ILLUSTRATIONS. 637 

32. A steam-boat, running at the rate of 10 miles per hour through the 
water, descends a river, the velocity of which is 4 miles per hour, and re- 
turns in 10 hours ; how far did she proceed ? 

Operation.— Let a? = distance required, ^ — time of going, 10 _ 4: z=time °f 

returning. 
Then, — + - — 10; 6x -f 14a? = S40 ; 20ar = 840; 840 -^ 20 = 42 miles. 

83. From Caldwell's to Newburg is 18 miles ; the current of the river is 
such as to accelerate a boat descending, or retard one ascending \% miles 
per hour. Suppose two boats, running uniformly at the rate of 15 miles 
per hour through the water, were to start one from each place at the same 
time, where will they meet ? 

Operation.—Let a? = the distance from N to the place of meeting; its distance 
from C, then, will be 18 — x. 

Speed of descending boat, 15 -f 1.5 = 16.5 miles per hour. . 

Speed of ascending boat, 15 — 1.5 = 13. 5 miles per hour. 

-?— — time of boat descending to point of meeting. 
10.5 

18 x 

- = time of boat ascending to point of meeting. 

13.5 

x 18 — x 
These times are of course equal; therefore, — — =— -r-^-. 

16.5 13.5 

Then, 13. 5a? = 297 — 16.5a?, and 13.5a? -f 16.5a? = 297, or 30a? = 297. 

297 
Hence x = •— - = 9. 9 miles the distance from Newburg. 

34. There is an island 73 miles in circumference ; 3 men start together 
to walk around it and in the same direction : A walks 5 miles per day, 
B 8, and C 10 ; when will they all come aside of each other again ? 

Operation. — It is evident that A and C will be together every round gone by A ; 
hence it remains to ascertain when A and B will he in conjunction at an even round, 
as 3 miles are gained every day hy B. Therefore, as 3 : 1 : : 73 : 24.33-)- ; but, as 
the conjunction is a fractional number, it is necessary to ascertain what number of a 
multiplier will make the division a whole number. 

73 -4- 24.33 -f- = 3, the number of days required in which A will go round 5 times, 
B 8, and C 10 times. 

35. Assume a cow, at the age of two years, to bring forth a cow-calf, and 
then to continue yearty to do the same, and every one of her produce to 
bring forth a cow-calf at the age of two years, and yearly afterward in 
like manner ; how many would spring from the cow and her produce in 
40 } r ears ? 

Operation. — The increase in the first year would be 0, in the second year 1, in the 
third 1, in the fourth 2, in the fifth 3, in the sixth 5, and so on to 40 years or terms, 
each term being = the sum of the two preceding ones. The last term, then, will be 
165 580 141 , from which is to be subtracted 1 for the parent cow, and the remainder, 
165 580 140, will represent the increase required. 

36. The interior dimensions of a box are required to be in the propor- 
tions of 2, 3, and 5, and to contain a volume of 1000 cubic inches ; what 
should be the dimensions ? 

I /1000 X 23 „ AO . , /1000 X 33 Q . . /1000X5J 

= 16 ins. 

And what for a box of one half the volume, or 500 cubic inches, and 
retaining the same proportionate dimensions ? 
30 

Operation.— -2 x 3 X 5 = 30, and — = 15. 



G37* 



MISCELLANEOUS ILLUSTRATIONS. 



37. The chances of events or games being equal, what are the odds for 
or against the following results ? 



Five Events. 



Against. 



Four Events. 



Against. 



31 to 1 All the 5 1 out of 5 

4% to 1 4 out of 5 2 out of 5 
5 to 3 in favor of the 5 events result- 
ing 3 and 2. 



15 tol 

2K tol 



All the 4 
3 out of 4 



1 out of 4 

2 out of 4 



Three Events. 



5 to 3 against 2 events only, or that 
the 4 events do not result 2 and 2. 

Two Events. 



Odds. 



Against. 



In favor. 



7 tol 
Even 



All the 3 
(2 or all out 
\ of 3 

3 to 1 in favor of the 3 events result- 
ing 2 and 1. 



1 out of 3 

2 or all out 
of 3 



3 tol 
Even 



1 out of 2 
(1 only out 
\ of 2 
Even that the events result 1 and 1. 



Both events 
only out 
"2 



7lou 
\ of 2 



38. Required the chances or probabilities in events or games, when the 
chances or probabilities of the results, or the players, are equal. 



Events or 


That a named event, 


Against a named 
event, or player, 


Against either event, 
or player (not named), 


Against each event, 

or player, occurring 

or winning an equal 

number of times. 


Games. 


wins a majority or 
more of times. 


occurring or winning 

an exact majority 

of times. 


occurring or winning 

an exact majority 

of times. 


21 


Even 


5 tol 


2 tol 




20 


1.33 to 1 






4.66 to 1 


19 


Even 


4.5 to 1 


l.S to 1 




18 


1.55 to 1 






4.4 to 1 


17 


Even 


4.4 to 1 


1.7 to 1 




16 


1.5 tol 






4.1 to 1 


15 


Even 


4 tol 


1.5 tol 




14 


1.5 tol 






3.S to 1 


IS 


Even 


3.7 to 1 


1.4 to 1 




12 


1.6 tol 






3.44 to 1 


11 


Even 


3.4 to 1 


1.2 to 1 




10 


1.7 to 1 






3.06 to 1 


9 


Even 


3 tol 


1.03 tol 




8 


1.75 to 1 






2.66 to 1 


7 


Even 


2.7 to 1 


.8 tol 




6 


2 to 1 






2.2 to 1 


5 


Even 


2.2 to 1 


.6 to 1 




4 


2.2 tol 






1.66 tol 


3 


Even 


1.66 to 1 


.33 to 1 




2 


3 tol 






Even 



The chances of consecutive events or results are as follows : 

11. 2047 to 1. 9. 511 to 1. 7. 127 to 1. 

10. 1023 to 1. 8. 255 to 1. 6. 63 to 1. 

Hence it will be observed that the chances increase with the number of events 
very nearly in a duplicate ra'io. 

Illustration — The chances of 11 consecutive events compared with 10, are as 
2017 to 1023, or 2 to 1. 

39. Required the chances or probabilities of events or results in a given 
number of times. 

The numerator of a fraction expresses the chance or probability either for the 
result or event to occur or fail, and the denominator all the chances or probabilities 
both for it to occur or fail. 

Thus, in a given number of events or games, if the chances are even, the proba- 

112 3 

bility of any particular result is as = _; ; , etc, being 1 out of 2, 

1-1-1 2 2 + 23 + 3 
2 out of 4, etc., or even. 



MISCELLANEOUS ILLUSTRATIONS. 637$* 

If the number of events or games are 3, then the probability of any par- 
ticular result, as 2 and 1, or 1 and 2, is determined as follows : 

The number of permutations of 3 events are 1x2x3 = 6, which represents the 
number of times that the number of events can occur, 2 and 1, or 1 and 2, to which 
is to be added the 2 times or chances they can occur all in one way or the reverse 
thereto. 

Hence, — - = — -, or 3 to 1 in favor of the result ; and the probability 

'2 + 6 4 4 — 3 1 F 

of one party naming or winning two precise events or results, as winning 2 out of 
the 3, is determined as follows : The number of permutations and chances, as before 

shown, are 8. Hence, the number of his chances being 3, = - = = _, or 

' ' & v 3 + 5 8 8 — 3 5 

3 to 5 in favor of the result; and the probability of one party naming or winning 
all, or the 3 events or results, is determined as follows : The number of permutations 
and chances being also, as before shown, 8. Hence, as there is but one chance of 

such a result, = - = = -, or 1 to 7 in favor of the result. 

'l + 7 8 8 — 1 7 

If the number of events, etc., are 4, then the probability of any par- 
ticular result, as 2 and 2, or of winning 2 or more of them, is determined 
as follows : 

The number of permutations and chances of 4 events are 16. Hence, as the number 

of chances of such a result are 11, — = — = = — , or as 11 to 5 in favor 

5 + 11 16 10—11 5 

of the result, and that the results do not occur precisely 2 and 2. The number of 

10 5 5 5 

chances of such a result being 10, == - = = -, or 5 to 3 against it. 

6 + 10 8 • S — 5. 3 1 & 

If the number of events, etc., are 5, then the probability of any par- 
ticular result, as 3 and 2, is determined as follows : 
The number of permutations and chances being 32, and the number of chances of 

i i* *, • on 20 10 10 10 5 K ' . . ] 

such a result being 20, = — = = — =_, or as 5 to 3 m favor of the 

12+20 16 16—10 6 3 
result; and that it may occur precisely 3 out of 5, the number of chances are 

10 10 5 5 ■" 5 oj , h . . .. 

-, or 11 to 5 against it. 



10 + 22 32 16 16 — 5 11 

40. What is the dilatation of the iron in a railway track per mile, be- 
tween the temperatures of — 20° and +130° ? 

Operation.— — 20° + 130° = 150°. The dilatation of wrought iron (as per table, 
page 530) is, from 32° to 212° = 180° = .0012575 times its length. 

Hence, as 180 : 150 : : .0012575 : .0010479 = Mm ™ of 52S0 (feet in a mile) = 5.53 

feet per mile. 



638 DETAILS OF AMERICAN AM> E.NGLI&H STEAMERS. 



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MARINE ENGINES. — SIDE WHEELS. 639 

IN" aval Steamer (Wood). 
w Powhatan," U. S. Navy — Inclined Engines. — Length upon. deck, 251.5 feet; be- 
tween perpendiculars, 250 feet; keel, 246 feet ; beam, 45 feet; hold, 2Q. 5 feet. 

Immersed Section at load-line, 675 square feet. 
Displacement 3600 tons, at load-draught of 18.5 feet. 

Cylinders Two, of TO ins. in diam. by 10 feet stroke of piston; volume of piston 

space, 534.5 cubic feet. Condensers. — Two, volume 190 cubic feet. Air-pumps. — 
Two, of 52>£ ins. in diam. by 42 ins. stroke of piston ; volun*e 103 cubic feet. Feed- 
pumps. — Four, of 8 ins. in diam. ; volume, 4.8 cubic feet. 

Water-ioheel Shafts.— Journals, 18.3 and 13 ins. in diam., and 20 and 15 ins. in 
length. Water-wheels.— Diam. 31 feet. Arms, 23. Blades (divided), 23; breadth 
of do., 10 feet ; depth of do., 26 ins. Dip at load-line, 5.5 feet. 

Boilers.— Four (vertical tubular). Heating surface, 12 000 square feet. Grates, 
338 square feet. Steam-room, 2980 cubic feet. Cross area of tubes, 53.5 square feet. 

Smoke-pipe. — Area 63.6 square feet, and 65 feet in height above the grate level. 

Pressure of Steam.— 10 lbs. per square inch, cut off at % the stroke of the piston, 
throttle X open. Revolutions, 13.5 per minute. Indicated Horses' Power, 1100. 

Fuel Bituminous coal, with a natural draught. Consumption, at load-line, mod- 
erate sea, and at pressure of 12 lbs. and 12 revolutions, 3950 lbs. per hour = 42.3 
tons per day. 

Speed, 10 knots per hour. Coal-bunker&, 800 tons capacity. 

Slip of Wheels from Centre of Pressure, 18.75 per cent. 

Note. — An average pressure of 17 lbs. and 16 revolutions have been obtained for 
70 hours, vessel drawing 17.75 feet, giving a speed of 12.6 knots per hour. 

Hull— Launching draught, 10.62 feet; displacement, 1585 tons. Angles of en- 
trance at 17.5 feet, 4S° ; at 18 feet, 51° 20' ; at 19.5 feet, 54° 40'. 

Centre of Displacement— In the vertical plane of the centre of the water-line, 
and at 18.5 feet draught, 8.86 feet below the water-line. Average per Inch. — From 
18 to 19 feet draught, 22.09 tons. Meta Centre. — Above centre of gravity of displace- 
ment 10.87 feet. 

Rig.— Full bark. Armament.— 3 10-inch chambered guns upon pivots, and 6 
8-inch chambered guns upon carriages. 

"Weight. Wheels. 

Cast iron 21 634 lbs. 1 Fitters and patterns . . . 2S9 days. 

Wrought iron 75S65 u | Planing 22 550 sq. ins. 

"Weight. Coal-Bunkers. 
Iron, brass, and copper. . . 117 367 lbs. | Fitters and laborers 150 days. 

Day's work (boilers, water-wheels, and coal-bunkers not included, as th^y were 
made by the pound, and omitting extra pieces and hoisting engine), 54 498. Turn- 
ing, boring, and planing, 742 348 square inches. 

"Weight. ^Boilers. 



Brass tubes 33 896 lbs. 

Cast-iron 12 000 " 

Iron 209410 " 



Smoke-pipe 23 978 lbs, 

Grate-bars, etc., of cast-iron 27 711 " 
Valves, cocks, etc., brass . . 8 531 " 



Total 315 526 

WEIGHTS OF ENGINES, BOILEKS, ETC. (finished). 



Engines, frames, flooring, etc. 491 282 lbs. 

Extra pieces 56135 " 

Boilers (iron and brass) 255 306 " 

Appurtenances 60 220 u 



Coal-bunkers, deck-plates . 117 367 lbs. 

Hoisting engine and boiler 4 569 u 

Water-wheels 97 499 " 

Water in boilers 189 000 " 



Total (567 tons) 1 271 378 

Cost. Engines. 



Cast iron 262 011 lbs. 

Copper 8 084 " 

Steel 996 " 

Wrought iron 171 761 " 



Fitters and patterns . . 39 824 days. 

Laborers 11 26S u 

Planing. 133 757 sq. ins. 

Turning and boring . . 545 515 " 



The cost of the engines, boilers, etc., etc., complete (1S49-51) was $135 pox ton^ 
C. II. (O. M.), and the cost of the vessel, including engines, etc., $300 per tonj.the. 
cost of the boilers proper in metal and labor was, 1S62, $ 

3 J 






640 MARINE ENGINES.— SIDE WHEELS. 

Passenger and. Cargo Steamers ("Wood.). 

"Adriatic," New York and Liverpool— Oscillating Engines Length upon 

deck, 351 feet ; length at load-line, 343. S3 feet; beam, 50 feet ; hold, 33. 16 feet. 

-Immersed Section at load-line 880 square feet. 

Displacement 5233 tons, at load-draught of 20 feet. 

Cylinders. — Two, of 101 ins. in diam. by 12 feet stroke of piston ; volume of piston 
space, 1335 cubic feet. ^Condensers. — Two, surface 24 000 square feet; tubes, % in. 
in diam. ; thickness, No. IT wire gauge. Air-pumps. — Two, double acting ; volume 
96 cubic feet. 

Water-wheels.— Diam. 40 feet. Arms, 32. Blades, 32; breadth of do., 12 feet ; 
depth of do., 3 feet Dip at load-line, 8.25 feet. 

Boilers. — Eight (vertical tubular). Healing surf ace, 31300 square feet. Grates, 
966 square feet. Steam-room, 9200 cubic feet. Cross area between tubes, 186 square 
feet. Tubes, 13 064, 2 ins. in diameter and 3.16 feet in length. 

Smoke-pipes Two, area 3S.5 sq. feet, and 48 feet in height above the grate level. 

Length of Engine and Boiler Space, including side coal-bunkers, 130 feet. 

Pressure of Steam. — 26 lbs. per square inch, cut off at X the stroke of the piston. 
Revolutions, 14 per minute ; at IS feet draught, 1T.5 per minuta Indicated. Horses' 
Power, 4S00. 

Speed At IS feet draught of water, 15.9 knots per hour. 

Fuel— Anthracite or Bituminous. Consumption, 96 tons per day. Coal-bunkers, 
1200 tons capacity. 

Rig — Brig. 

weights of engines, boilers, etc. 



Engines 825000 lbs. 

Coal 265S000 " 

Boilers 6T4000 " 

Water in boilers 1075 200 " 



Hull, launching draught 

9.96 feet 4173120 lbs. 

Spars, Sails, Anchors, etc. " 

Cargo 1772000 " 



Displacement. — Average per inch, from 10.16 to 17.125 feet, (light load-line) 26-43 
tons; from 17.125 to 20 feet, (load-line) 2S.75 tons; from 20 to 21.5 feet, 31.5 tons. 
Accommodation.— Passengers, cabin, 350 ; 2d cabin, 200 ; steerage, 100.' 
Freight. — S00 tons measurement. 

M Costa Rica," New York and Aspinw all— Vertical Beam Engine. — Length 

at load-line, 214c feet; beam, 39 feet; hold, 19.3 feet; do. to spar deck, 21 feet. 

Immersed Section at load-line, 550 square feet. 

Displacement, 2300 tons at load-draught of 15 feet. 

Cylinder.— One, of 81 ins. in diam. by 12 feet stroke of piston; volume of piston 

space, 419 cubic feet. Condenser Volume, 209 cubic feet. Air-pump. — Volume, 

85.5 cubic feet. 

Water-wheels.— Diam. 33 feet. Blades, 28 of 12 and 20 ins. in depth ; breadth of 
do., 8.25 feet. Dip, at load-line, 5.25 feet. 

Boilers. — Two (horizontal tubular). Heating Surface, 9090 square feet. Grates, 
273 square feet. Area of tubes, 3360 square inches. Tubes, 3.25 inches by 7-75 feet. 

Smoke-pipe. — Area 23.75 square feet, and 56 feet in height above the grate level. 

Pressure of Steam. — 20 to 25 lbs. per square inch, cut off at .45 of the stroke of 
the piston. Revolutions, IS per minute. Indicated Horses' 1 Power, 1950. 

Fuel. — Anthracite or Bituminous. Consumption, 3000 lbs. per hour. 

Coal-bunkers.— Capacity, 550 tons. Ri g. —Fore-topsail schooner. 

weights of engine, boilers, etc. 



Engine 526 704 lbs. 

Boilers : 225060 " 

Smoke-pipe.....'. 12 000 " 



Grates, floors, etc 16S 000 lbs. 

Water 200 000 " 

Total 9S06241bs. 



MARINE ENGINES. — SIDE WHEELS. 641 

Passenger and. Deck Cargo Steamers (Wood). 

"City of Boston," New York and Norwich— Vertical Beam Engine.— Length 
upon load-line, 320 feet; beam, 39 feet; hold, 12.6 feet. 

Immersed Section at load-line, 288 square feet. 

Displacement 1450 tons, at load-draught of 8.25 feet. 

Cylinder. — One, of 80 ins. in diam. by 12 feet stroke of piston ; volume of piston 
space, 419 cubic feet. 

Water-wheels.— Diam. 3T feet 8 ins. Arms, 36. Blades, 37; breadth of do. 10 
feet ; depth of do., 30.5 ins. Dip at load-line, 4.25 feet. 

Boilers. — Two (return flue). Heating surface, 9200 square feet. Grates, 192. X 
square feet. 

Pressure of Steam. — 35 lbs. per square inch, cut off at % the stroke of the piston. 
Revolutions (maximum), 19.75 per minute. Indicated Horses' Power, 2500. 

Fuel. — Anthracite, with a blast. Consumption, at ordinary speed, 5200 lbs. per 
hour. 

weights of engine, boilers, etc. 



Engines. 

Cast iron 203 500 lbs. 

Wrought iron 171700 " 

Brass 12 440 " 

Steel and Lead 575 " 



Boilers and Appurtenances. 

Boilers 157 SOS lbs. 

Cast iron 27 060 " 

Wrought iron 17 050 " 

Brass and Copper 620 " 



Total 590 753 lbs. 

Hull. — 800 tons. Light draught of hull and machinery without fuel, water, or 
furniture, 7 feet. 

" Wm. H. Webb," Towing, N. Y. Harbor and Coast— Vertical Beam Engines. 
—Length upon deck, 185.5 feet; beam, 30.25 feet; hold, 10.8 feet. 

Immersed Section at load-line, 194 square feet. 

Displacement 498.25 tons, at load-draught of 7.25 feet. 

Cylinders. — Two, of 44 ins. in diam. by 10 feet stroke of piston ; volume of piston 
space, 211 cubic feet. Condensers. — Two, volume 105 cubic feet. Air-pumps. — 
Two, volume 45 cubic feet. 

Water-wheels. — Diam., 30 feet. Blades (divided), 21; breadth of do., 4.6 feet; 
depth of do., 2.33 feet. Dip at load-line, 3.75 feet. 

Boilers. — Two (return flue). Heating surface, 3280 square feet. Grates, 147.5 
square feet. 

Smoke-pip>e. — Area 11.6 square feet, and 35 feet in height above the grate level. 

Pressure of Steam.— 35 lbs. per square inch, cut off at % the stroke of the piston. 
Revolutions, 22 per minute. Indicated Horses' Power, 1500. 

Fuel. — Anthracite or Bituminous. Consumption, 16S0 lbs. per hour. 

weights of engines, boilers, eto. 
Engines, Wheels, Frame, and Boilers 310 579 lbs. 

"Banshee", Holyhead to Dublin (Eng.) — Length between perpendiculars, 1S9 
feet; beam, 27.16 feet; hold, 14.75 feet. 

Immersed Section at load-line, 190 square feet. 

Displacement 770 tons, at light draught of water of 9 feet. 

Cylinder 72 ins. in diam. by 5.5 feet stroke of piston; volume of piston space, 

155.5 cubic feet. 

Water-wheels. — Diam. 25 feet by 9 feet in width. Blades, area 33.75 feet. 
Pressure of Steam. — 14 lbs. per square inch. Revolutions, 30 per minute. 
Speed. — 18.62 knots per hour at trial, and 13.84 average. 
Indicated Horsed Power, at trial, 1660. Rig. — Schooner. 



642 MARINE ENGINES. SIDE WHEELS. 

Passenger and. Cargo Steamers (Iron). 

"Cleopatra" (English) — Oscillating Engines — Length upon load-line, 202 
feet; beam, 21 feet; depth at sides, 10.5 feet; Hull, 138.28 tons; Engine room 
123.53 tons; Builder's measurement, 453.4 tons. 

Immersed Section at load-line, 122 square feet. 

Displacement 453 tons, at load-draught of 6.25 feet. 

Cylinders.— T 'wo, of 40 ins. in diam. by 4 feet stroke of piston; volume of piston 
space, 80 cubic feet. 

Wafer-wheels (Feathering).— Diam. 16 feet. Blades 10; breadth of do., 8 feet,- 
depth of do., 2.5 feet. 

Boilers.— Two (tubular), length 15 feet, breadth 9 feet, height 8.5 feet. Heating 
surface, 3320 square feet. Grates, 15) square feet. 

Pressure of Steam Average 25 lbs. per square inch, cut off at % the stroke of 

the piston. Revolutions, average 42 per minute. 

Speed.— Average 14.78 knots per hour. Indicated Horses' power, 882. 

Wet Surface of Hull, 4573 square feet. 

itfgr.—Schooner. Sails, 490 square yards. 

Bulkheads, 5. Coal-bunkers, 35 tons' capacity. 

Saloons.— Three, of 17.12 and 10 feet by 14.17 and 19 feet by 6.25 and 7.25 feet. 

WEIGHTS OF ENGINES, BOILEE8, ETC. 

Hull (iron) 212 S00 lbs. Engine 230 720 lbs. 

Boilers 38 030 u Water in boilers 44 800 " 

Cargo 80 640 " Fuel 78 400 " 



" Ly-ee-moon" (ENGLisn)— Oscillating Engines Length upon load-line, 270.5 

feet; beam,21.Sfeet ; hold, 15.25 feet. 

Immersed Section at load-line, 282.6 square feet. 

Displacement at load-draught of 12.5 feet, 1317.7 tons ; displacement per inch be- 
tween light and load lines, 12.74 tons. 

Cylinders. — Two, of 70 ins. in diam. by 5.5 feet stroke of piston ; volume of piston 
space, 290 cubic feet. 

Water-wheels.— Diam. 22 feet. Blades, breadth, 10 feet; depth of do., 416 feet 

Boilers.— Four (tubular). 

Pressure of Steam— 25 lbs. per square inch. 

Speed, 16 knots p:r hour. 



MARINE ENGINES. — SCREW PROPELLER. 643 

TsTaval Steamer (Iron Clad). 

"Warrior," R.N. — Trunk Engines.— Length between perpendiculars, 380.1 feet) 
beam, 58 feet; hold, 37. 33 feet. 

Immersed Section at load-line, 1193 square feet. 

Displacement 8997 tons, at load-draught of 2Qfeet. 

Tonnage (Eng.), C109. Height out of water, 20 feet. 

Cylinders.— Two, of 112 ins. in diam. by 4 feet, effective diam. = 104 ins. ; volume 
of piston space, 500 cubic feet. 

Boilers.— Ten (horizontal tubular). Heating surface, 23 197 square feet. Grates, 
8C8 square feet. 

Pressure of Steam.— 21 lbs. per square inch. Revolutions, 43 per ruinuto. In- 
dicated Horses' Power, 5409. 

Speed, 13.49 knots per hour. 

Armor Plates, 4.5 ins. thick by 1.5 ins. in width. 

WEIGHTS OP AEMOE, ENGINES, AND PROPELLER. 

Armor plates 1 792 000 lbs. | Engines and Propeller 593 600 lbs. 

Fuel 2 128 000 lbs. 



Naval Steamer (Wood). 

"General Admiral," R. I. N. — Horizontal Back-action Engines Length 

upon deck, 313.7 feet; length upon load-line, 302.83 feet; beam, 54.5 feet; hold, 
33.6 feet. 

Immersed Section at load-line, 1090 square feet. 

Displacement 5200 tons, at load-draught of 23 feet. 

Cylinders. — Two, of 84 ins. in diam. by 3.75 feet stroke of piston ; volume of piston 
space, 282.8 cubic feet. Air-pumps. — Two, double acting; volume, 27.6 cubic feet. 

Propeller.— Diam. 19 feet. Blades 2. Pitch, 31.5 feet. Area of disc, 71.39 feet. 

Boilers.— Six (horizontal tubular). Heating surface, 19 500 square feet. Grates, 
700 square feet. 

Smoke-pipe. — Area 95 square feet, and 65 feet in height above the grate level. 
Pressure of Steam. — 18 lbs. per square inch, cut off at % the stroke of the piston. 
Revolutions, 42 per minute ; maximum, 52. Indicated Horses' Power, 2000. 
Fuel. — Anthracite coal, with a natural draught. Consumption, 7960 lbs. per hour. 
Sliced, 11 knots per hour. Coal-bunkers, 650 tons' capacity. 
Rig.— Ship. 

WEIGHTS OF ENGINES, BOILERS, ETC. 

Engine 535 600 lbs. I Mountings and fixtures for do. 26 6S0 lbs 

Propeller 27 900 " | Boilers 452 964 " 

Total 1 043 144 lbs. 

a i* 



644 MARINE ENGINES. SCREW PROPELLER, AUXILLIARY. 

Naval Steamers (Wood). 

"Brooklyn," U. S. N Horizontal Direct Engines.— Length upon load-lins to 

forward stern-post, 233 feet; to stern-post, 243 feet; beam, 43 feet; hold, 22.6G 
feet. Tonnage, O. M., 2G60. 

Immersed Section at load-line, 551 square feet. 

Displacement 2532 tons, at load-draught of 15.5 feet. 

Cylinders Two, of 61 ins. in diam. by 33 ins. stroke of piston; volume of piston 

space, 112 cubic feet. Condensers. — Two, volume 65.2 cubic feet. Air-pumps. — 
Two, of 19 ins. in diam. by 33 ins. stroke of piston, double acting ; volume 11 cubic 
feet. Steam Valves, 162 square ins. Exhaust, 234 square ins. Feed-pumps. — Four, 
of 5.5 ins. in diam. ; volume, 1.8 cubic feet. 

Propeller Shaft. — Journals, 12 and IS ins. in length. 

Boilers Two (vertical water tubular, brass tubes). Heating surface, 7S00 square 

feet. Grates, 252 square feet. Steam-room, 1150 cubic feet. 

Smoke-pipe. — Area 3S.5 square feet, and 50 feet in height above the grate level. 

Pressure of Steam. — IS lbs. per square inch, cut off at % the stroke of the piston, 
full throttle. Revolutions, 51 per minute. Indicated Horses' Power, 706- 

Fuel. — Anthracite coal, with a natural draught. Consumption, at load-line, and 
at a pressure of 20 lbs., and 50 revolutions, 22.4 tons per day. 

Speed, 9 knots per hour. Coal-bunkers, 360 tons' capacity. 

Slip of Propeller, 26 per cent. 

Hull. — Weight, 1350 tons. Angle of entrance at load-line, 56°. 

Centre of Displacement. — (Gravity of) at 16 feet 2 ins. draught, 6.5 feet below the 
water-line. Average per Inch, at load-line, 19.3 tons. Meta Centre, above centre 
of displacement (gravity of), 10.44 feet. 

Rig. — Full bark. Sails, 22 430 square feet. 

Armament. — 2 10-inch chambered guns upon pivots, and 16 9-inch do. upon car- 
riages. 

WEIGHTS OF ENGINES, BOILERS, ETO. 

Engines and extras 356 035 lbs. i Water in boilers 135 370 lbs. 

Boilers 157 670 " Grates 15120 " 

Smoke-pipe S 443 " | Armament 406 113 " 

Total (Engine^, Boilers, etc.) 537 265 lbs. 



u Wyoming," U. S. N.— Horizontal Direct Engines Length upon deck, 209.75 

feet; between perpendiculai s, l$S.5feet; keel, 188 feet; beam, 33 feet: holds 
15.83/**. 

Immersed Section at load-line, 391 square feet. 

Displacement 1475 tons, at load-draught of 12.83 feet. 

Cylinders.— -Two, of 50 ins. in diam. by 2.5 feet stroke of piston ; volume of piston 

space, 67 cubic feet. Condenser Surface 3000 square feet; 3000 tubes, % in. in 

diam. by 6 feet in length. Vacuum, 22.5 ins. Air-pumps.— (Fresh water.) Two, 
of 11 ins. in diam. by 2.5 feet stroke of piston ; (Salt water J Two, of 10 ins. in diam. 
by 2.5 feet stroke of piston. 

Propeller Shaft — Journals, 11.57 and 10 ins. in diam., and 16, 18, 18, and 24 ins. 
in length. 

Propeller (true screw).— Diam. 12.25 feet. Blades, 4. Pitch, 19 feet; length, 2.5 
feet. Surface, 81 square feet. 

Boilers.— Three (vertical tubular, brass tubes), length, including fire-room, 29 
feet ; breadth (lengthwise of the vessel), 24.75 feet ; height, 10. 16 feet. Heatinq sur- 
face, 7SP0 square feet; tubes, 4280 of 2 ins. external diam. by 31^ ins. in length. 
Grates, 242 square feet. 

Smoke-pipe.— Are& 36.7 square feet, and 52 feet in height above the grate level. 

Pressure of Steam.— 1S.C5 lbs. per square inch, cut off at .38 the stroke of the 
piston, throttle wide open. Revolutions, 74.5 per minute ; attainable, 85. Indicated 
Horse*? Power t 793. 



MARINE ENGINES. — SCREW PROPELLER. 645 

Fuel.— Anthracite coal, with a natural draught. Consumption, 1710 lbs. per hour, 
or about 2.1G lbs. per indicated horses' power = 18.3 tons per day. 

Speed, 9.87 knots per hour. Coal-bunkers, 235 tons' capacity. 

Slip of Propeller, 20 per cent. 

Note. — A pressure of 27 lbs. per square inch, cut off at % the stroke of the piston, 
throttle valve wide open, and 80.5 revolutions per minute have been attained; 
draught of water, 13.25 feet; speed, 11.25 knots per hour. 

Evaporation, 9.32 lbs. water per lb. of fuel. 

Rig Full bark. Sails.— Area 9705 square feet. 

Armament. — 2 11-inch pivot guns, and 4 32-pounder chambered guns. 

Stores, 6 months. Provisions, 3 months. 

WEIGHTS OF ENGINES, BOILERS, ETC. 



Engines and dependencies. . 132 907 lbs. 

Propeller, shafting, etc 51 231 " 

Coal-bunkers and bulkheads 31 274 " 



Boilers, Pipe, etc 177 995 lbs. 

Water in boilers 89 040 " 

Tools, Extra pieces, etc 14 15S " 



Total 496 605 



Yacht (Wood). 

"Campaneea" (English) — Horizontal Direct Engine. — Length upon load-line, 
108.5 feet; beam, 21 feet; depth at side, 11.5 feet 

Immersed Section at load-line, 125 square feet. 

Displacement 151 tons, at load-draught of 7 feet. 

Cylinder.— One, of 12 ins. in diam. by 1.5 feet stroke of piston; volume of piston 
space, 1.18 cubic feet. 

Propeller. — Two-bladed; diam. 9 feet. Revolutions, 108 per minute. Indicated 
Horses' Power, 50 ; Nominal, 35. 

Fuel Bituminous coal. Consumption, 420 lbs. per hour. 

Speed, 9 knots per hour. Coal-bunkers, 32 tons' capacity. 

Stowage, 36 tons. Sails, 506 square yards. 

Weights. — Engine, Boiler, and Water, 53 760 lbs. 



Passenger and. Cargo Steamer (Iron). 

"Australasian" (British) — Vertical Direct Engines. — Length upon load-line, 
314. 5 feet ; beam, 42. 15 feet ; depth at sides, 31. 25 feet. 

Immersed Section at load-line, 764 square feet. 

Displacement 4447 tons, at load-draught of 22 feet. 

Cylinders. — Two, of 90 ins. in diam. by 3.5 feet stroke of piston; volume of piston 
space, 331.3 cubic feet. 

Propeller Three-bladed, diam. 19 feet. Pitch, 34 feet. 

Boilers. — Six (tubular). 

Pressure of Steam — 20 lbs. per square inch, cut off at % the st"oke of the piston. 
Revolutions, 46 per minute. Indicated Horses' Power ', 2500. 

Speed, 12 knots per hour. 

Fuel. — Bituminous coal. Consumption, 80 tons per day. 

Wet Surface of Hull, 18 370 square feet. Slip of Propeller, 28.5 per cent. 

Rig Bark. Sails, 3210 square yards. 

Passengers, 142. 

Weights.— H ull, 1050 tons. Cargo, 1100 tons. Fuel, 1250 tons. 



646 RIVER ENGINES. SIDE WHEELS. 

Passenger and. Deck Cargo Steam-boat (Wood). 

" Clifton, No. 2," New York to Staten Island — Vertical Beam Englnb.— « 
Length upon deck, ISO feet; beam, 32 feet; hold, 12.5 feet. 

Immersed Section at load-line, 147 square feet. 
Displacement 465 tons, at load-draught of 6 feet. 

Cylinder. — One, 43 ins. in diain. by 10 feet stroke of piston ; volume of piston 
space, 101 cubic feet. 

Water-wheel Shafts. — Journals, 11.25 and 8 ins. in diam., and 13 and 10 ins. in 
length. 

Water-wheels.— Diam. 22 feet. Blades (divided), 20 ; breadth of do., 8 feet ; depth 
of do., 2 feet by 2.25 ins. thick. Dip at load-line, 24 ins. Centres, 3, of 5 feet in 
diam. Rims (iron), 3X>£\ %)4XK, and4x% ins. Braces, 10, IX ins. in diam. 

Beam, length 19 feet ; depth 9 feet. End centres, 4.5 ins. in diam. Main centre, 
C.5 ins. in diam. Strap, least section, 5 by 3 ins. 

Engine Frame, Yellow-pine, 12 by 16 ins. at foot, and 12 by 12 ins. at head. 

Boiler. — One (return flue), 12 feet front by 24 feet in length. Shell, diam. 10 feet; 
height of front 10>£ feet Furnaces, 3, 6.16 feet in length. Steam Chimney, 12 
feet in height. Heating surface, 1724 square feet. Grates, 65.2 feet 

"Pressure of Steam 2S lbs. per square inch, cut off at % stroke. Revolutions, 

26 per minute. Speed, 13.5 knots per hour. Indicated Horses' Power, 570. 

Hull. — Floors, molded, 16 ins. ; sided, 12 ins. Launching draught, 4 feet. 

Passenger Steam-boats (Wood). 

41 Daniel Drew," New Yokk to Albany — Vertical Beam Engine.— Length upon 
deck, 251M feet; do. at load-line, 244 feet; beam, SI feet; hold, 9.25 feet. 

Immersed Section at load-line, 136 square feet. 

Displacement 380 tons, at load-draught of4=.S3feet. 

Cylinder. — One, 60 ins. in diam. by 10 feet stroke of piston; volume of piston 
space, 196 cubic feet. Condenser, volume 6S cubic feet. Air-pump, volume 20 
cubic feet. 

Water-wheels. —Diam. 29 feet. A rms, 24. Blades, 24 ; breadth of do. , 9 feet ; depth 
of do., 26 ins. Dip at load-line, 2.33 feet. 

Boilers — Two (return flue), 29 feet in length by 9 feet in width at furnace. Shell, 
diam. S feet. Heating surface, 3350 square feet. Grates, 105 square feet Cross 
area of lower flues, 15.5 square feet ; of upper, 13 square feet. 

Smoke-pipes.— Two, area 25.13 sq. feet, and 32 feet in height above the grate level. 

Pressure of Steam.— 35 lbs. per square inch, cut off at >£ the stroke of the piston. 
Revolutions (maximum), 26 per minute. Indicated Horses' Power, 1720. 

Fuel— Anthracite coal, with a blast. Consumption, 3S00 lbs. per hour. 

Speed, 22.3 miles per hour. Slip of Wheels from Centre of Pressure, 12.5 per cent 

Frames. — Molded, 15% ins.; sided, 4 ins., and 20 ins. apart at centres. 

Weight of boilers, SO 650 lbs. 

"Seth Geosvenor," African Coast and River — Steeple Engine. — Length upon 
deck, 95 feet; beam, 17.2 feet; hold, 5 feet. 

Immersed Section at load-line, 43 square feet. 
Displacement 73 tons, at load-draught ofS.25feet. 
Cylinder.— 2S ins. in diam. by 3 feet stroke of piston ; volume, 12. S cubic feet. 
Water-wheels.— Diam. 13.5 feet Blades, 14; breadth of do., 3 feet; depth of do., 
125 feet. 

Boiler (return tubular). Heating surface, 540 square feet Grates, 22.5 square 
feet Area of tubes, 367 square inches. Indicated Horses' Power, 90. 

Weights.— Boiler, Engine, Wheels, and Frame, 61 556 lbs. =27.4 tons. 

The operation of this vessel was in every way successful, being very fast, economical in fuel, etc., 
and she would have been improved if the hull* had had 15 feet additional length, all other dimett 
sions and capacities remaining the same. 



EIVER ENGINES. — SIDE WHEELS. 647 

Passenger and. Cargo Steam-boats (Wood). 

" Buckeye State," Ohio River— Horizontal Engines {Son-condensing). — Length 
upon deck, 2Q0feet; beam, 30.3 feet; depth of hold, 6.5 feet. 

Immersed Section at load-line, 143 square feet. 
Displacement 530 tons, at load-draught of 5 feet. 

Cylinders. — Two, of 29.5 ins. in diara. by 8 feet stroke of piston ; volume of piston 
space, T6 cubic feet. 

Water-wheel Shafts Journals, 17 ins. in diam. 

Water-wheels.— Diara. 31 feet 2 ins. Blades, 20 ; breadth of do., 12 feet ; depth of 
do., 2.5 feet. 

Boilers.— Five (cylindrical fiued), 42 ins. in diam. by 30 feet in length, with 2 re- 
turn flues in each, 17% ins. in diam. Heating surface, 2394 square feet. Grates, 
S6.1 square feet. 

Smoke-pipes.— Two, area 47.5 sq. feet, and 76 feet in height above the grate level. 

Pressure of Steam. — 140 lbs. per square inch, cut off at %^ the stroke of the piston. 
Revolutions (maximum), 19 per minute. Indicated Horses' Power, 2000. 

Fuel.— Bituminous coal or Wood. Consumption, 42S0 lbs. per hour. 

Freight 180 tons at load-draught. Light draught of water, 3.5 feet. 

Hull. — Keelson, 11x17 ins. ; bilge do., 3x10.5 ins.; false do., 3x8 ins. ; bottom 
plank, 4 ins. ; deck beams, 3.5x6.25 ins. ; plankshear, 2.5x25 ins., and deck plank, 
2 ins. Frames. — Throats, 7 ins. ; sides, 3.5 ins. ; distance apart from centres, 17 ins. 

Notes Areas of immersed section of hull at light draught (142.7 square feet), and 

of cross blade surface, are as 1 to 2.38. Areas of grate and heating surface, as 1 to 
27.8. Fuel consumed upon each square feet of grate, 50 lbs. per hour. Areas of 
smoke-pipes, 47 feet ; of flues, 13.2 feet ; and of bridge wall, 27 feet. 



"Magnolia," Mississippi River — Horizontal Engines (Son -condensing). — Length 
upon deck, 295 feet; beam, So feet; over wheel guards, 72 feet; hold, 9 feet 

Immersed Section at light-draught of '4 feet, 132 square feet. 

Displacement 1000 tons, at load-draught of 7 feet. 

Cylinders. — Two, of 30 ins. in diam. by 10 feet stroke of piston; volume of piston 
space, 98 cubic feet. 

Water-wheel Shafts.— Journals, 18 ins. in diam. 

Water-wheels.— Diam. 40 feet. Blades, 26 ; breadth of do., 12.5 feet ; depth of do., 
2.33 feet. 

Boilers — Six (cylindrical flued), 3.5 feet in diam. by 30 feet in length, with two 
return flues in each, 16 ins. in diam. Heating surface, 2716 square feet, Area at 
bridge, 77 square feet. Area of Flues, 16.7 square feet. Grates, 9S.4 square feet. 
Furnace, 2 feet, and Bridge wall, 10 ins. in depth below boiler. 

Smoke-pipes — Two, area 39.25 sq. feet, and SI feet in height above the grate level. 

Pressure of Steam. — 130 lbs. per square inch, cut off at .6 the stroke of the piston, 
throttle wide open. Safety-valve*, two, of 9^ ins. in diam. Revolutions, 16 per 
minute. Indicated Horses' Power, 1380. 

Evaporation.— 10.9 lbs. fresh water per square foot of heating surface per hour, 
and 7.05 lbs. per pound of coal. 
Speed, 15. 13 miles per hour. 

Consumption of Fuel — 2.9 cords Fine wood per hour, weighing 2700 lb3, per cord, 
or 3986 lbs. Bituminous coal. 

Freight.— 4590 bales of cotton, or 800 tons in weight. 
Weights. -EDgines, Wheels, Boilers, and Water, 550 000 lbs, 



648 RIYER ENGINES. SIDE WHEELS. 

Passenger Stearrx-tooat (Iron). 

"Lee" (English)— Oscillating Engines. — Length upon deck, 160 feet; beam, 
17.5 feet ; hold, 1.5 feet; draught of water at load-line, 3.5 feet Quarter-deck 
raised 2.5 feet above main deck. 

Immersed Section at load-line, 60 square feet. 

Displacement 130 tons, at load-draught of 3.5 feet. 

Cylinders.— Two, of 36 ins. in diam. by 3 feet stroke of piston ; volume of piston 
space, 42 cubic feet. 

Water-wheels (Feathering).— Diam. 14.5 feet. Blades, 10 ; breadth of do., 10 feet ; 
depth of do., 2 feet. 

Boilers Two (tubular). Tubes 376, of brass, 2.75 ins. in external diam. by 6 

feet in length. Grates, 90 square feet. Speed, 13.7 knots per hour. 

Hull Keel, 5x4x.5ins. Stew, 4x1.25 ins. Sternpost, 3.5x1.5 ins. Frames, 

L 2.5X2.5X.3125 ins., 2 feet apart from centres at body, and 2.5 feet at ends. Cross 
floors, one to each frame, .25 in engine and boiler space, and .1875 forward and aft ; 
reversed frames L, one on every cross floor, 2.5 x 2.5 X .25 in engine and boiler 
space ; and 2x2x .25, forward and aft, all running up each side of hull, 1 foot above 
junction of cross floor and frame. 

Engine plates, .3125 in. thick, with angle iron L, 2.5x2.5x.3125 ins. thick. 

Deck beams, L 3 X 2.5 X. 25 ins., one upon every frame, with triangular plate knees 
at ends, 1 foot each in length and depth. 

Engine bearers, of .3125 in. plate, and L 2.5x2.5x.3125 ins. Water-wheel bear- 
ers, I 8X.5 ins. 

Covering Plates, for 75 feet amidships, 12 X .25 ins. ; forward and aft, 12X.1S75 ins. 
Gunwale, L 2-.5x2.5x.25 ins. 

Bulkheads.— Three, .1875 in. thick, with L 2.5x2.5x.25 ins., set 2.5 feet apart 

Plating. — Garboard strake, .3125 in.; second strake throughout, and bottom to 
turn of bilge for 60 feet amidship, .25 in. ; remainder of plating .1875 in., except 
gunwale strake of .25 in. 

Wheel-houses.— Inboard siding .125 in., with L 2.25x2.25x.lS75 in3., set 2.5 feet 
apart. Brackets for supporting outboard plumber-blocks, at side of hull (wheels 
overhung), of .375 in. plates and L 3X3X.375 in. 

Rivets. — Keel, Stem, Sternpost, and Garboard strake single riveted with .625 in. 
rivets, 2.5 ins. apart from centre to centre. Plating rivets, .5 iu. and 2 ins. apart; 
frame rivets, 4.5 ins. apart. 

Cost of Hull and Engines, 1S61, $26 250. 



"Alabama," Mobile Bay— Yeetical Beam Engine. — Length between perpendicu- 
lars, 225 feet; beam, 32 feet ; hold, 10 feet ; launching draught, 2.33 feet. 

Immersed Section at load-line, 130 square feet. 

Displacement 440 tons, at load-draught of -Lb feet. 

Cylinder.— One, 50 ins. in diam. by 10 feet stroke of piston; volume of piston 
ipace, 136.4 cubic feet. 

Water-wheels.— Diam. 30 feet; breadth of do., 10 feet. 
Boiler. — One (return flue). 

Hull.— Keel U, .625 in. thick. Keelsons, two box, and five single I. Frame L, 
3.5x3.5x.375 ins. thick, 1.5 feet apart from centres. Cross floors T, 12 ins. deep 
by .3125ths thick, with angle iron on top, 3X3X.3125 ins. thick. Garboard strake, 
.5 in. thick; to bilge, .3125; bilge, .375 in. ; wales, .375 in. ; clamps, .375 in. by 16 
ins. deep. 

Bulkheads. — Three, water-tight. Deck beams, white pine, 5.5x3.5 ins., 2 feet 
apart. Deck, white pine, 2.5 ins. Suspension frames over gunwale. 

HulL— Weight, 336 000 lbs. 



RIVER ENGINES.— SIDE WHEELS. 649 

Passenger Steam-boat (Iron). 

(British) Horizontal Engines {Non-condensing).— Length upon deck, 250 feet; 
beam, 30 feet. 

Immersed Section at load-line, 58 square feet. 

Displacement 260 tons, at load-draught of '2 feet. 

Cylinders.— Two, of 26 ins. in diam. by 6 feet stroke of piston; volume of piston 
space, 44 cubic feet. 

Pressure of Steam.— 100 lbs. per square inch. Revolutions, 35 per minute. 

Indicated Horses' Power, 1020. 

Water-wheels Diam. 20 feet. Blades. 16 ; breadth of do., 10 feet ; depth of do., 

1.5 feet. 

Speed, 14 miles per hour. 



Xjaixxich. (Wood). 

Launch of a 1st Class Ship-of-the-Line. 

"Experiment"— Direct Acting Engines (Non-condensing).— Length, 24 feet. 

Light draught of water, 2 feet. 

Cylinders.— Two, of 4 ins. in diam. by 6 ins. stroke of piston ; volume of piston 
space, .087 cubic feet. 

Propellers.— Two, diam. 2 feet. Pitch, 3.3T5 feet. 

Pressure of Steam.— 60 lbs. per square inch. Revolutions, 290 per minute. 

Indicated Horses' Power, 1.3. 

Speed, 6.T4 knots per hour. 

Engines and Boiler occupy a space of 6 feet 11 ins. by 4 feet 4 ins. With one pro- 
peller, 340 revolutions were attained, and a speed of 4.61 knots. 



Critter (Iron). 

w La Bonita" — Inclined Engine (Non-condensing). — Length upon deck, 42 feet ; 
beam, 9 feet; hold, 3 feet. 

Immersed Section at load-line, 8.75 square feet. 

Displacement S3S6 lbs., at load-draught of 1.3 feet. Tons, 9.65, O. M. 

Cylinder. — S ins. in diam. by 1 foot stroke of piston ; volume of piston space, .35 
cubic foot. 

Water-wheels. — Diam. 5.66 feet. Blades, 7 ; breadth of do., 2.3 feet ; depth of do., 
7 ins. 

Boiler.— One (horizontal tubular). Heating surface, 95 square feet.. Grates, 6 
square feet. 

Fuel Coal or Wood. Exhaust draught. 

Pressure of Steam. — 65 lbs. per square inch, full stroke. Revolutions, 54 per min- 
ute. Indicated Horses' Power, 9. 

Hull. — Corrugated and galvanized plates, .0625 in. thick. 

WEIGHTS OF HULL, ENGINE, BOILER, ETC. 

Hull 2876 lbs. I Boiler 2260 lbs. 

Engine and wheels 2400 u | Pipes, grates, etc 750 " 



650 RIVER ENGINES. STERN WHEELS. 

STERN WHEELS. 

Passenger and Cargo Steam-boats (Wood). 

" Vencedor," Magdalena River — Horizontal Engines (Son - condensing").— 
Length between perpendiculars, lbOfeet; beam, 24 feet; hold, 5 feet. 

Immersed Section at load-line, 90 square feet. 

Displacement 230 tons, at load-draught of 4 feet. 

Cylinders Two, 16 ins. in diam. by 6 feet stroke of piston; volume of piston 

space, 16. S cubic feet. 

Water-wheel Shaft. — Journal, 8.75 ins. in diam. 

Wheel.— One, diam. 16 feet. Blades, 15; breadth of do., 17 feet; depth of do., 
1.25 feet. 

Boiler. — One (horizontal tubular) (locomotive). Tubes, 138 of 3 ins. diam. by 12 
feet in length. Heating surface, 1500 square feet. Grates, 42 square feet. 

Hull. — Frame, yellow pine, molded 6 ins., sided 4 ins., and 2 feet apart at centres ; 
bottom plank 2>£ ins. thick. 



Horizontal Engines {Son-condensing). — Length upon deck, 90 feet; beam, 16 feet ; 
hold, 3.5 feet. 

Immersed Section at load-line, 60 square feet. 

Displacement 100 tons, at load-draught of 4 feet. 

Cylinders.— Two, of 12 ins. in diam. by 3 feet stroke of piston ; volume of piston 
space, 2.36 cubic feet. 
Boiler.— One (horizontal tubular). 
Launching draught, 9 ins. 



Horizontal Engines (Son-condensing). — Length upon deck. 56 feet; beam, 12 feet; 
hold, Z.Sfeet. 

Immersed Section at load-line, 24 square feet. 

Displacement 23 tons, at load-draught of 2.16 feet. 

Cylinders.— Two, of 10 ins. in diam. by 2.5 feet stroke of piston ; volume of piston 
space, 2.73 cubic feet. 
Boiler. — One (horizontal tubular). 



Iron. 

Horizontal 'Engines {Non-condensing). — Length upon deck, 110 feet ; beam^ 14 
feet {deck projecting over, 4 feet) ; hold, 3.5 feet. 

Immersed Section at load-line, 10.25 square feet. 

Displacement 33 tons, at load-draught of 1.1 feet. 

Cylinders. — Two, of 10 ins. in diam. by 3 feet stroke of piston; volume of piston 
space, 1.6 cubic feet. 

Wheel.— Diam. 13 feet. Blades, 13 ; breadth of do., S.5 feet ; depth of do., S ins. 

Revolutions, 33 per minute. 

Boiler. — One (horizontal tubular). Tubes, 100 of 2 ins. in diam. 

Fuel. — Bituminous coal. Consumption, 4480 lbs. in 24 hours. 

Hull.— Plates, keel, No. 3; bilges, No. 4; bottom, No. 5; sides, Noe. 6 and 7. 
Frames, 2.5X.5 ins., and 20 ius. apart from centres. 



ELEMENTS OF VESSELS AND BOATS. 651 

Passenger and. Cargo Steamers (Wood). 

* l Asia," Ccnard Line — Side Lever Engines. — Length of keel and fore rake, 267 
feet; beam, 46.5 feet; over W. W. guards, 63.5 feet; depth of hold, 21.5 feet. 
Load draught of water, 19.5 feet. 

Hull.— 2128 tons, O. M. Engine space, 92.5 feet in length ; volume, 812 tons, O. M. 

" Gorgon," R. N.— Length between perpendiculars, 178 feet; beam, 37 feet; 
depth of hold, 2^ feet 

WEIGHTS. 



Hull 630 tons. 

Masts and rigging 25 " 

Anchors and chains 50 u 

Crew, etc., etc 133 " 



Engines and water 270 tons. 

Coal 300 u 

Stores and provisions 7S " 

Total 1486 u 



Comparison of Tonnage of Engine and Boiler Space to total Tonnage (English). 
Side lever Engine 44 per cent. | Direct acting Engine 34 per cent. 

(Iron). 

English.— Length upon deck, 178 feet; do. at mean load-line of 19.16 feet, 177 feet; 
keel, I'll feet; beam, ZZ.SSfeet; depth of keel (mean), 2.75 feet; hold, 21.75 feet. 

Immersed Section at load-line, 387 square feet. 

Displacement 1385 tons, at load-draught of 19.16 feet ; and, in proportion to its 
circumscribing parallelopipedon, .524, and 1495 tons, at deep load-draught of 20 feet. 

Load-line. — Area at load-draught, 4557 square feet. Angle of entrance, 57° ; of 
clearance, 64°. Area in proportion to its circumscribing parallelogram, .784. 

Immersed Section. — Area in proportion to its circumscribing parallelogram, .737. 

Centre of Gravity, 6.416 feet below mean load-line. 

Centre of Displacement (gravity of), 6.25 feet below the. load-line ; and 4.33 feet 
before the middle of the length of the load-line. 

Immersed Surface.— Bottom, 7370 square feet. Keel, 1130 square feet. 

Rig.— Ship. Sails, 13 2S2 square feet. 

Meta Centre, 6.66 feet above centre of gravity of displacement. 

Centre of Effort before centre of displacement, 3.5 feet ; height of do. above mean 
load-line, 55.5 feet. 

Surface in proportion to immersed section, 34.32 to 1; do. in proportion to dis- 
placement in tons, 9.59 to 1. 

Surf Boat (Wood), New York — Length over all, 30 feet; beam, 8 feet; hold, 

2.75 feet. 

Breadth and Height at Sections of 5 feet. 

1. 2. 3. 4. 5. 

Breadth of Beam. . 5 feet 10 ins. 7 feet 2 ins. 8 feet ins. 6 feet 10 ins. 3 feet 9 ins. 
Breadth of Floor . . 1 " 3 " 2 " 9 4< 3 " 6 " 2 " 4 " " 3 " 
Height of Gunwale 2 " 7 " 2 " 4 " 2 " 4 " 2 " 5 " 2 " 6 " 
Rake of stem, 2 feet ; of stern, 3 feet. Oars.— 6 single, or 10 double banked. 

Life Boat (Wood)— English — Length over all, 36 feet; at load-line, 31 feet; beam, 
9.5 feet; hold, o.Sfeet; shear, dfeet; keel, 8 inches. 

Capacity — 300 cubic feet of air-vessel, equal to 8.5 tons or 70 persons. 
Thwarts. — 7 of 2.25 feet apart and 7 inches below gunwale. 
Oars.— 12 double banked, with pins and grommets. 
Weight, 7500 lbs. 

Race Boat.— Length, 46 feet. Breadth, 20 ins. ; at water line, IS ins. Depth, 
8 ins. 

Oars, 4. Weight (of Cedar), 150 lbs. 

3K 






652 ELEMENTS OF VESSELS AXD BOATS. 

Sailing Vessels. 

"Amebica,'' Yactt (Woof, [--Length over all, QSfeet; upon deck, Ufeet; at loadr 

line, 90.5 feet; beam 22.5 feet, at load-line 22 feet; depth of hold, 9.25 feet 

feet^aft a \5fee{ TOm ""^ ***** ° f garhoard strake i U f eeU Sheer , forward, 3 
Immersed Section at load-line, 121.8 square feet. 
Displacement at load-draught of S.5 feet, from under side of garhoard stroke and 
of 11 feet aft, 191 tons; and, in proportion to Volume of circumscribing parallelo- 

Displacement at 4 feet (from garboard stroke), 43 tons; at 5 feet, 66 tons; at 6 
feet, 93 tons; at 7 feet, 12T tons; and at Sfeet, 167 tons. 

line* 2% f^ oad - line ^ 12S0 s( * uare feet - Mean Sirths of immersed section to load- 

u^lf S iZZ g LT FoTW&Id ' 491 feet; aft ' 1L5 feet Rake * Stem > 17 feet from 

*j£ul~o¥? in ™ a8 £ 81 fe V n lensth hj - 2 ins - in diam - Foremast, 79.5 feet in 

&fi2 Y tlT 9? • diam - *JT 6 ?T' 5 l feet * len ^ th - Gaff, 2S feet .Fore G«jf, 
At feet. itafo, 2.7 ins. per foot. Drag of Keel, 3 feet. 

Ton*.— O. M., U. S., 170.56; O. M., English, 210. 

J%£\ % Gramty-LongituainBny, 1.75 feet aft of centre of length upon load-line 

o^dy 19 feet aff x^nr^'^ V, °/ F T ^ 14 ' 25 feet fo ™^ and of A^er 
ooay, iy leet aft. Meta Centre, 6.72 feet above centre of gravity. 

te^%&'g£X fr ° m l0ad " line * ^ 0/ *" W *'* 1 *^ *» 



9 5/f / f ' -^ ' ^"^ °^ lcator > 6 /««* forward; aft, 

Keel, 22 ins. in depth. Mse fcjeZ, 12 ins. in depth at centre 

*3^££T^*3&'5* K *™ f ' T6feet - *** ^-,46 feet 
Totw — N. M., 46.32. 



''^^°^f^'w IP r^ SmP o ( . f ^ n) — ^^ °^ M a ^/°™ rait,, 199/atf; 
JK? £^5tf&l^ W* o/W, 47 

M^lS7S f i;?itt^: 5 feet deep b - v 175 feet in - idth * *** «* 

JVamw—L, 5 X 3 X .625 ins. thick. Bottom Plates, clincher laid. 
Floors, .4375 thick by 22 ins. in depth, and 15 ins. apart at centres. 
Beams.-Lovrer deck, 9 ins. in depth ; Main deck, S ins. ; and Poop, 6 ins. 
Masts, iron. Decks. -Two, with top-gallant forecastle. 



"Twilight," Clippee Smr (Iron) Length of keel and fore rake, 160feet; upon 
deck, lbl.2o feet; beam, 30 feet; hold, 20 feet. : . ' \f 

Load-draught, 17.5 feet aft; 1J feet forward. 
Bulkheads.— Three. Capacity, 700 tons cargo. 
Decks.— Two, with top-gallant forecastle. 



ELEMENTS OF ENGINES AND MACHINES. 653 

BLOWING ENGINES. 

Furnace. — One, diam. 14 feet. Lonaconing (Md.). 

Engine {Non-condensing). — Cylinder, 18 ins. in diam. by 8 feet stroke of piston. 

Boilers (plain cylindrical). — Five, 3 feet in diam. and 24 feet in length. Grates^ 
50 square feet. 

Pressure of Steam 50 lbs. per square inch, cut off at % the stroke oi the piston. 

Revolutions, 12 per minute. 

Blowing Cylinders. — Two, 5 feet in diam. and 8 feet stroke of piston. Pressure, 
2 to 2)4 lbs. per square inch. Volume of Air, 3770 cubic feet per minute. 

Furnaces. — Four, diam. Ik feet. Mount Savage (Md.). 
100 Tons Pig Iron per Week. 

Engine (Condensing). — Cylinder, 56 ins. in diam. by 10 feet stroke of piston. 

Boilers. — Six (cylindrical flue), 60 ins. in diam. and 24 feet in length; 1 flue and 
return, 22 ins. in diam. Grates, 198 square feet. 

Revolutions, 15 per minute. 

Blowing Cylinder, 126 ins. in diam. by 10 feet stroke of piston. Pressure, 4 to 5 
lbs. per square inch. Area of Pipes, 2300 square inches = .2 ofcylinder. 

Furnaces. — Two. Fineries, tivo. (England.) 
240 Tons Forge Pig Iron per Week. 

Engine (Non-condensing). — Cylinder, 20 ins. in diam. by S feet stroke of piston. 

Boilers. — Six (plain cylindrical), 36 ins. in diam. and 2S feet* in length. Grates, 
100 square feet. 

Blowing Cylinders Two, 62 ins. in diam. by 8 feet stroke of piston. Pressure, 

2.17 lbs. per square inch. Revolutions, 22 per minute. 

Pipes, 3 feet in diam.*r=. 166 of the cylinder. 

Tuyeres. — One Furnace, 2 of 3 ins. in diam. and 1 ef %% ins. ; and one, 3 of 3 ins. 
One Finery, of \% ins. ; and one, 4 of 1% ins. 

Temperature of Blast, 600°. Ore, 40 to 45 per cent, of iron. 

Furnace. — One. Esther, Peru {Clinton Co., N, Y.). 
37 Tons Pig Iron per Week. 

Breast Wheel.— 20 feet in diam. by 39 ins. face. Fall, 14 feet. Volume of Water,, 
350 cubjc feet per minute. Revolutions, 9 per minute. 

Blowing Cylinders. — Two (single acting), 7 feet in diam. and IS ins. stroke of piston. 
Volume of Air, 1039 cubic feet per minute. 

Furnaces. — Eight, diam. 16 to 18 feet. Dowlais Iron Works (England). 

1300 Tons Forge Iron per Week ; discharging 44 000 Cubic Feet of Air 
per minute. ' 

Engine (Non-condensing).— Cylinder, 55 ins. in diam. by 13 feet stroke of piston. 

Pressure of Steam.— $0 lbs. per square inch, cut off at % the stroke of the piston. 
Valves, 120 ins. in area. 

Boilers. — Eight (cylindrical flue, internal furnace), 7 feet in diam. and 42 feet in 
length; one flue 4 feet in diam. Graces, 288 square feet. 

Fly Wheel Diam. 22 feet; weight, 25 tons. 

Blowing Cylinder, 144 ins. in diam. by 12 feet stroke of piston. 

Revolutions, 20 per minute. Blast, 3% lbs. per square inch. Discharge pipe, 

diam. 5 feet, and 420 feet in length. Valves Exhaust, 56 square feet ; Delivery, 

16 square feet. 

* 40 feet would have afforded economy in fuel. 



lure 



654 STEAM FIRE ENGINE AND SUGAR MILLS. 

STEAM FIRE ENGINE. 
1st Class. (Amoskeag, IN*. H.) 

Steam Cylinder. — Two of 1% ins. in diam. by S ins. stroke of piston. 

Witter Cylinder.— Two of 4>£ ins. in diam. 

Boiler (vertical tubular). — Heating surface, 175 square feet. Grates, 4.75 squ 
feet. 

Pressure of Steam. — 100 lbs. per square inch. Revolutions, 200 per minute. 

Discharges. — Two gates of 2^ ins., through hose, one of 1% in. and two of 1 in. 

Projection Horizontal, 1% in. stream, 311 feet; two 1 in. streams, 253 feet 

Vertical, 1^ in. stream, 200 feet. 

Water Pressure With \ x /i in. nozzle, 200 lbs. 

Time of Raising Steam. — From cold water, 25 lbs., 4 min. 45 sec. 

Weights Engine complete, 6000 lbs. ; water, 300 lbs. 



2d Class. "Portland. Co. ("Portland, Me.) 

Steam Cylinder 9% ins. in diam. by 10 ins. stroke of piston; volume of piston 

space, .421 cubic feet. 

Water Cylinder.— 4% ins. in diam. by 10 ins. stroke of pi-ton; volume of piston 
space, .1025 cubic foot. 

Boiler (vertical tubular). — Heating surface, 134. 3 square feet. Grates, 5. 5 sq. feet. 

Pressure of Steam SO lbs. per square inch. Revolutions, 200 per minute. 

Pressure of water in cylinder double that of steam when«discharging through 500 
feet of hose. 

Discharges. — Two of % in. ; one of \% ins. 

Projection.— Horizontal, two % in. streams, 200 feet ; one \}i ins., 260 feet. Ver- 
tical, one \y & ins., 100 feet hose, 210 feet. 

Weights.— Complete, 5000 lbs. ; water, 1000 lbs. 

Time of Raising Steam. — From cold water, 6 to S minutes ; from water at 130°, 
4 minutes. 



SUGAR MILLS. 
Exjiressing 20 000 lbs. Cane-juice per Day. 

Engine {Non-condensing). — Cylinder, 15 ins. in diam. by 4 feet stroke of piston. 

Boiler (cylindrical flue) — 62 ins. in diam. by 30 feet in length; two return flues 
IS ins. in diam. Grates, 36 square feet. Weight, 15500 lbs. 

Pressure of Steam.— 50 lbs. per square inch, cut off at X the stroke of the piston. 
Revolutions, 30 per minute. 

Rollers.^ Two sets of 3 each, 24 ins. in diam. by 5 feet in length; geared 2.5 to 3C 
of engine. Speed of peripheries, 15.5 feet per minute. 

Fly Wheel, IS feet in diam. ; weight, 11 200 lbs. 

Note. — The arrangement of a second set of rolls is for the purpose of distributing 
the cane over an increased surface of rollers, reducing their speed, and affording 
more time for the juice to run off; an increase of 20 per cent, is effected by it. 



SUGAR MILLS AND FLOUR MILLS. 655 

sugar mills. — Continued, 

Expressing 40 000 lbs. Cane-juice per day, or for a Crop o/*5000 Boxes of 
450 lbs. each in four Months' Grinding. 

Engine (Non- condensing). — Cylinder, 18 ins. in diam. by 4 feet stroke of piston. 

Boihr (cylindrical flue).— 64 ins. in diam. and 36 feet in length ; two return flues, 
20 ins. in diam. Heating surface, 660 square feet. Grates, 30 square feet. 

Pressure of Steam. — 60 lbs. per square inch, cut off at X the stroke of the piston. 
Revolutions, 40 per minute. 

Rollers.— One set of 3, 28 ins. in diam. by 6 feet in length ; geared 1 to 14. Shafts^ 
11 and 12 ins. in diam. 

Spur Wheel, 20 feet in diam. by 1 foot in width. 
Fly Wheel, 18 feet in diam. ; weight, IT 400 lbfc. 

WEIGHTS OP ENGINE, BOILER, ETC. 



Engine 61 460 lbs. 

Sugar Mill 65 730 " 

Spur Wheel and Connecting 
Machinery to Mill 28 6S0 " 



Boiler 18 520 lbs. 

Appendages 6 T30 " 

Total 181 120 " 



FLOUR MILLS. 

30 Barrels of Flour per Hour. 

Water -tdheels— Overshot.— Five, diam. 18 feet by 14.5 feet face. Buckets, 15 ins. 
in depth. 

Water — Head, 2.5 feet. Opening, 2.5 ins. by 14 feet in length over each wheel. 

. 5 Barrels of Flour per Hour, and Elevating 400 Bushels of Grain 86 feet. 

Water-wheel — Overshot. — Diam. 22 feet by 8 feet face. Buckets, 52 of 1 foot in 
depth. 

Water.— Head, from centre of opening, 25 ins. Opening, 1% ins. by 80 ins. in 
length. 

Revolutions, 3% per minute. Stones, 3 of \% feet ; revolutions, 130. 

Three Run of Stones, Diameter 4 feet. 
Water-wheel— Overshot Diam. 19 feet by 8 feet face. Buckets, 14 ins. in depth 

Or, 

Steam-engine (Non-condensing). — Cylinder, 13 ins. in diam. by 4 feet stroke of 
pision. 

Boiler (cylindrical flue) — Diam. 5 feet by 30 feet in length; two flues 20 ins. in 
diameter. 

3K* 



656 PRESSES. — PILE-DRIVING. HOISTING ENGINE. 

HYDROSTATIC PRESS (CottO?l). 
30 Bales of Cotton per Hour. 

Engine {Non-condensing). — Cylinder, 10 ins. in diam. by 3 feet stroke of piston. 

Pressure of Steam.— 50 lbs. per square inch, full stroke. Revolutions, 45 to CO 
per minute. 

Presses.— Two, with 12-inch rams; stroke, 4.5 feet. 

Pumps — Two, diam. 2 ins. ; stroke, 6 ins. 



For 1000 Bales in 12 Hours. 

Engine {Non-condensing). — Cylinder, 14 ins. in diam. by 4 feet stroke of piston. 

Boilers. — Three (plain cylindrical), 30 ins. in diam. by 26 feet in length. Grates, 
32 square feet. 

Pressure of Steam. — 40 lbs. per square inch. Revolutions, 60 per minute. 

Presses.— Four, geared 6 to 1, with two screws, each of 7.5 ins. in diam. by 1.625 
in pitch. 

Shaft (wrought iron). — Journal, 8% ins. 

Fly Wheel, 16 feet in diam. ; weight, 8960 lbs. 

PILE-DRIVING. 

Driving Two Piles. 

Engine (Non-condensing).— Cylinders, two, 6 ins. in diam. by IS ins. stroke of 
piston. 

Boiler (horizontal tubular). — Shell, diam. 3 feet by 6 feet in length. Furnace end 
3.75 feet in width, 3.5 feet in length, and 6 feet in height. 

Pressure of Steam. — 60 lbs. per square inch, full stroke. Revolutions, 60 to SO 
per minute. 

Frame, S.5 feet in width by 26 feet in length. Leaders. 3 feet in width by 24 feet 
in height. 

Hams.— Two, 1000 lbs. each, lifted 5 times in a minute. 

Driving One Pile. 

Engine {Non-condensing). — Cylinder, 6 ins. in diam. by 1 foot stroke of piston. 

Boiler (vertical tubular).— 32 'ins. in diam. by 6.166 feet in height. Grates, 3.7 
square feet. Furnace, 20 ins. in height. Tubes, 35, 2 ins. in diam., 4.5 feet in 
length. 

Revolutions, 150 per minute. Drum, 12 ins. in diam., geared 4 to 1. Leader, 
40 feet in height. 
Ram.— One, 2000 lbs., 2 blows per minute. Fuel, 30 lbs. coal per hour. 

HOISTING ENGINE. 







Lid 


gerwood's. 


(For all Purj 


oses.) 








Single Engine. 






Double Engine. 




Horse- 


Diameters. 


Capacity, 
single 


Horse- 


I> i a in e t e 


r s. 


Capacity, 
single 


power. 


Cylinder. 


Drum. 


Boiler. 


rope. 


power. 


Cylinder, j Drum. 


Boiler 


rope. 


No. 


Ins. 


Ins. 


Ins. 


lbs. . 


No. 


Ins. 


Ins. 


Ins. 


•lbs. 


4 


5 


10 


2S 


1200 


8 


5 


12 


32 


1800 


6 


6 


10 


30 


1500 


12 


6 


• 14 


36 


2500 


10 


7 


12 


34 


2000 


20 


7 


14 


40 


5000 


12 


8 


14 


86 


3000 


35 


8 


14 


42 


7000 


15 


8.5 


16 


40 


5000 


40 


8.5 


IS 


48 


10000. 


25 


10 


16 


42 


6000 


50 


10 


IS 


50 


12000 



PUMPING ENGINE AND COTTON FACTOEIES. 657 

pumping engine (Lynn, Mass.). 

For Elevating 200 000 Gallons of Water per Hour. 

Engine (Compound).— Cylinders, 17.5 and 3G in?, in diam. by 7 feet stroke of pis- 
ton: Volume of piston space, 61.2 cubic feet. Air Pump (double-acting), 11.25 ins. 
in diam. by 49.5 ins. stroke of piston. 

Pump Plunger, 18.5 ins. in diam. by 7 feet stroke. 

Pressure of Steam, 90.5 lbs. ; average in high-pressure cylinder, 86 lbs., cut off at 
1 foot, or to an average of 44.5 lbs. ; average in low-pressure cylinder, 27 lbs., cut off 
at 6 ins., or to an average of 10.8 lbs. 

Revolutions, IS. 3 per minute. 

Boilers.— T«o (return flue) horizontal tubular; diam. of shell, 5 feet; drum, 3 
feet ; tubs, 3 ins. Length of shell, 16 feet. Grates, 27.5 square feet. 

Fly Wheel— Weight, 24 000 lbs. 

Evaporation of Water, 4644 lbs. per hour. Loss of action by Pump, 4 per cent. 

Consumption of Coal. — Lackawanna, 291 lbs. per hour. 

Duty, 205 772 gallons of water per hour, under a load and frictional resistance of 
73.41 lbs. per square inch, equal to 103923 217 foot-pounds for each 100 lbs. of coal. 

cotton factoeies {English). 

'For driving 13000 Spindles {Mules and Throstles), with 256 Looms for 
Cloth three quarters wide. No. 30. 

Engine {Condensing) —Cylinders, two ; 22 ins. in diam. by 3 feet stroke of piston ; 
volume of piston space, 15.8 cubic feet. 

Pressure of Steam. — 15 to 45 lbs. per square inch, cut off at ~% the stroke of the 
piston. Revolutions, 50 per minute. 

For driving 22 060 Hand-mule Spindles, with preparation, and 2G0 Looms, 
with common Sizing. 

Engine (Condensing)— Cylinder, 37 ins. in diam. by 7 feet stroke of piston; vol- 
ume of piston space, 53.6 cubic feet. 

Pressure of Steam. — (Indicated average) 16.73 lbs. per square inch. Revolutions, 
17 per minute. 

Friction of Engine and Shafting.— (Indicated) 4.75 lbs. per square inch of piston. 

Indicated Horses' Power, 1-5. 

Total power = 1. Available, deducting friction = . 717. 

Notes. — Each Indicated Horse's power will drive 

305 hand-mule spindles, with preparation, 
* or 230 self-acting " " 

or 104 throstle " " 

or 10.5 looms, with common sizing. 

Including preparation : 

1 throstle spindle = 3 hand-mule, or 2.25 self-acting spindles. 
1 self-acting spindle = 1.2 hand-mule spindles. 
Exclusive of preparation, taking only the spindle: 

1 throstle spindle = 3.5 hand-mule, or 2.56 self-acting spindles. 
1 self-acting spindle = 1.375 hand-mule spindles. 
The throstles are the common, spinning 34 twist for power-loom weaving; the 
spindles revolve 4000 per minute. The self -acting mults are, one half spinning 36's 
weft, spindles revolving 4S00 ; the other half spinning 36's twist, spindles revolving 
5200. The hand-mules spinning about equal quantities of 36's weft and twist. Weft 
spindles 4700, and twist spindles 5000 revolutions per minute. 

Average breadth of looms 37 ins. (weaving 37 ins. cloth), making 123 picks per min- 
ute. All common calicoes about 60 reed, Stockport count, and 68 picks to the inch. 
No power consumed by the sizing. When the yarn is dressed instead of sized, 
one horse's power can not drive so many looms, as the dressing machine will absorb 
from .17 to .14 of th*e power. 



658 BOWING. 



ROWING. 

NOTE.— Xo performances but sucn as nave been made over properly measured courses are-here given*, 
hence New York Harbor, Hudson and Harlem rivers, and like courses, are omitted, except when 
the effect of a tidal or fluvial current has been compensated by a " turn," i. e. } an equal course with 
and against the current. 

Scmlls. 

TWO MILES. 

1561, J. D. Parker and Carpenter, Boston, Mass., Shell, double sculls, one turn, in 
12 min. 54% sec. 

1S71, Miss Amelia Shean, Harlem, N. Y., Working boat, single sculls, 3 turns, in 
18 min. 32% sec. 

1ST6, James Riley, Saratoga, N. Y., Shell, single sculls, one turn, in 13 min, 21^ 
sec. 

1S76, F. E. Yates and C. E. Courtney (amateurs), Saratoga, N. Y., one turn, in 12 
min. 16 sec. 

THREE MILES. 

1ST2, Edioard Smith, Staten Island, N. Y., Shell, single sculls, one turn, in 21 min. 
51% sec. ; and Geo. Engelhart, in 22 min. 21% see. 

1ST7, C. E. Courtney, Owego, N. Y., Shell, single sculls, one turn, in 20 min. 
14% sec. 

1S74, E. Smith and F. C. Eldred, Saratoga, N. Y., Shell, double sculls, one turn, in 
21 min. 52% sec. 

FOUR MILES. 

1S71, Jos. H. Sadler, of England, Saratoga, N.Y., Shell, single sculls, one turn, in 
30 rain. 1S% sec. 

FIVE MILES. 

1562, James Hamill, Philadelphia, Penn., Shell, single sculls, one turn, in 37 min. 
39 sec. 

1.374, George Brown, near St. John's, N. B., Shell, single sculls, one turn, in 37 min. 

six MILES. 

1S74, O'Briens and T. Twigg Crew, Boston, Mass., Shell, 4 oars, bne turn, in 40' 
min. 2 sec. 

Oars. 

• TWO MILES. 

1871, Ward Brothers Crew, Saratoga, N. Y., Shell, 4 oars, straight course, in 11 min. 
20 sec. 

THREE MILES. 

1860, Union Boat Club, of Boston, Worcester, Mass., Shell, 4 oars, one turn, in 19 
min. 41 sec. 

1868, Harvard University Crew?, Worcester, Mass., 6 oars, one turn, in IT min. 
48% sec. 

1868, Ward Brothers Crew, Worcester, Mass., Shell, 6 oars, one turn, in 17 min. 
40% sec. 

•1872, Amherst University Crew, Springfield, Mass., 6 oars, straight course, in 16 
min. 32.8 sec. 

1S74, Beaverwyck Club, of Albany, N.Y., Saratoga, N.Y., 4 oars, one turn, in IS 
min. 34 sec. 

FOUR MILES. 

1S71, Ward Brothers Crew, Saratoga, N. Y., Shell, 4 oars, one turn, in 24 min. 40 sec. 

FIVE MILES. 

1867, "«/. F. Tapley," Springfield, Mass., Shell, 6 oars, one turn, in 33 min. 1% sec. 

SIX MILES. 

1S67, Ward Brothers Crew, of New York, Springfield, Mass., Shell, 4 oars, one turn, 
in 39 min. 28 sec. 

English College Races. 

18C9, Oxford University Crew, Putney to Mortlake, Eng., 4% miles, 8 oars, favora- 
ble current, in 20 min. 6% sec. 

1873, Cambridge University Crew, Mortlake to Putney, Eng., 4% miles, S oars, 
favorable current, in 19 min. 35 sec. 



ROWING. 659 

American College Races. 

1S65, Yale University Crew, Worcester, Mass., Shell, 3 miles, 6 oars, one turn, in 17 
min. 42% sec* 

186S, Harvard University Crew, Worcester, Mass., Shell, 3 miles, 6 oars, one turn, 
in 17 min. 48% sec. 

1872, Amherst College Crew, Springfield, Mass., Shell, 3 miles, 6 oars, straight 
course, in 16 min. 32.8 sec. 

1874, Columbia College Crew, Saratoga, N. Y., Shell, 3 miles, 6 oars, straight 
course, in 10 min. 42% sec. 

1876, Yale University Crew, Springfield, Mass., 4 miles, S oars, one turn, in 22 min. 
2 sec. 

1878, Harvard University Crew, New London, Conn., 4 miles, 8 oars, one turn, in 
20 min. 44% sec. 



Various Distances and. Performances. 

Note.— In the following cases the effects of the direct and varying currents have been impracticable 
of attainment. 

SCULLS AND OARS. 

1330, F. Creswell and William Lewis, of Eng., Thames wherry, Billingsport to 
Gravesend, up to Richmond Bridge and down to Old Swan, Eng , 96 miles, in 11 
hours 50*mm. 

1817, George Gyngell, of Millwall, London, Eng., Thames wherry, Thames River, 
1 000 miles, in 20 days. 

1S32, J. Williams, single sculls. London, Eng., to Gravesend, to Richmond, and 
back to London, 91 miles, in 11 hours 29 min. 3 sec, without leaving his seat. 

1860, James Ward, Poughkeepsie, N. Y., Shell, single sculls, one turn, 10 miles, 
in 1 hour 23 min. 

1866, Kingston Club, Henley, Eng., Barge, S oars, straight course, favorable cur- 
rent, 1.3125 miles, in 7 min. 21 scc.—\ mile in 5 min. 36 sec. 

1869, G. W. Chambers, around Staten Island, N.Y., Working boat, 33>£ miles, in 6 
hours 48 min. 

1S74, William B. Curtis, Calumet River, 111., Paper Shell, 50 miles, rough water, in 
10 hours 11 min. 55 sec. 

1860, Biglin Brothers and Leary, Harlem River, N. Y.,4 oars, 5 miles, one turn, 
in 30 min. 44% sec. 

1869, Tyne Crew, Putney to Mortlake, Eng., 4% miles, four oars, favorable cur^ 
rent, in 20 min. 43 sec. 

1876, Geo. Tarryer, Putney to Mortlake, Eng., 4% miles, single sculls, favorable 
current, in 23 min. 4 sec. 



International Races. 

1863, R. Chambers, of England, Thames, Eng., Shell, 4.375 miles, straight course, 
favorable current, in 25 min. 23 sec. 

1866, Henry Kelley, of England, Newcastle, Eng., Shell, 4.429 miles, straight 
course, favorable current, in 33 min. 29 sec 

18Q9,Walter Brown, of Maine, U. S., NeAvcastle, Eng., Shell, 3.405 miles, straight 
course, favorable current, in 21 min. 50 sec. 

1871, Paris Crew, of St. John's, N. B., Kennebecassis River, N. B., Shell, 6 miles, 
4 oars, one turn, in 38 min. 50 sect 

1S71, Taylor -Winship Crew, of England, Halifax, N. S., 6.977 miles, 4 oars, one 
turn, in 44 min. 28 sec. 

1S71, Ward Brothers Crew, of New York, Saratoga, N.Y.. Shell, 4 miles, 4 oars, one 
turn, in 24 min. 40 *ec ; and to turn, 2 miles in 11 min. 2v sec. 

1S79, E. Hanlan, of Canada, Newcastle, Eng., Shell, 3.319 miles, straight course, 
favorable current, in 21 min. 1 sec. 

* Time disputed. t Engelhardt gives 39 min. 20^ see. 



660 



MISCELLANEOUS NOTES. 



MISCELLANEOUS NOTES. 



Solutions of Questions "by a Graphic Operation 
1. If a man walks 5 miles in 1 hour, how far will he walk in 4 hours ? 

Operation. — Draw a horizontal line and divide it into 
equal parts, as 1, 2, 3, and 4, representing hours. From each 



; : -nnlH H* these points let fall the vertical lines A C, 1 1, etc., and 

£- - -f— - v divide A C into miles, as 5, 10, 15, and 20, and from these 




points draw equidistant and horizontal lines parallel thereto. 

Hence, the horizontal lines represent time or hours, and the 
vertical distance or miles. 

Therefore, as any inclined line in the diagram represents 
both time and distance, the course of the man walking 5 miles 
in an hour is represented by the diagonal Ae; and if he walks 
fur 4 hours, continue the time to 4, and read off from the ver- 
tical line A C the distance = 20 miles. 

2. If a second man were to walk 10 miles in 1 hour, or twice as far as 
the first, how far will he walk in 2 hours ? 

His course is shown by the line A o, representing 20 miles. 

3. If two men start from a point at the same time, one walking at the 
rate of 5 miles in an hour and the other at 10 miles, how far apart will 
they be at the end of 2 hours ? 

Their courses being shown by the lines A r and A o, the distance r o represents the 
difference of their distances, or 10-^20 = 10 miles. 

4. How long have they been walking ? 

Their courses are now shown by the lines A o and A 4, the distance 2 4 represents 
the difference of their times, or 2a;4 = 2 hours. 

5. When they are 10 miles apart, how long have they been walking? 

Their courses are again shown by the lines A r and A o, the distance r o repre- 
sents the difference of their distances of 10 miles, and A 2, 2 hours. 

6. If a man walks a given distance at the rate of 3.5 miles per hour, and 
then runs a part of the distance back at the rate of 7 miles per hour, and 
walks the remainder of the distance in 5 minutes, occupying 25 minutes 
of time in all, how far did he run ? 

Operation. — Draw a horizontal line, as A C, 
representing the whole time of 25 minutes ; set 
off the point e representing a convenient frac- 
tion of an hour (as 10 minutes), and a i equal 
to the corresponding fraction of 3.5 miles (or 
.5833) ; draw the diagonal A n, produced in- 
" «i definitely to O, and it will represent the rate of 
3.5 miles per hour. 

Set off C r equal to 5 minutes, upon the 
same scale as that of A C ; let fall the vertical 
r s, and draw the diagonal C u at the same 
angle of inclination as that of A n; then from 
the point u draw the diagonal u O, inclined at such a rate as to represent 7 miles 
per hour ; thus, if i n represents the rate of 3.5 miles, s O, being one half of the dis- 
tance, will represent 7 miles. 

The whole distance between the two points is thus determined by C ar, and the 
distance ran by u «j measured by the Fcale of miles employed. 

Verification. — The distances A e and A i are respectively 10 minutes = % of 
an hoiir, and .5S33 miles = % of 3 - 5 mile*. Henc?, C x = .875 miles, and u 8 
= .5833 miles. Consequently, the man walked A <) = .S75 miles = 15 minutes, ran 
u = .5333 miles = 5 minutes, and walked »C = .2910 miles. 




MISCELLANEOUS NOTES. 661 

7. If a second man were to set out from C at the same time the man re- 
ferred to in the preceding question started from A, and to walk to A and 
return to C, at a uniform rate of speed and occupying the same time of 
25 minutes, at which points and times will he meet the i;rst man ? 

Operation. — As A C represents the whole time, and C x the distance between the 
two points, v z and z x will represent the course of the second man walking at a 
uniform rate, and he will meet the first man, on his outward course, at a distance 
from his starting-point of A, represented by A o, and at the time A a; and on his 
return course at the distance A v, x m, and at the time Ac. 

Matter. " izonta ' 

The unit of the plrysicist is a molecule, and a mans of matter is com- 
posed of them, having the same plrysical properties as the parent mass. 

It exists in three forms, knownas solid, liquid, and gaseous. Solids 
have individuality of form, and they press downward alone. Liquids 
have not individualit}^ of form, except in the spherical form of a drop, 
and they press downward and sideward. Gases are wholly deficient in 
form, expanding in all directions, and consequently they press upward, 
downward, and sideward. 

Liquids are compressible to a very moderate degree. Water has been 
forced through the pores of silver, and it may be compressed bj T a pressure 
of one pound per square inch to the 3 300 000th part of its volume. 

Gases may be liquified by pressure or by the reduction of heat. 

Combustible matter (as coal) maybe burned, a structure (as a house) 
may be destroyed as such, and the fluid of an ink may be evaporated, j T et 
the matter of which the coal and the house were composed, although dis- 
sipated, exists, and the water and coloring matter of the ink are yet in 
existence. 

The spaces between the particles of a body are termed pores. 

All matter is porous. Polished marble will absorb moisture, as evi- 
denced in its discoloration by the presence of a colored fluid, as ink, 
etc. 

Silica is the base of the mineral world, and carbon of the organized. 

HVIiiixiteness of* !M!atter. 

A piece of metal, stone, or earth, divided to a powder, a particle of it, 
however minute, is } r et a piece of the original material from which it was 
separated, retaining its identity, and is termed a molecule. 

It is estimated that there are 120 000 000 corpuscles in a drop of blood 
of the musk-deer. 

The thread of a spider's web is of a cable form, and is but one sixth the 
diameter of a fibre of silk. 

One imperial gallon (277.24 cubic inches) of water will be colored b} r 
the mixture therein of a grain of carmine or indigo. 

A grain of platinum can be drawn out the length of a mile. 

The film of a soap-and-water bubble is estimated to be but the 300 000th 
part of an inch in thickness. 

It is computed that it would require 12 000 of the insect known as the 
twilight monacl to fill up a line one inch in length. 

A drop of waiter, or a minute volume of gas, however much expanded — 
even to the volume of the earth — would present distinct molecules. 

Gold leaf is t he 280 000th part of an inch in thickness. 

Volumes. 
Permanent gaises, as air ? etc., are diminished in their volume in'a ratio 
direct with that of the pressure applied to them. With vapor, as steam, 
etc., this rule is >yaried in consequence of the presence of the temperature 
of vaporization. , 

3L 



662 MISCELLANEOUS NOTES. 

Minerals. 

Minerals have varying degrees of hardness, as illustrated in the fol. 
lowing order, from the hardest to the softest: 

1. Diamond. 6. Apatite. 

2. Sapphire or ruby. 7. Fluor-spar. 

3. Topaz. 8. Calc-spar. 

4. Quarts 9. Gvpsum. 

5. Feldspar. 10. Talc. 

Hence a minfrel that can impress gypsum, and would be impressed by 
fluor-spar, would be termed H 8. 

Metals. 
• 

Metals have five degrees of lustre — splendent, shining, glistening, glim- 
mering, and dull. 

All metals can be vaporized, or exist as a gas, by the application to 
them of their appropriate temperature of conversion. 

The repeated hammering of a metal renders it brittle ; reheating it re- 
stores its tenacity. 

The repeated melting of iron renders it harder, and up to the twelfth 
time it becomes stronger. 

Platinum is the most ductile of all metals. 

Elasticity. 

Elasticity is the term for the capacity of a body to recover its former 
volume, after being subjected to compression by percussion or bending. 

Glass, ivory, and steel are the most elastic of all bodies, and clay and 
putty are illustrations of bodies almost devoid of elasticity. Caoutchouc 
(India rubber) is but moderately elastic ; it possesses contractility, how- 
ever, in a great degree. 

Impenetrability. 

Impenetrability expresses the inability of two or more bodies to occupy 
the same space at the same time. 

A mixture of two or more fluids may compose a less volume than that 
due to the sum of their original volume, in consequence of a denser or 
closer occupation of their molecules. This is evident in the mixture of 
alcohol and water in the proportion of 10% volumes of the former to 25 
of the latter, when there is a loss of one volume. 

Charcoal. 

Charcoal as yet has not been liquified, nor has alcohol been solidified. 

Temperature. 

Sulphuric acid and water produce a much greater proportionate con- 
traction than alcohol and water. Both of these mixtures, however low 
their temperature, produce heat which is in a direct proportion to their 
diminution in volume. 

At the depth of 45 feet, the temperature of the earth is uniform through- 
out the year. 

Momentum. 

Momentum is the quantity of motion, and is the product of the mass 
and its velocity. Thus the momentum of a cannon-ball is the product of 
its velocity in feet per second and its weight, and is denominated foot- 
pounds. 

A foot-pound is the power that will raise one pound one foot. 

Sound. 
Sound passes in water at a velocity of 4708 feet per second. 






MISCELLANEOUS NOTES. 663 

Metal Boring and. Turning. 

Boring cast iron. Divide 25 by the diameter of the cylinder in inches 
for the revolutions per minute. 

Wrought iron. — The speed is one fifth greater than for cast iron. 

Brass. — The speed is one half that for cast iron. 

Turning cast iron. — The speed is twice that of boring. 

Wrought iron. — The speed is one fifth greater than that for cast iron. 

Brass.— The speed is twice that of boring. 

Vertical boring. — The speed may be twice that of horizontal boring. 

The feed depends upon the stability of the machine and depth of the 
cut. 

"W^ell Boring. 

At Coventry, Eng., 750 000 galls, of water per day are obtained by two 
borings of 6 and 8 inches, at depths of 200 and 300 feet. 

At Liverpool, Eng., 3 000000 galls, of water per day are obtained by a 
bore 6 inches in diameter and 161 feet in depth. 

This large yield is ascribed the existence of a fault near to it, and extending to 
a depth of 4S4 feet. 

At Kentish Town, Eng., a well is bored to the depth of 1302 feet. 

At Passy, France, a well with a bore of 1 metre in diameter is sunk to a 
depth of 1804 feet, and for a diameter of 2 feet 4 ins. it is further sunk to a 
depth of 109 feet 10 ins., or 1903 feet 10 ins., from which a yield of 5 582 000 
gallons of water are obtained per da}*. 

Tempering IBoring Instruments. 

Heat the tool to a blood-red heat; hammer it until it is nearly cold; 
reheat it to a blood-red heat, and plunge it into a mixture of 2 oz. each of 
vitriol, soda, sal ammoniac, and spirits of nitre, 1 oz. of oil of vitriol, % 
oz. of saltpetre, and 3 gallons of water, retaining it there until it is cool. 

Bridges. 

Bridges should have between parapets or rails, for a 

Turnpike or public road 35 feet. 

Public side road 25 " 

Private road 12 " 

The parapets should be 4 feet high. 

The approaches should not have a greater grade than, for a 

Turnpike or public road 1 in 30 feet. 

Public side road 1 in 20 " 

Private road 1 in 16 " 

Ifcoads. 

The maximum grade of a turnpike road is 1 in 30 feet. An ascent is 
easier for draught if taken in alternate ascents and levels, than in one con- 
tinuous rise, although the ascents may be steeper than in a uniform grade. 

Traction on Roads. — The traction of a wheeled vehicle is to its weight 
upon various roads as follows : 

Asphalted road, as .1 to 100 Macadamized road, as 1 to 45 

Plank road 1 to 100 do. stones loose. 1 to 31 

Stone trackway 1 to 77 Gravel road 1 to 15 

Stone pavement 1 to GO Sandy road 1 to 7 

Hence, a horse that can draw 140 lbs. at a walk, can draw upon a 
gravel road 140 x 15 = 2100 lbs. 
The above is a mean and a low estimate. 



664 



MISCELLANEOUS NOTES. 



Assuming the maximum load that a horse can draw on a gravel road 
as a standard, he can draw, 

On a well-made pavement 4.5 times as much. 

On a stone trackway 11 u 

On a railway 18 

Hence, the track of an iron railway compared with a plank road is as 
270 to 100. 

Inclination of Roads. — The power of draught at different inclinations 
and velocities is as follows. {Sir Henry Parnell.) 





Angles. 


Feet per Mile. 


Pow 


er in Miles per 


Hour. 


Inclination. 


Six Miles. 


Eight Miles. 


Ten Miles. 


1 in 20 


2° 52' 


264 


26S 


290 


318 


1 in 2G 


2° 12' 


203.4 


213 


219 


225 


1 in 30 


1°55' 


176 


165 


196 


200 


1 in 40 


1° 26' 


132 


160 


166 


172 


1 in 60 


W%' 


8S 


111 


120 


128 



Assuming the load that a horse can draw on a level 100, he can draw 
upon inclinations as follows : 

1 in 100 .... 90 1 in 40 .... 72 1 in 20 .... 40 

1 in 50 81 1 in 30. . . .64 1 in 10. . . .28 

Hence, upon an inclination of 1 foot in 44, or 120 feet in a mile, he can 
draw but % as much as he can upon a level ; upon 1 in 24, or 220 feet in 
a mile, but % as much ; and upon 1 in 10, or 528 feet in a mile, but % as 
much. 

Rock Blasting. 

The proper charge of powder for a blast is determined by dividing the 
cube of the line of least resistance in feet by 32. Thus, a line of 4 feet 



will require 4 3 -f- 32 : 



32 



or, for every ton of rock to be removed 



there is required % lb. of powder. 
A hole 1 inch in diameter will contain 5 oz. in each foot of its depth. 

Filing. 
Piles may be loaded to 1000 lbs. per square inch of head, if driven to 
firm bottom. 

In sandy soil, the greatest force of a pile-driver will not drive a pile 
over 15 feet. 

Masonry. 
Concrete or Beton should be thrown, or let fall from a height of at least 
10 feet, or well beaten down. 

The average weight of brickwork in mortar is about 102 lbs. per cubic 
foot. 

Dredging. 
A steam dredge will raise 6 cubic yards, or 8.5 tons, per hour per horse- 
power. 

Pavement. 
Paving-blocks, as the Belgian, etc., where the crest of the street or 
area of pavement does not exceed 1 inch in 7)^ feet, should taper slightlv 
toward the top, and the joints be well filled ("blinded") with gravel. 
The common practice of tapering them downward is erroneous. 

^N"ew York and Brooklyn Bridge. 

Height of under side above high water, 119 feet at side and 135 in centre. 

Midland Railway Station. {London.) 
Span of roof, 240 feet. 



MISCELLANEOUS NOTES. 665 

Plastering. 
In measuring plasterers' work all openings, as doors, windows, etc., 
are computed at one half of their areas, and cornices are measured upon 
their extreme edges, including that cut off by mitring. 

Grlazing. 
In glaziers' work, oval and round windows are measured as squares. 

Corn IVEeasrire. 
Two cubic feet of corn in the ear will make a bushel of corn when 
shelled. 

Tides. 

The difference in time between high water averages about 49 minutes 
each day. 

A. Hj-uminoxis 3?oint. 

To produce a visual circle, a luminous point must have a velocity of 10 
feet in a second, the diameter not exceeding 15 inches. 
All solid bodies become luminous at 800 degrees of heat. 

Current of Rivers. 
A fall of .1 of an inch in a mile will produce a current in rivers. 

3VEelted Snow 
Produces from % to % of its bulk in water. 

IVIagnetic Bearings of jN"ew York. 
The avenues of the city of New York bear 28° 50' 30" east of north. 

Service Train of a Quartermaster. 

The Quartermaster's train of an army averages 1 wagon to every 24 
men ; and a well-equipped army in the field, with artillery, cavalry, and 
trains, requires 1 horse or mule, upon the average, to every 2 men." 

Dimensions of Drawings for Patents. 

United States, 8.5x12 inches. 

Candles. 

A Spermaceti candle .85 of an inch in diameter consumes an inch in 
length in 1 hour. 

UVXalleaDle or Aluminum Bronze. 

By weight: copper, 90; aluminum, 10. This composition may be 
forged either when heated or cooled, and becomes extremely dense. Its . 
tensile strength is 100 000 lbs., and when drawn into wire~128 000 lbs., 
and its elasticity one half that of wrought iron. Specific gravity, 7700. 

A.tlantic and Pacific Oceans. 
The height of the Atlantic Ocean at the Isthmus of Darien is 6.56 feet 
above the Pacific Ocean. 

Hoofs. 

Midland Railway Station, London. 240ft. I Union Railway Station, Glasgow.. 195ft. 
Imperial Riding-School, Moscow. . . 235 u | Grand Central Stataion, N. Y 200 u 

New York and Brooklyn Bridge. 

New York approach 15 >2 feet. 

Brooklyn u 971 " 

River space 1595 c ' 

Land spaces each 935 



3L* 



Roading above high water 119 feet. 

Tower «« " 2G5 « 

Length, total 50C3 " 

Height from water 346 w 



666 OKTHOGRAPHY OF TECHNICAL AVOKDS AXD TEEMS, 



ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. 

The orthography in ordinary use of the following words and terms 
is so varied, that they are here giveu for the purpose of establishing a 
more general uniformity of expression. 

Abut. To meet, to adjoin to at the end, to border upon. The alut end of a log, 
etc., is that having the greatest diameter or side. 

But and Butt end, when applied in this manner, are corruptions. 

Amidships. The middle or centre of a vessel, either fore and aft or athwart ships. 
The amidship frame of a vessel is at 0. 

Arabesque. Applied to painted and carved or sculptured ornaments of imaginary 
foliage and animals, in which there are no perfect figures of either. Synonymous 
with Moresque. 

Arbor. The principal axis or spindle of a machine of revolution. 

Bagasse. Sugar-cane in its crushed state, as delivered from the rollers of a mill 

Baluster. A small column or pilaster ; a collection of them, joined by a rail, forms 
a balustrade. 

Banister is a corruption of balustrade. 

Bark. A ship without a mizzen-topsail, and formerly a small ship. 

Bateau. A light boat, with great length proportionate to its beam, and wider at 
its centre than at its ends. 

Bevel. A term for a plane having any other angle than 45° or 90°. 

Binacle. The case in which the compass or compasses is set on board of a vessel. 

Bit. The part of a bridle which is put into an animal's mouth. In Carpentry, a 
boring instrument. 

Bitter End. The inboard end of a vessel's cable. 

Bitts. A vertical frame upon the deck of a vessel, around or upon which is secured 
cables, hawsers, sheets, etc 

Boomkin. A short spar projecting from the bow or quarter of a vessel, to extend 
the tack of a sail to windward. 

Bowlder. A stone rounded by natural attrition ; a rounded mass of rock trans- 
ported from its original bed. 

Breast-summer. A lintel beam in the exterior wall of a building. - 

Buhr-stone. Mill-stone which is nearly pure silex, full of pores and cavit ; 

Burden. A load. The quantity that a ship will carry. Hence burdens** a pile 

Cag. A small cask, differing from a barrel only in size. Commonly wrii 

Calibers. A compass with arched legs, to measure the diameters of spheres, or 
the exterior and interior diameters of cylinders, bores, etc. 

Callipers is a corruption. 

Calk. To stop seams and pay them with pitch, etc. To point an iron shoe so as 
to prevent its slipping. 

Cam. An irregular curved instrument, having its axis eccentric to the shaft upon 
which it is fixed. 

Camboose. The stove or range in which the cooking in a vessel is effected. The 
cooking-room of a vessel ; this term is usually confined to merchant vessels ; in ves- 
sels of war it is termed Galley. 

Cantle. A fragment; a piece ; the raised portion of the hind part of a saddle. 

Capstan. A vertical windlass. 

Caravel. A small vessel (of 25 or 30 tons burden) used upon the coast of France in 
herring fisheries. 

Cartings. Pieces of timber set fore and aft from the deck beams of a vessel, to re* 
ceive the ends of the ledges in framing a deck. 

Carvel built. A term applied to the manner of construction of small boats, to sig- 
nify that the edges of their bottom planks are laid to each other like to the manner 
of planking vessels. Opposed to the term Clincher* 



ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. 661 

Caster. A small phial or bottle for the table ; one of a set of Casters. 

Castors. Small wheels placed upon the legs of tables, etc., to allow them to be 
moved with facility. 

Catamaran. A small raft of logs, usually consisting of three, and designed for use 
in an open roadstead and upon the sea-coast. 

Chamfer. A slope, groove, or small gutter cut in wood, metal, or stone. To 
Chamfer is to slope, to channel, or to groove. 

Chimney. The flue of a fire-place or furnace, constructed of masonry in houses 
and furnaces, and of metal, as in a steam boiler. See Pipe. 

Chinse. To chinse, is to calk slightly with a knife or chisel. 

Chock. Small pieces of wood used to make good any deficiency in a piece of tim- 
ber, frame, etc. See Furrings. 

Choke. To stop, to obstruct, to block up, to hinder, etc. 

Cleats. Pieces of wood or metal of various shapes, according to their uses, either 
to resist or support weights or strains, as shoar cleats, beam cleats, etc. 
fr Clincher built. A term applied to the construction of vessels, when the lower 

. e of the bottom planks overlays the next under it. 

Coa'k. In Mechanics, a cylinder, cube, or triangle of hard wood let into the ends 
or faces of two pieces of plank or timber to be secured together. The metallic eyes 
in a sheave through which the pin runs. In Naval Architecture, the oblong ridges 
ban led on the masts of ships. 

Coamings. Raised borders around the edges of hatches. 

Coble. A small fishing-boat. 

Coccoon. The case which certain insects make for a covering during the period 
of their metamorphosis to the pupa staie. 

Cog. In Mechanics, a short piece of wood or other material let into the faces of a 
body to impart motion to another. A term applied to a tooth in a wheel when it is 
made of a different material than that of the wheel. 

Colter. The fore iron of a plow that cuts earth or sod. 

Compass. In Geometry, an instrument for describing circles, measuring figures, etc. 

To say, A pair of compasses, is superfluous and improper. 

Contrariwise. Conversely, opposite. 

Crossways is a corruption. 

Corridor. A gallery or passage in or around a building, connected with various 

c ^tments, sometimes running within a quadrangle : it may be opened or inclosed. 

'■'fications, a covert way. 
. q quinerie. Inlaying in metal, 
^no-th : t ^ snort hoom fitted to hoist an anchor or boat. 
j_ wai. To fasten two boards or pieces together by pins inserted in their edges. 

This is very similar to coaking, but is used in a diminutive sense. An illustration of it is had in 
the manner a Cooper secures two or more pieces in the head of a cask. 

Draught. A representation by delineation. The depth which a vessel or any 
floating body sinks into water. The act of drawing. A detachment of men from 
the main body, etc., etc. 

Edgewise. An edge being put into a particular direction. Hence endwise and 
sidewise have similar significations with reference to an edge and a side. 
Edgeways is a corruption. 

Felloe, Felloes. The pieces of wood which form the rim of a wheel. 

Flange. A projection from an end or from the body of an instrument, or any 
part composing it, for the purpose of receiving, confining, or of securing it to a sup- 
port or to a second piece. 

Frap. To bind together with a rope, as to frap a fall, etc. 

Frustrum. The part of a solid next the base, left by the removal of the top or 
segment. 

Furrings. Strips of timber or boards fastened to frames, joists, etc., in order to 
bring their faces to the required shape or level. 
Galcts. Pieces of stone chipped off by the stroke of a chisel. See SpalU 



G68 ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. 

Galeting. Putting galets into pointing-mortar or cement. 

Galiot. A small galley built for speed, having one mast, and from sixteen to 
twenty thwarts for rowers. A Dutch-constructed brigantine. 

Gate. In Mechanics, the hole through which molten metal is poured into a mold 
for casting. U1U 

Geat and Gett are corruption*. 

Gearing. A series of teethed or cogged wheels for transmitting motion To near 
a machine is to prepare to connect its parts as by an articulation. 

Gingle. To shake so as to produce a sharp, clattering noise. 

Girt. The circumference of a tree or piece of timber. Girth. The band or *tra D 
bv which a saddle or burden is secured upon the back of an animal, by passing around 
his belly. In Printing, the bands of a press. y g uiounu 

Beaming.' BmQin3 ° ff ¥W» suells ' etc " from a shi P' s ^oni. Synonymous with 

Grommet. A wreath or ring of rope. 

Gymbal Ring A circular rynd for the connection of the upper millstone to tho 
spindle by which the stone is suspended, so that it may vibrate upon ill side. 

Jtoefllee iafgTig*™^ l ° ^ W ° f * VG3Sel when *"* ends dro P bel ™ h * r 

Horsing. In JVamZ Architecture, calking with a large maul or beetle. 

Jaw. To press, to crowd, to wedge in. In Nautical Language, to squeeze tight. 

Jamb. A pier ; the sides of an opening in a wall. 

«on!L Th A G pi :?l ecfcin S beam of a crane fl '°m which the pulleys and weight are bus- 
pended. A sail in a vessel. ° c aia 

of a ? boom T ° Shift a b00m " sail from one tack t0 an( >ther; hence Jibing, the shifting 

anf exactly ove'tlS ?*** * ^ ^ UPOn the ^^ ° f the fl °° r timber3 > 

Cavil d ' LarSG W °° den Cleats t0 b2lay bawsers and r °Pes to, commonly written 

Lacquer. A spirituous solution of lac. 

Laitance. A pulpy, gelatinous fluid washed from the cement of concrete deposited 
in m ater. " 

noffloattr^it u \^\ expressive of the condition of a vessel or any body when it will 

Leat. A trench to conduct water to or from a mill-wheel. 

Leech. In Nautical language, the perpendicular or slanting edge of a sail when 
not secured to a spar or stay. b 6 eau wnen 

Luf. The fullest part of the bow of a vessel. 

Mall. A large double-headed wooden hammer, 
of f chimne^ ^^ t0 T^- Mantle ^ icce ^ the shelf over a fire-place in front 

Marquetry. Checkered or inlaid work in wood. 

Matrass. A chemical vessel with a body alike to an egg and a tapering neck. 

3fattress. A quilted bed ; a bed stuffed with hair, moss, etc., and quilted. 
. X2SFSL i lQ Mechanics > cut t0 an angle of 45°, or two pieces joined so as to make 

Mizzen-mast. The aftermost mast in a three-masted vessel. 

Mold. In Mechanics, a matrix in which a casting is formed. «A number of nieces 
of vellum or like substance, between which gold and silver are laid L fo? the rurpo'e 
of being beaten. Thin pieces of materials cut to curves or any required figure In 
Isaval Architecture, pieces of thin board cut to the lines of S SterTetc' 

J^SSSiSS^tS!^^^ soil * A 8ubstance which forms upon bodies in warm 

This orthography i 8 by analogy, as gold, sold, old, bold, cold, fold, etc. 

Molding. In A rchitecture, a projection beyond a wall, from a column, wainscot etc 
Moresque. See Arabesque. 



I 






ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. 669 

Mortise. A hole cut in any material to receive the end or tenon of another piece. 

Net. Clear of deductions, as net weight. 

Newel. An upright post, around which winding stairs turn. 

Nigged. Stone hewed with a pick or pointed hammer instead of a chisel. 

Ogee. A molding with a concave and convex outline, like to an S. 

Pagcting. In Architecture, rough plastering, alike to that upon chimneys. 

Paillasse. Masonry raised upon a floor. A bed. 

Parquetry. Inlaying of wood in figures. See Marquetry, 

Pawl. The catch which stops, or holds, or falls on to a ratchet wheel. 

Peek. The upper or pointed corner of a sail extended by a gaff, or a yard set ob- 
liquely to a mast. To peek a yard is to point it perpendicularly to a mast. 

Pendant. A short rope over the head of a mast, for the attachment of tackles 
thereto ; a tackle, etc. 

Pennant. A small pointed flag. 

Pile. In Engineering, spars pointed at one end and driven into soil to support a 
superstructure or holdfast. 

Spile is a corruption. 

Pipe. In Mechanics, a metallic tube. The flue of a fire-place or furnace when 
constructed of metal; usually of a cylindrical form. 

The term or application of Stack (which refers solely to masonry) to a metallic pipe is a misappli- 
cation of terms. 

Piragua. A small vessel with two masts and boom sails. 

Commonly termed Perry Augur. 

Plastering. In Architecture, covering with plaster cement or mortar upon walls 
or laths. In England, termed laying, if in one or two coat work; and pricking up, 
if in three-coat work. 

Plumber block. A bearing to receive and support the journal of a shaft. 

Polacre. Masts of one piece, without top3. 

Poppets. In Naval Architecture, pieces of timber set perpendicular to a vessel's 
bilge-ways, and extending to her bottom, to support her in launching. 

Porch. An arched vestibule at the entrance of a building. A vestibule supported 
by columns. A portico. 

Portico. A gallery near to the ground, the sides being open. A piazza encom- 
passed with arches supported by columns, where persons may walk ; the roof may 
be flat or vaulted. 

Prize. In Mechanics, to raise with a lever. 

To pry and a pry are corruptions. 

Puzzuolana. A loose, porous, volcanic substance, composed of silicious, argilla-- 
ceous, and calcareous earths and iron. 

Rebate. In Mechanics, to pare down an edge of a board or a plate for the purpose 
of receiving another board or plate by lapping. To lap and unite edges of boards 
and plates. In Naval Architecture, the grooves in the side of the keel for receiving 
the garboard strake of plank. 

Commonly written Rabbet. 

Rarefaction. The act or process of distending bodies, by separating their parts 
and rendering them more rare or porous. It is opposed to Condensation. 

Rendering. In Architecture, laying plaster or mortar upon mortar or walls. 
Rendered and Set refers to two coats or layers, and Rendered, Floated, and Set, to 
three coats or layers. 

Resin. The residuum of the distillation of turpentine. 

Roain is a corruption. 

Riband. In Naval Architecture, a long, narrow, flexible piece of timber. 

Rimer. A bit or boring tool for making a tapering hole. In Mechanics, To Rime 
is to bevel out a hole. Riming. The opening of the seams between the planks of a 
vessel for the purpose of calking them. 

Rotary. Turning upon an axis, as a wheel. 



670 ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. 

Rynd. The metallic collar in the upper mill-stone by which it is connected to the 
spindle. 

Sagging. A term applied to the hull of a vessel when her centre drops below her 
ends. The converse of Hogging. 

Scallop. To mark or cut an edge into segments of circles. 

Scarf. To join, to piece : to unite two pieces of timber at their ends by running 
the end of one over and upon the other, and bolting or securing them together. 

Sennit. Flat-braided cordage. 

Sewerage. The system of sewers. Sewage. Matter borne off by a sewer. 

Shaky. Cracked, or split, or as timber loosely put together. 

Shammy. Leather prepared from the skin of a chamois goat. 

Sheer. In Naval Architecture, the curve or bend of a ship's deck or sides. Ta 
sheer, to slip or move aside. 

Sheers. Elevated spars connected at the upper ends, and used to elevate heavy 
bodies, masts, etc. 

Shoal. A great multitude ; a crowd ; a multitude of fish. 

School is a corruption. 

Shoar. An oblique brace, the upper end resting against the substance to be sup- 
ported. 

Sholes. Pieces of plank under the heels of shoares, etc. 

Shoot. A passage-way on the side of a steep hill, down which wood, coal, etc., arc 
thrown or slid. The artificial or natural contraction of a river. A young pig. 
Sidewise. See Edgewise. 
Signaled. Communicated by signals. 

Signalized, when applied to signals, is a misapplication of words. 

Sill. A piece of timber upon which a building rests ; the horizontal piece of tim» 
ber or stone at the bottom of a framed case. 

Siphon. A curved tube or pipe designed to draw fluids out of vessels. 

Skcg. The afterpart of a keel ; the part upon which the stern-post is set. 

Slantwise. Oblique ; not perpendicular, 

Sleek. To make smooth. Refuse; small coal. 

Sleeker. A spherical-shaped, curved, or plane-surfaced instrument with which to 
smooth surfaces. 

Slue. The turning of a substance upon an axis within its figure. 

Snying. A term applied to planks when their edges at their ends are curved or 
rounded upward, as a strake to the ends of a full-modeled vessel. 

Spall. A piece of stone, etc., etc., chipped off by the stroke of a hammer or the 
force of a blow. Spoiling, Breaking up of ore into small pieces. 

Sponson. An addition to the outer side of the hull of a steam vessel, commencing 
near the light water-line and running up to the wheel guards ; applied for the pur- 
pose of shielding them from the shock of a sea. 

Sponson-sided. The hull of a vessel is so termed when her frames have the ou t- 
line of a sponson, and the space afforded by the curvature is included iu the hold. 
* Spending and Sponsing, etc., etc , are corruptions. 

Stack. In Masonry, a number of chimneys or pipes standing together. The 
chimney of a blast furnace. 

The application of this word to the 6moke-pipe of a steam boiler is erroneous. 

Stivwg. The elevation of a vessel's bowsprit, cathead, etc., etc. 

Strake. A breadth of plank. 

Strut. An oblique brace to support a ratter. 

Surcingle. A belt, band, or girth, which passes over a saddle or blanket upon 
a horse's back. 

Swage. To bear or force down. An instrument having a groove on its under side 
for the purpose; of giving shape to any piece subjected to it when receiving a blow 
from a hammer. 



ORTHOGRAPHY OF TECHNICAL WORDS AND TERMS. 671 

Scend. The settling of a vessel below the level of her keel. 

Syphered. Overlapping the chamfered edge of one plank upon the chamfered 
edge of another in such a manner that the joint shall be a plane surface. 

Template. In Architecture, a wooden bearing to receive the end of a girder to 
distribute its weight. 

Templet. A mold cut to an exact section of any piece or structure. 

Tenon. The end of a piece of wood, cut into the form of a rectangular prism, de- 
signed to be set into a cavity of a like form in another piece, which is called the 
mortise. 

Terring. The earth overlying a quarry. 

Tester. The top covering of a bedstead. 

Tholes. The pins in the gunwale of a boat which are used as row-locks. 

Thwarts. The athwartship seats in a boat. 

Tire. The metal hoop that binds the felloes of a wheel. 

Tompion. The stopper of a piece of ordnance. The iron bottom to which grape< 
dhot are secured. 

Treenails. Wooden pins employed to secure the planking of a vessel to the frames. 

Trepan. In Mining, the instrument used in the comminution of rock in earth- 
boring at great depths. 

Trestle. The frame of a table ; a movable form of support. In Mast-making, two 
pieces of timber set horizontally upon opposite sides of a mast-head. 

Trice. In Seamanship, to haul or tie up by means of a rope or tricing-line. 

Tue-iron or Tuyere. The nozzle of a bellows or blast-pipe in a forge or smelting- 
furnace. 

Vice. In Mechanics, a press to hold fast any thing to be worked upon. 

Voyal. In Seamanship, a purchase applied to the weighing of an anchor, leading 
to a capstan. 

Wagon. An open or partially inclosed four-wheeled vehicle, adapted for the trans- 
portation of persons, goods, etc. 

Whipple-tree. The bar to which the traces of harness are fastened. 

Windrow. A row or line of hay, etc., raked together. 

Woold. To wind ; particularly to bind a rope around a spar, etc. 



Fault. In Mining, a break of strata, with displacement, which interrupts opera- 
tions. Also, fissures traversing the strata. 

Selvagee. A strap made of rope yarns, without being twisted or laid up, and re- 
tained in form by knotting it at intervals. 

Skeg. The extreme after-part of the keel of a vessel ; the portion that supports 
the rudder-post, 

Squilgee. A wooden instrument, alike to a hoe, its edge faced with leather or 
vulcanized rubber, used to facilitate the drying of the wet decks of a vessel. 

Dock. In Marine Architecture, an inclosure in a harbor or shore of a river, etc., 
for the reception, repair, or security of vessels. It may be wholly or only partially 
inclosed. 

When applied to a pier or jetty it is a misapplication. 



THE END. 



'08 



